Grade 7 Mathematics
Expanded Textbook, 2026 Edition
HOMESCHOOL MATHEMATICS: GRADE 7
Real-World Problem Solving Complete Student Edition
GLOBAL SOVEREIGN UNIVERSITY
"Building a Bridge to Freedom Through Education Grade 4 Mathematics: Real-World Problem Solving Student Edition
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Why This Book Is Different
🏆 NOBEL PRIZE RESEARCH SAYS:
"Students who score 96% on classroom math tests succeed only 1% of the time in real life."
— Banerjee & Duflo, Nature, February 2025
The Research That Proves Traditional Math Education Fails
In February 2025, Nobel Prize-winning economists published a groundbreaking study in Nature, the world's most respected scientific journal. They tested over 1,400 children in India and discovered something shocking:
Children who ace classroom math CANNOT use it in the real world.
Students who scored 96% on standard school math problems could solve simple real-world market calculations only 1% of the time. Meanwhile, children who worked in markets—without formal education—solved the same problems with 96% accuracy.
The researchers' conclusion was clear: "Schools need to build a bridge between math learned in the classroom and math encountered in real-life situations."
The Problem: Abstract Doesn't Transfer to Applied
The Nobel laureates found that schoolchildren learned procedures but not understanding. They could solve "24 × 8" on a worksheet by adding 24 eight times. But give them a real problem—"How much do 8 items cost at $24 each?"—and they froze.
Market children used efficient mental strategies: breaking 11 × 43 into (10 × 43) + 43, rounding for easier calculation, and working with meaningful quantities. Schoolchildren followed rigid, slow methods that worked for tests but failed in life.
It's not that school kids were less intelligent. It's that their education never connected the math to reality.
The GSU Solution: Real-World Math
This textbook does exactly what the Nobel laureates recommend. Instead of abstract problems, every calculation has a
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HOMESCHOOL MATHEMATICS: GRADE 7
professional context:
Complete Pre-Algebra Textbook for Seventh Grade
PART I: THE NUMBER SYSTEM
Chapter 1·RATIONAL NUMBERS
What You'll Learn
★ Understanding what makes a number rational ★ Converting between fractions, decimals, and percents ★ Adding and subtracting rational numbers ★ Multiplying and dividing rational numbers ★ Order of operations with rational numbers ★ Comparing and ordering rational numbers
Why This Matters
Rational numbers are everywhere in daily life. When you split a pizza, calculate a discount, measure ingredients, or figure out how much you've saved toward a goal, you're working with rational numbers. Mastering operations with positive and negative fractions and decimals prepares you for algebra and gives you tools you'll use throughout your life.
LESSON 1.1: What Is a Rational Number?
The Definition
A rational number is any number that can be written as a fraction a/b, where a and b are integers and b ≠ 0.
The word "rational" comes from "ratio"—a rational number is any number that can be expressed as a ratio of two integers.
Examples of Rational Numbers
Integers are rational:
5 = 5/1 ✓ -3 = -3/1 ✓ 0 = 0/1 ✓
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Fractions are rational: 3/4 ✓ -7/8 ✓ 15/2 ✓ Terminating decimals are rational:
0.75 = 75/100 = 3/4 ✓
2.5 = 25/10 = 5/2 ✓
-0.125 = -125/1000 = -1/8 ✓
Repeating decimals are rational:
0.333... = 1/3 ✓
0.666... = 2/3 ✓
0.142857142857... = 1/7 ✓
What Is NOT Rational?
Numbers that cannot be written as fractions are called irrational numbers.
Examples of irrational numbers:
π = 3.14159265358979... (never ends, never repeats)
√2 = 1.41421356237... (never ends, never repeats)
√3 = 1.73205080757... (never ends, never repeats)
Key Idea: If a decimal terminates (ends) OR repeats a pattern, it's rational. If it goes on forever without repeating, it's irrational.
Converting Terminating Decimals to Fractions
Example 1: Convert 0.75 to a fraction.
Step 1: Count decimal places. 0.75 has 2 decimal places.
Step 2: Write over the appropriate power of 10. 0.75 = 75/100
Step 3: Simplify. 75/100 = ¾
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HOMESCHOOL MATHEMATICS: GRADE 7
Answer: 0.75 = 3/4
Example 2: Convert 2.125 to a fraction.
Step 1: 2.125 has 3 decimal places.
Step 2: Write as fraction: 2125/1000
Step 3: Simplify by finding GCF (125): 2125 ÷ 125 = 17
1000 ÷ 125 = 8
Answer: 2.125 = 17/8 or 2 1/8
Converting Repeating Decimals to Fractions
This requires algebra! Let's see how it works.
Example 3: Convert 0.333... to a fraction.
Step 1: Let x = 0.333...
Step 2: Multiply both sides by 10: 10x = 3.333...
Step 3: Subtract the original equation: 10x = 3.333...
x = 0.333... __________________ 9x = 3
Step 4: Solve for x: x = 3/9 = 1/3
Answer: 0.333... = 1/3
Example 4: Convert 0.454545... to a fraction.
Step 1: Let x = 0.454545...
Step 2: The pattern "45" has 2 digits, so multiply by 100: 100x =
45.454545... Step 3: Subtract: 100x = 45.454545...
x = 0.454545... __________________99x = 45
Step 4: Solve: x = 45/99 = 5/11
Answer: 0.454545... = 5/11
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Example 5: Convert 0.1666... to a fraction.
This is trickier because only part of the decimal repeats.
Step 1: Let x = 0.1666...
Step 2: Multiply by 10 to move past the non-repeating part: 10x = 1.666... Step 3: Multiply by 100: 100x = 16.666...
Step 4: Subtract: 100x = 16.666...
10x = 1.666...__________________ 90x = 15
Step 5: Solve: x = 15/90 = 1/6
Answer: 0.1666... = 1/6
✏️ Try This
Convert each to a fraction in simplest form:
1. 0.4
2. 0.625
3. 1.75
4. 0.222...
5. 0.363636...
6. 0.8333...
LESSON 1.2: Adding and Subtracting Rational Numbers
The Rules for Signs
When adding numbers with the same sign:
★ Add the absolute values ★ Keep the common sign
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HOMESCHOOL MATHEMATICS: GRADE 7
When adding numbers with different signs:
★ Subtract the smaller absolute value from the larger ★ Use the sign of the number with the larger absolute value
Example 6: Add: -5 + (-3)
Same signs (both negative):
Add absolute values: 5 + 3 = 8
Keep the negative sign
Answer: -5 + (-3) = -8 Example 7: Add -7 + 4
Different signs:
Subtract: 7 - 4 = 3
The negative number has larger absolute value, so answer is negative Answer: -7 + 4 = -3
Example 8: Add: 6 + (-2)
Different signs:
Subtract: 6 - 2 = 4
The positive number has larger absolute value, so answer is positive Answer: 6 + (-2) = 4
Subtracting = Adding the Opposite
To subtract a number, add its opposite.
a - b = a + (-b)
Example 9: Subtract: 5 - 8
Rewrite: 5 + (-8)
Different signs, larger absolute value is negative: 5 - 8 = 5 + (-8) = -3
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Example 10: Subtract: -4 - 6
Rewrite: -4 + (-6)
Same signs (both negative): -4 + (-6) = -10
Example 11: Subtract: -3 - (-7)
Rewrite: -3 + 7
Different signs, larger absolute value is positive: -3 + 7 = 4
Adding and Subtracting Fractions
Remember: You need a common denominator to add or subtract fractions. Example 12: Add: 2/3 + 1/4 Step 1: Find LCD. LCD of 3 and 4 is 12. Step 2: Convert fractions: 2/3 = 8/12 1/4 = 3/12 Step 3: Add numerators: 8/12 + 3/12 = 11/12 Answer: 2/3 + 1/4 = 11/12 Example 13: Add: -2/5 + 3/10 Step 1: LCD of 5 and 10 is 10. Step 2: Convert: -2/5 = -4/10 3/10 = 3/10 Step 3: Add (different signs): -4/10 + 3/10 = -1/10 Answer: -2/5 + 3/10 = -1/10
Example 14: Subtract: 5/6 - 2/3
Step 1: LCD of 6 and 3 is 6.
Step 2: Convert: 5/6 = 5/6 2/3 = 4/6
Step 3: Subtract: 5/6 - 4/6 = 1/6
Answer: 5/6 - 2/3 = 1/6
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HOMESCHOOL MATHEMATICS: GRADE 7 Example 15: Subtract: -1/4 - 3/8
Step 1: Rewrite as addition: -1/4 + (-3/8)
Step 2: LCD is 8: -1/4 = -2/8 -3/8 = -3/8
Step 3: Add (same signs): -2/8 + (-3/8) = -5/8
Answer: -1/4 - 3/8 = -5/8
Adding and Subtracting Decimals
Line up the decimal points!
Example 16: Add: -3.7 + 2.4
Different signs:
Subtract: 3.7 - 2.4 = 1.3
Larger absolute value is negative Answer: -3.7 + 2.4 = -1.3
Example 17: Subtract: -2.5 - 1.8
Rewrite: -2.5 + (-1.8)
Same signs:
Add: 2.5 + 1.8 = 4.3
Both negative, so answer is negative
Answer: -2.5 - 1.8 = -4.3
Your Turn
Calculate:
1. -8 + 5 = _____
2. -6 + (-4) = _____
3. 7 - 12 = _____
4. -9 - (-3) = _____
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5. 3/4 + (-1/2) = _____
6. -2/3 - 1/6 = _____
7. -4.2 + 1.7 = _____
8. 3.5 - (-2.1) = _____
LESSON 1.3: Multiplying and Dividing Rational Numbers
The Sign Rules
For multiplication and division:
Signs Result
Positive × Positive = Positive
Negative × Negative = Positive
Positive × Negative Negative
Negative × Positive Negative
Key Idea: Same signs = positive result. Different signs = negative result.
This works for both multiplication AND division!
Example 18: Multiply: (-4)(6)
Different signs → Negative
4 × 6 = 24
Answer: (-4)(6) = -24
Example 19: Multiply: (-5)(-7)
Same signs → Positive
5 × 7 = 35
Answer: (-5)(-7) = 35
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HOMESCHOOL MATHEMATICS: GRADE 7 Example 20: Divide: -20 ÷ 4
Different signs → Negative
20 ÷ 4 = 5
Answer: -20 ÷ 4 = -5
Example 21: Divide: -36 ÷ (-9)
Same signs → Positive
36 ÷ 9 = 4
Answer: -36 ÷ (-9) = 4
Multiplying Fractions
Multiply numerators, multiply denominators, then simplify.
Example 22: Multiply: (2/3) x (4/5) Multiply: (2 × 4)/(3 × 5) = 8/15
Answer: (2/3)(4/5) = 8/15
Example 23: Multiply: (-3/4)(2/5)
Different signs → Negative
Multiply: (3 × 2)/(4 × 5) = 6/20 = 3/10
Answer: (-3/4)(2/5) = -3/10 Example 24: Multiply: (-2/3)(-9/10)
Same signs → Positive motional cap
Before multiplying, simplify by cross-canceling: 2 and 10 share factor of 2
3 and 9 share factor of 3
(-2/3)(-9/10) = (-1/1)(-3/5) = 3/5
Answer: (-2/3)(-9/10) = 3/5
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Dividing Fractions
To divide by a fraction, multiply by its reciprocal. a/b ÷ c/d = a/b × d/c
Example 25: Divide: (3/4) ÷ (2/5)
Multiply by reciprocal:
(3/4) × (5/2) = 15/8 = 1 7/8
Answer: (3/4) ÷ (2/5) = 15/8
Example 26: Divide: (-5/6) ÷ (2/3)
Different signs → Negative
Multiply by reciprocal:
(5/6) × (3/2) = 15/12 = 5/4
Answer: (-5/6) ÷ (2/3) = -5/4
Example 27: Divide: (-4/9) ÷ (-2/3)
Same signs → Positive
Multiply by reciprocal:
(4/9) × (3/2) = 12/18 = 2/3
Answer: (-4/9) ÷ (-2/3) = 2/3
Multiplying and Dividing Decimals
Example 28: Multiply: (-2.5)(0.4) Different signs → Negative
2.5 × 0.4 = 1.0
Answer: (-2.5)(0.4) = -1.0
Example 29: Divide: -7.2 ÷ (-0.9)
Same signs → Positive
7.2 ÷ 0.9 = 8
Answer: -7.2 ÷ (-0.9) = 8
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HOMESCHOOL MATHEMATICS: GRADE 7
Your Turn
Calculate:
1. (-8)(7) =______
2. (-6)(-9) =______
3. 42 ÷ (-7) =______
4. -56 ÷ (-8) =______
5. (3/5)(-2/3) =______
6. (-4/7) ÷ (2/3) =______
7. (-1.5)(0.6) =______
8. -4.8 ÷ (-1.2) =______
LESSON 1.4: Order of Operations with Rational Numbers
PEMDAS/GEMDAS Review
P/G - Parentheses/Grouping symbols first
E - Exponents
MD - Multiplication and Division (left to right)
AS - Addition and Subtraction (left to right)
Example 30: Evaluate: -3 + 4 × 2
Multiply first: 4 × 2 = 8
Then add: -3 + 8 = 5
Answer: -3 + 4 × 2 = 5
Example 31: Evaluate: (-2)³ + 5 Exponent first: (-2)³ = (-2)(-2)(-2) = -8
Then add: -8 + 5 = -3
Answer: (-2)³ + 5 = -3
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⚠️ Watch Out! (-2)³ = -8, but -2³ = -(2³) = -8. In this case they're the same, but be careful with even exponents: (-2)² = 4, while -2² = -4.
Example 32: Evaluate: 12 ÷ (-3) - 4 × (-2)
Division and multiplication from left to right:
12 ÷ (-3) = -4
4 × (-2) = -8
Now subtract: -4 - (-8) = -4 + 8 = 4
Answer: 12 ÷ (-3) - 4 × (-2) = 4
Example 33: Evaluate: (1/2 + 1/3) × 6
Parentheses first:
1/2 + 1/3 = 3/6 + 2/6 = 5/6
Then multiply:
5/6 × 6 = 5
Answer: (1/2 + 1/3) × 6 = 5
Example 34: Evaluate: -2(3 - 7) + 4²
Parentheses: 3 - 7 = -4
Exponent: 4² = 16
Multiply: -2(-4) = 8
Add: 8 + 16 = 24
Answer: -2(3 - 7) + 4² = 24
Example 35: Evaluate: [(-6) + 2] ÷ [(-1)(-2)]
Brackets first:
(-6) + 2 = -4
(-1)(-2) = 2
Divide:
-4 ÷ 2 = -2
Answer: [(-6) + 2] ÷ [(-1)(-2)] = -2
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HOMESCHOOL MATHEMATICS: GRADE 7
Your Turn
Evaluate:
1. 5 - 3 × 4 =______
2. (-4)² - 10 =______
3. -6 ÷ 2 + 8 ÷ (-4) =______
4. (2/3 - 1/2) × 12 =______
5. 3(-4 + 6) - 2³ =______
6. [-8 + 2(-3)] ÷ 7 =______
LESSON 1.5: Comparing and Ordering Rational Numbers
Using a Number Line
On a number line:
Numbers increase from left to right
Negative numbers are to the left of zero
The further left, the smaller the number
Comparing Fractions To compare fractions, find a common denominator OR convert to decimals.
Example 36: Compare: -3/4 and -2/3
Method 1: Common denominator -3/4 = -9/12
-2/3 = -8/12 Since -9 < -8, we have -9/12 < -8/12
Answer: -3/4 < -2/3
Method 2: Convert to decimals. -3/4 = -0.75
-2/3 = -0.666... -0.75 < -0.666...
Answer: -3/4 < -2/3
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Example 37: Order from least to greatest: 1/2, -3/4, 2/3, -1/3
Convert to decimals:
1/2 = 0.5 -3/4 = -0.75
2/3 ≈ 0.667
-1/3 ≈ -0.333
Order: -0.75, -0.333, 0.5, 0.667
Answer: -3/4, -1/3, 1/2, 2/3
Example 38: Order from greatest to least: -2.5, -2.05, -2.55, -2.1
All negative, so smaller absolute value = greater number.
Absolute values: 2.5, 2.05, 2.55, 2.1
Order of absolute values (least to greatest): 2.05, 2.1, 2.5, 2.55
So original numbers (greatest to least): -2.05, -2.1, -2.5, -2.55
Your Turn
1. Compare: -5/8 ___ -3/5 (use < or >)
2. Compare: -0.45 ___ -0.405
3. Order from least to greatest: -0.7, -3/4, -0.65, -7/10 ____________
4. Order from greatest to least: 2/5, -1/2, 0.35, -0.6 ____________
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HOMESCHOOL MATHEMATICS: GRADE 7
Chapter 1 Practice Problems
Section A: Converting Between Forms
Convert each decimal to a fraction in simplest form: 1. 0.8
2. 0.35 ____________
3. 1.25 ____________
4. 0.125 ____________
5. 0.555... ____________
6. 0.272727... ____________
7. 0.4166... ____________
8. 2.333…____________
Section B: Adding and Subtracting
Calculate:
9. -15 + 8 ____________
10. -7 + (-12) ____________
11. 6 - 14 ____________
12. -8 - (-5) ____________
13. 3/4 + (-1/2) ____________
14. -5/6 + 2/3 ____________
15. -2/5 - 3/10 ____________
16. 7/8 - (-1/4) ____________
17. -3.6 + 2.1 ____________
18. -5.2 - 3.8 ____________
19. 4.5 + (-7.3) ____________
20. -0.75 - (-1.25) ____________
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Section C: Multiplying and Dividing
Calculate:
21. (-9)(8) ____________
22. (-7)(-11) ____________
23. -48 ÷ 6 ____________
24. -63 ÷ (-9) ____________
25. (2/3)(-3/4) ____________
26. (-5/8)(-4/15) ____________
27. (3/5) ÷ (-2/3) ____________
28. (-7/10) ÷ (-14/15) ____________
29. (-2.4)(0.5) ____________
30. (-0.8)(-1.5) ____________
31. 5.4 ÷ (-0.9) ____________
32. -3.6 ÷ (-0.4)____________ Section D: Order of Operations
Evaluate:
33. 8 - 12 ÷ 4 ____________
34. -5 + 3 × (-2) ____________
35. (-3)² - 4 × 2 ____________
36. 15 ÷ (-3) + (-2)³ ____________
37. (4 - 10) ÷ 2 + 5 ____________
38. -2(5 - 8) + 4 ____________
39. [(-12) ÷ 4] × [(-2)(-3)] ____________
40. (1/2)² + 1/4 ____________
41. 2/3 × (3/4 - 1/2) ____________
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HOMESCHOOL MATHEMATICS: GRADE 7
42. (-1/2)³ + 3/8 ____________
Section E: Comparing and Ordering
Compare using < or >:
43. -7/8 ___ -5/6
44. -0.625 ___ -5/8
45. 3/5 ___ 0.59
46. -1.333... ___ -4/3
Order from least to greatest:
47. 0.5, -0.6, 1/3, -1/2 ____________
48. -3/4, -0.8, -7/10, -0.72 ____________
49. 2/3, 0.65, 5/8, 0.7 ____________
50. -1.25, -5/4, -1.2, -6/5 ____________
Section F: Word Problems
51. The temperature was -4°F in the morning. It rose 12°F by noon, then dropped 7°F by evening.
What was the evening temperature? ____________
52. A submarine was at -250 feet. It rose 75 feet, then dove 120 feet. What is its new depth? ____________
53. Maria has $45.50 in her account. She writes checks for $23.75 and $18.00, then deposits
$15.25. What is her new balance? ____________
54. A recipe calls for 2/3 cup of sugar. If you want to make 1 1/2 times the recipe, how much
sugar do you need? ____________
55. A stock lost 3/4 point on Monday, gained 1 1/2 points on Tuesday, and lost 5/8 point on
Wednesday. What was the total change? ____________
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೦೦ Challenge Problems
1. If a × b = -12 and a + b = 1, find two possible pairs of values for a and b. ____________
2. The average of five numbers is -3. Four of the numbers are 2, -7, 4, and -6. What is the fifth number? ____________
3. Simplify: (-1/2)(-2/3)(-3/4)(-4/5)(-5/6) ____________
4. If x = -2/3 and y = 3/4, evaluate: (x + y)/(x - y) ____________ 5. The temperature drops 2.5°F every hour. If it's currently 18°F, what will the temperature be in 10 hours? When will it reach 0°F? ____________
Chapter 1 Summary
Key Concepts
Rational Numbers:
Can be written as a/b, where a and b are integers and b ≠ 0
Include integers, fractions, terminating decimals, repeating decimals
Adding/Subtracting:
Same signs: Add absolute values, keep the sign
Different signs: Subtract absolute values, use sign of larger absolute value
Subtracting = adding the opposite
Multiplying/Dividing:
Same signs → Positive result
Different signs → Negative result
Order of Operations: PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction)
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HOMESCHOOL MATHEMATICS: GRADE 7
Key Formulas and Rules
Operation Same Signs Different Signs
Add Add, keep sign Subtract, larger sign Operation Same Signs Different Signs
Subtract Add the opposite.
Multiply Negative
Divide Positive Negative
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Add opposite
Positive
This chapter provides the foundation for all algebraic work with rational numbers. CHAPTER 2: COMPLEX EQUATIONS AND INEQUALITIES
What You'll Learn
Solving multi-step equations with complex terms
Working with equations involving fractions and decimals
Solving compound inequalities
Solving absolute value equations
Graphing solutions on number lines
Why This Matters As mathematical problems become more complex, so do the equations we need to solve. These advanced equation-solving skills are essential for algebra and beyond. Inequalities help us describe ranges of solutions—like budget limits, acceptable measurements, or qualifying scores—that occur constantly in real life.
LESSON 2.1: Multi-Step Equations with Fractions
Strategy
When equations contain fractions, you can:
1. Work directly with fractions, OR
2. Multiply both sides by the LCD to eliminate fractions first
Method 1: Working with Fractions
Example 1: Solve: (2/3)x + 1/4 = 5/6
Step 1: Subtract 1/4 from both sides. (2/3)x = 5/6 - 1/4
Step 2: Find common denominator (12): 5/6 = 10/12 1/4 = 3/12 (2/3)x = 10/12 - 3/12 = 7/12
Step 3: Multiply both sides by reciprocal of 2/3 (which is 3/2): x = (7/12) × (3/2) = 21/24 = 7/8
Check: (2/3)(7/8) + 1/4 = 14/24 + 6/24 = 20/24 = 5/6 ✓
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HOMESCHOOL MATHEMATICS: GRADE 7
Method 2: Clearing Fractions
Example 2: Solve: (2/3)x + 1/4 = 5/6 (same equation, different method)
Step 1: Find LCD of 3, 4, and 6. LCD = 12
Step 2: Multiply every term by 12: 12 × (2/3)x + 12 × (1/4) = 12 × (5/6)
8x + 3 = 10 Step 3: Solve the simpler equation: 8x = 7 x = 7/8
Example 3: Solve: (x/2) - (x/5) = 3
LCD = 10. Multiply every term by 10: 10(x/2) - 10 (x/5) = 10 (3) 5x - 2x = 30 3x = 30 x = 10
Check: 10/2 - 10/5 = 5 - 2 = 3 ✓
Example 4: Solve: (x + 3)/4 = (x - 1)/3
Cross multiply:
3(x + 3) = 4(x - 1)
3x + 9 = 4x - 4
9 + 4 = 4x - 3x
13 = x
x = 13
Check: (13 + 3)/4 = 16/4 = 4 and (13 - 1)/3 = 12/3 = 4 ✓
✏️ Try This
Solve:
1. (1/2)x + (1/3) = 5/6 ____________
2. (x/3) + (x/4) = 7 ____________
3. (2x - 1)/5 = (x + 2)/3 ____________
4. (3/4)x - 2 = (1/2)x + 1 ____________
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LESSON 2.2: Equations with Decimals
Strategy
Multiply by a power of 10 to eliminate decimals. Decimal places Multiply by
1 (tenths) 10
2 (hundredths) 100
3 (thousandths) 1000
Example 5: Solve: 0.3x + 1.2 = 2.7
Method 1: Direct 0.3x = 2.7 - 1.2 0.3x = 1.5 x = 1.5 ÷ 0.3 = 5
Method 2: Clear decimals Multiply by 10: 3x + 12 = 27 3x = 15 x = 5
Example 6: Solve: 0.05x + 0.25 = 0.45
Multiply by 100 (2 decimal places): 5x + 25 = 45 5x = 20 x = 4
Check: 0.05(4) + 0.25 = 0.20 + 0.25 = 0.45 ✓
Example 7: Solve: 0.4(x - 3) = 0.2x + 1.8
Distribute: 0.4x - 1.2 = 0.2x + 1.8
Multiply by 10: 4x - 12 = 2x + 18 2x = 30 x = 15
Check: 0.4(15 - 3) = 0.4(12) = 4.8 0.2(15) + 1.8 = 3 + 1.8 = 4.8 ✓
✏️ Try This
Solve:
1. 0.5x - 0.3 = 1.7 ____________
2. 0.08x + 0.04 = 0.2 ____________
3. 0.25(x + 4) = 0.5x - 1 ____________
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HOMESCHOOL MATHEMATICS: GRADE 7
LESSON 2.3: More Complex Multi-Step Equations
Equations with Variables on Both Sides and Grouping
Example 8: Solve: 3(2x - 4) - 2(x + 5) = 2(x - 1)
Step 1: Distribute on both sides. 6x - 12 - 2x - 10 = 2x - 2
Step 2: Combine like terms on the left side. 4x - 22 = 2x - 2
Step 3: Move variables to one side. 4x - 2x = -2 + 22 2x = 20 Step 4: Solve. x = 10
Check: 3(20 - 4) - 2(10 + 5) = 3(16) - 2(15) = 48 - 30 = 18 2(10 - 1) = 2(9) = 18 ✓
Example 9: Solve: 5 - 2(3x - 1) = 4(x + 2) - 3x
Distribute:
5 - 6x + 2 = 4x + 8 - 3x
Combine like terms:
7 - 6x = x + 8
Move variables: -6x - x = 8 - 7 -7x = 1 x = -1/7
Example 10: Solve: (1/2)(4x + 6) = (2/3)(6x - 9)
Distribute:
2x + 3 = 4x - 6
Move variables: 3 + 6 = 4x - 2x 9 = 2x x = 9/2 or 4.5
Special Cases Review
No Solution: Variables cancel, leaving a false statement.
Example 11: Solve: 2(x + 3) = 2x + 8
2x + 6 = 2x + 8
6 = 8 ✗ (False!)
No solution
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Infinite Solutions: Variables cancel, leaving a true statement.
Example 12: Solve: 3(x - 2) + 6 = 3x
3x - 6 + 6 = 3x
3x = 3x
0 = 0 ✓ (Always true!)
All real numbers are solutions (identity)
✏️ Try This
Solve:
1. 4(x - 2) - 3(x + 1) = x - 5____________
2. 2(3x + 1) = 3(2x - 4) + 14 ____________
3. (1/3)(6x - 9) = 2(x + 1) - 5 ____________
LESSON 2.4: Solving Inequalities Review
Inequality Symbols
Symbol Meaning Graph
< Open circle > Open circle ≤ Closed circle
≥ Greater than or equal to Closed circle
Solving One-Step Inequalities
Same rules as equations, BUT:
⚠️ When you multiply or divide by a negative number, FLIP the inequality sign!
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Less than
Greater than
Less than or equal to
HOMESCHOOL MATHEMATICS: GRADE 7
Example 13: Solve and graph: x + 5 < 12
x < 12 - 5
x < 7
Graph: Open circle at 7, arrow pointing left
←——————○
Example 14: Solve and graph: -3x ≥ 15
Divide by -3 (FLIP the sign!):
x ≤ -5
Graph: Closed circle at -5, arrow pointing left
←——————●
-5
Solving Multi-Step Inequalities
Example 15: Solve: 4x - 7 > 13
4x > 20 x > 5 Example 16: Solve: -2x + 5 ≤ 11
-2x ≤ 6 x ≥ -3 (flip the sign!)
Example 17: Solve: 3(x - 2) + 4 ≥ 2x - 5
3x - 6 + 4 ≥ 2x - 5 3x - 2 ≥ 2x - 5 x ≥ -3
✏️ Try This
Solve and graph:
1. x - 4 > 2 ____________
2. -5x < 20 ____________
3. 2x + 3 ≤ 11 ____________
4. -3(x - 1) ≥ 9 ____________
Global Sovereign University Press Page: 27
LESSON 2.5: Compound Inequalities
"And" Compound Inequalities
"And" means BOTH conditions must be true. The solution is the intersection (overlap).
Example 18: Solve: x > 2 AND x < 7
This can be written as: 2 < x < 7
Graph:
○——————○
2 7
All numbers between 2 and 7 (not including 2 and 7).
Example 19: Solve: -3 ≤ x + 2 < 5
Solve as a "sandwich" inequality—do the same thing to all
three parts: Subtract 2 from all parts: -3 - 2 ≤ x < 5 - 2 -5 ≤
x < 3
Graph:
●——————○
-5 3
Example 20: Solve: 1 < 2x - 3 ≤ 7 Add 3 to all parts:
4 < 2x ≤ 10
Divide all parts by 2: 2 < x ≤ 5
"Or" Compound Inequalities
"Or" means AT LEAST ONE condition must be true.
The solution is the union (combined).
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HOMESCHOOL MATHEMATICS: GRADE 7
Example 21: Solve: x < -2 OR x > 3
Graph:
←——○ ○——→
-2 3
Two separate regions: everything less than -2 OR everything greater than 3.
Example 22: Solve: 3x + 1 < -5 OR 2x - 3 > 7
Solve each separately:
First inequality: 3x + 1 < -5 3x < -6 x < -2
Second inequality: 2x - 3 > 7 2x > 10 x > 5
Solution: x < -2 OR x > 5
✏️ Try This
Solve and graph:
1. x > -1 AND x ≤ 4 ____________
2. -2 < x + 3 < 6 ____________
3. 2x - 1 < 5 OR 2x - 1 > 9 ____________
4. -4 ≤ 3x + 2 ≤ 11 ____________
LESSON 2.6: Absolute Value Equations
Understanding Absolute Value
Absolute value is the distance from zero on a number line.
|a| = a if a ≥ 0
|a| = -a if a < 0
|5| = 5 and |-5| = 5 (both are 5 units from zero) Solving |x| = c
If |x| = c (where c > 0), then x = c OR x = -c
Global Sovereign University Press Page: 29
Example 23: Solve: |x| = 7
x = 7 OR x = -7
Solutions: x = 7, x = -7
Solving |ax + b| = c
Step 1: Set up two equations:
ax + b = c
ax + b = -c
Step 2: Solve each equation.
Example 24: Solve: |x - 3| = 5
Equation 1: x - 3 = 5 → x = 8 Equation 2: x - 3 = -5 → x = -2
Solutions: x = 8, x = -2
Check: |8 - 3| = |5| = 5 ✓ and |-2 - 3| = |-5| = 5 ✓
Example 25: Solve: |2x + 1| = 9
Equation 1: 2x + 1 = 9 → 2x = 8 → x = 4 Equation 2: 2x + 1 = -9 → 2x =
-10 → x = -5 Solutions: x = 4, x = -5
Example 26: Solve: |3x - 4| + 2 = 11
First, isolate the absolute value:
|3x - 4| = 9
Equation 1: 3x - 4 = 9 → 3x = 13 → x = 13/3
Equation 2: 3x - 4 = -9 → 3x = -5 → x = -5/3
Solutions: x = 13/3, x = -5/3
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HOMESCHOOL MATHEMATICS: GRADE 7
Special Cases
Example 27: Solve: |x + 2| = -3
Absolute value is NEVER negative!
No solution Example 28: Solve: |4x - 8| = 0
Absolute value equals zero only when the expression inside equals zero:
4x - 8 = 0 4x = 8 x = 2
Only one solution.
✏️ Try This
Solve:
1. |x| = 12 ____________
2. |x + 4| = 6 ____________
3. |2x - 5| = 11 ____________
4. |x - 1| + 3 = 8 ____________
5. |3x + 6| = -2 ____________
LESSON 10.7: Writing and Solving Real-World Inequalities
Setting Up Inequalities
Phrase Symbol At most, no more than, maximum ≤ At least, no less than, minimum ≥ More than, greater than >
Less than, fewer than <
Global Sovereign University Press Page: 31
Example 29: You need at least 80 points to pass. You have 53 points. How many more points (p) do you need?
53 + p ≥ 80
p ≥ 27
You need at least 27 more points.
Example 30: A phone plan costs $35 plus $0.10 per text. You can spend at most $60. How many texts can you send?
35 + 0.10t ≤ 60
0.10t ≤ 25 t ≤ 250
You can send at most 250 texts.
Example 31: The temperature must stay between 65°F and 75°F. Write and solve an inequality if the current temperature is 60°F and it rises r degrees.
65 ≤ 60 + r ≤ 75 65 - 60 ≤ r ≤ 75 - 60 5 ≤ r ≤ 15
The temperature must rise between 5 and 15 degrees.
✏️ Try This 1. A roller coaster requires a height of at least 48 inches. Write an inequality for acceptable heights (h).
2. You have $25 and want to buy notebooks costing $3 each. How many can you buy?
3. A quiz has 10 questions worth 10 points each. You need more than 70 points to pass. How many must you get right?
Chapter 2 Practice Problems
Section A: Equations with Fractions
Solve:
1. (1/2)x + (2/3) = 7/6 ____________
2. (x/4) - (x/6) = 2 ____________
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HOMESCHOOL MATHEMATICS: GRADE 7
3. (x + 2)/3 = (x - 4)/2 ____________
4. (2x + 1)/5 = (3x - 1)/4 ____________
Section B: Equations with Decimals
Solve:
5. 0.6x + 0.4 = 2.2 ____________
6. 0.15x - 0.05 = 0.4 ____________
7. 0.3(x - 2) = 0.5x + 1.4 ____________
Section C: Complex Multi-Step Equations
Solve:
8. 5(x - 3) - 2(x + 4) = 10 ____________
9. 4(2x - 1) = 3(x + 2) + 5x____________
10. (1/2)(6x + 4) = (1/3)(9x - 6) ____________
Identify: one solution, no solution, or all real numbers?
11. 3(x + 2) = 3x + 6 ____________
12. 2(x - 4) = 2x - 5 ____________
13. 5x - (2x + 3) = 3x - 3 ____________
Section D: Solving Inequalities
Solve and graph:
14. x + 7 > 3 ____________
15. -4x ≤ 12 ____________
16. 3x - 5 < 10 ____________
17. -2(x + 3) ≥ 8 ____________
Global Sovereign University Press Page: 33
Section E: Compound Inequalities
Solve:
18. x > 1 AND x < 6 ____________
19. -5 < 2x + 1 < 9 ____________
20. x ≤ -3 OR x ≥ 2 ____________
21. 3x - 2 < 4 OR 3x - 2 > 10 ____________
22. -1 ≤ (x - 3)/2 < 4 ____________
Section F: Absolute Value Equations
Solve:
23. |x| = 15 ____________
24. |x - 5| = 8 ____________
25. |3x + 2| = 14 ____________
26. |2x - 1| - 3 = 6 ____________
27. |4x + 8| = 0 ____________
28. |x - 2| = -5 ____________
Section G: Word Problems
29. A student has test scores of 78, 85, and 82. What score is needed on the fourth test to have an average of at least 85?
30. A rental car costs $40/day plus $0.25/mile. You have $100. If you rent for 1 day, how many miles can you drive?
31. The sum of three consecutive odd integers is between 50 and 70. Find all possible sets of integers.
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HOMESCHOOL MATHEMATICS: GRADE 7 ೦೦ Challenge Problems
1. Solve: |2x - 3| = |x + 4| ____________
2. Solve: (2x + 1)/3 - (x - 2)/4 = (x + 5)/6 ____________
3. For what values of k does |x - 3| = k have exactly one solution? ____________
4. Solve the compound inequality: 2 ≤ |x - 1| ≤ 5 ____________
5. A store sells items for $12 each with a $5 shipping fee OR items for $14 each with free
shipping. For how many items is the first option cheaper? ____________
Chapter 2 Summary
Strategies for Complex Equations
Fractions: Multiply by LCD to clear fractions
Decimals: Multiply by power of 10 to clear decimals
Grouping: Distribute first, then combine like terms
Compound Inequalities
Type Meaning Graph
AND Overlap (bounded region)
OR At least one true Union (two regions)
Absolute Value Equations
|ax + b| = c (where c > 0)
Solve: ax + b = c AND ax + b = -c
Two solutions (usually)
|ax + b| = 0 → One solution (ax + b = 0)
|ax + b| = negative → No solution
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Both true
Key Reminders
⚠️ Flip inequality sign when multiplying/dividing by negative
⚠️Absolute value is never negative
⚠️Always check solutions in original equation Mastering complex equations and inequalities is essential preparation for algebra and advanced mathematics. CHAPTER 3: INTRODUCTION TO LINEAR EQUATIONS
What You'll Learn
Understanding linear equations and their graphs
Finding and interpreting slope
Writing equations in slope-intercept form
Graphing linear equations
Understanding parallel and perpendicular lines
Writing equations from graphs and points
Why This Matters
Linear relationships are everywhere—from phone bills to distance traveled to temperature
conversions. Understanding how to graph and write linear equations helps you analyze trends, make predictions, and solve real-world problems. This chapter builds the foundation for all of
algebra.
LESSON 3.1: Understanding Linear Equations
What Is a Linear Equation?
A linear equation in two variables creates a straight line when graphed.
Standard forms:
y = mx + b (slope-intercept form)
Ax + By = C (standard form)
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HOMESCHOOL MATHEMATICS: GRADE 7
Identifying Linear Equations
Linear equations:
Have variables only to the first power (no x², x³, √x, etc.)
Have no products of variables (no xy) Linear Not Linear
y = 3x + 2 y = x² + 1
2x + 5y = 10 xy = 6
y = -4 y = √x
x = 7 y = 1/x
Solutions to Linear Equations
A solution is an ordered pair (x, y) that makes the equation true.
Example 1: Is (2, 5) a solution to y = 2x + 1?
Substitute: 5 = 2(2) + 1 = 4 + 1 = 5 ✓
Yes, (2, 5) is a solution.
Example 2: Is (3, -1) a solution to 2x + 3y = 3?
Substitute: 2(3) + 3(-1) = 6 - 3 = 3 ✓
Yes, (3, -1) is a solution.
Global Sovereign University Press Page: 37
Example 3: Find three solutions to y = 2x - 1.
Choose x values and calculate y:
x y = 2x - 1 Point
0 (0, -1)
1 (1, 1)
2 2(2) - 1 = 3 (2, 3)
✏️ Try This
1. Is (1, 4) a solution to y = 3x + 1? ____________
2. Is (-2, 5) a solution to 3x + y = -1? ____________
3. Find three solutions to y = -x + 4.____________
LESSON 3.2: Understanding Slope
What Is Slope?
Slope measures the steepness of a line—how much y changes for each unit change in x.
Slope = Rise / Run = Change in y / Change in x
The Slope Formula
For two points (x₁, y₁) and (x₂, y₂):
m = (y₂ - y₁) / (x₂ - x₁)
Example 4: Find the slope of the line through (1, 2) and (4, 8).
m = (8 - 2) / (4 - 1) = 6/3 = 2
The line rises 2 units for every 1 unit to the right.
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2(0) - 1 = -1
2(1) - 1 = 1
HOMESCHOOL MATHEMATICS: GRADE 7
Example 5: Find the slope through (-2, 5) and (3, -5).
m = (-5 - 5) / (3 - (-2)) = -10/5 = -2
The line falls 2 units for every 1 unit to the right.
Example 6: Find the slope through (1, 4) and (5, 4).
m = (4 - 4) / (5 - 1) = 0/4 = 0
A horizontal line has slope 0.
Example 7: Find the slope through (3, 1) and (3, 6).
m = (6 - 1) / (3 - 3) = 5/0 = undefined
A vertical line has an undefined slope.
Types of Slope
Slope Line Direction
Positive (m > 0) / Negative (m < 0) \ Zero (m = 0) —
Undefined Vertical
Interpreting Slope
Example 8: A car travels 240 miles in 4 hours. What does the slope
represent? slope = 240 miles / 4 hours = 60 miles per hour
The slope (60) represents the speed—60 miles per hour.
Global Sovereign University Press Page: 39
Goes up left to right
Goes down left to right
Horizontal
✏️ Try This
Find the slope:
1. Through (2, 3) and (6, 11) ____________
2. Through (0, 5) and (4, -3) ____________
3. Through (-1, 2) and (4, 2) ____________
4. Through (5, -1) and (5, 7) ____________
LESSON 3.3: Slope-Intercept Form
The Form
y = mx + b
Where:
m = slope
b = y-intercept (where the line crosses the y-axis)
Identifying m and b
Example 9: Identify the slope and y-intercept: y = 3x - 4
Slope (m) = 3 Y-intercept (b) = -4 → Point (0, -4)
Example 10: Identify slope and y-intercept: y = -2x + 5
Slope (m) = -2 Y-intercept (b) = 5 → Point (0, 5)
Example 11: Identify slope and y-intercept: y = (1/2)x
Slope (m) = 1/2 Y-intercept (b) = 0 → Point (0, 0) Converting to Slope-Intercept Form
Example 12: Rewrite 2x + 3y = 12 in slope-intercept form.
Solve for y:
3y = -2x + 12
y = (-2/3)x + 4
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HOMESCHOOL MATHEMATICS: GRADE 7
Slope = -2/3, y-intercept = 4
Example 13: Rewrite 4x - y = 7 in slope-intercept form.
-y = -4x + 7
y = 4x - 7
Slope = 4, y-intercept = -7
✏️ Try This
Identify slope and y-intercept:
1. y = 5x + 2
2. y = -x - 3
3. y = (3/4)x
Convert to slope-intercept form:
4. 3x + y = 9
5. 2x - 4y = 8
LESSON 3.4: Graphing Linear Equations
Method 1: Using Slope and Y-Intercept
Steps:
1. Plot the y-intercept (0, b)
2. Use the slope (rise/run) to find another point
3. Draw the line through both points
Example 14: Graph y = 2x - 1
Step 1: Y-intercept is -1. Plot (0, -1).
Step 2: Slope is 2 = 2/1 (rise 2, run 1). From (0, -1): go up 2, right 1 → (1, 1) Step 3: Draw a line through (0, -1) and (1, 1).
Global Sovereign University Press Page: 41
Example 15: Graph y = -(2/3)x + 4
Step 1: Y-intercept is 4. Plot (0, 4).
Step 2: Slope is -2/3 (down 2, right 3). From (0, 4): go down 2, right 3 → (3, 2)
Step 3: Draw a line through (0, 4) and (3, 2).
Method 2: Using a Table
Example 16: Graph y = -x + 3 using a table.
x y = -x + 3 Point
-1 (-1, 4)
0 (0, 3)
1 (1, 2)
2 -(2) + 3 = 1 (2, 1)
Plot these points and connect them with a line.
Method 3: Using Intercepts
Example 17: Graph 3x + 2y = 6 using intercepts.
X-intercept (set y = 0): 3x + 0 = 6 → x = 2 → Point (2, 0)
Y-intercept (set x = 0): 0 + 2y = 6 → y = 3 → Point (0, 3)
Plot (2, 0) and (0, 3), then connect.
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-(-1) + 3 = 4
-(0) + 3 = 3
-(1) + 3 = 2
HOMESCHOOL MATHEMATICS: GRADE 7
✏️ Try This
Graph each equation:
1. y = 3x + 1 (use slope and y-intercept)
2. y = -(1/2)x + 2 (use slope and y-intercept)
3. 2x + y = 4 (use intercepts) LESSON 3.5: Writing Equations of Lines
Given Slope and Y-Intercept
Just substitute into y = mx + b.
Example 18: Write the equation of a line with slope 4 and
y-intercept -2. y = 4x - 2
Given Slope and a Point
Use point-slope form, then convert to slope-intercept.
Point-slope form: y - y₁ = m(x - x₁)
Example 19: Write the equation of a line with slope 3 passing through (2, 5). y - 5 = 3(x - 2) y - 5 = 3x - 6 y = 3x - 6 + 5 y = 3x - 1
Example 20: Write the equation of a line with slope -2 passing through (-1, 4). y - 4 = -2(x - (-1)) y - 4 = -2(x + 1) y - 4 = -2x - 2 y = -2x - 2 + 4 y = -2x + 2
Given Two Points
Step 1: Find the slope using m = (y₂ - y₁)/(x₂ - x₁)
Step 2: Use point-slope form with either point
Example 21: Write the equation through (1, 3) and (3, 9).
Step 1: m = (9 - 3)/(3 - 1) = 6/2 = 3
Step 2: Using point (1, 3): y - 3 = 3(x - 1) y - 3 = 3x - 3 y = 3x
Example 22: Write the equation through (-2, 5) and (4, -1).
Step 1: m = (-1 - 5)/(4 - (-2)) = -6/6 = -1
Step 2: Using point (4, -1): y - (-1) = -1(x - 4) y + 1 = -x + 4 y = -x + 3
Global Sovereign University Press Page: 43
✏️ Try This
Write the equation:
1. Slope 2, y-intercept 5 ____________
2. Slope -3, through (1, 4) ____________
3. Through (2, 1) and (4, 5)____________
4. Through (0, 3) and (6, 0) ____________
LESSON 3.6: Parallel and Perpendicular Lines
Parallel Lines
Parallel lines never intersect and have the same slope.
If line 1 has slope m, a parallel line also has slope m.
Example 23: Line A has equation y = 2x + 3. Line B is parallel to A and passes through (1, 5). Find the equation of Line B.
Parallel means same slope: m = 2
Using point (1, 5): y - 5 = 2(x - 1) y - 5 = 2x - 2 y = 2x + 3
Wait—that's the same line! Let me reconsider.
y - 5 = 2(x - 1)
y - 5 = 2x - 2
y = 2x + 3
Actually, (1, 5) IS on line A: 2(1) + 3 = 5 ✓
Let's redo with point (1, 7): y - 7 = 2(x - 1) y - 7 = 2x - 2 y = 2x + 5
Example 24: Write an equation parallel to y = -3x + 1 through (2, 4).
Same slope: m = -3
y - 4 = -3(x - 2) y - 4 = -3x + 6 y = -3x + 10
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HOMESCHOOL MATHEMATICS: GRADE 7
Perpendicular Lines
Perpendicular lines intersect at 90° angles. Their slopes are negative reciprocals.
If line 1 has slope m, a perpendicular line has slope -1/m.
Original slope Perpendicular slope
2 -1/2
-3 1/3
1/4 -4
-2/5 5/2
Example 25: Line A has equation y = 4x - 2. Find the slope of a perpendicular line.
Original slope: 4
Perpendicular slope: -1/4
Example 26: Write an equation perpendicular to y = (2/3)x + 1 through (4, 5).
Original slope: 2/3
Perpendicular slope: -3/2
y - 5 = -(3/2)(x - 4) y - 5 = -(3/2)x + 6 y = -(3/2)x + 11
Checking If Lines Are Parallel or Perpendicular
Example 27: Are y = 3x + 2 and y = 3x - 5 parallel, perpendicular, or neither?
Both have slope 3. Same slope → Parallel
Example 28: Are y = 2x + 1 and y = -(1/2)x + 3 parallel, perpendicular, or neither?
Slopes: 2 and -1/2
Product: 2 × (-1/2) = -1
Slopes multiply to -1 → Perpendicular
Global Sovereign University Press Page: 45
✏️ Try This
1. Write an equation parallel to y = 4x - 3 through (1, 2).
2. Write an equation perpendicular to y = 2x + 5 through (6, 3). 3.
Are y = -2x + 4 and y = (1/2)x - 1 parallel, perpendicular, or neither? ____________
Chapter 3 Practice Problems
Section A: Solutions and Linear Equations
1. Is (3, 7) a solution to y = 2x + 1? ____________
2. Is (-1, 4) a solution to 3x + 2y = 5? ____________
3. Find three solutions to y = -2x + 5. ____________
Section B: Finding Slope
Find the slope:
4. Through (1, 4) and (5, 12) ____________
5. Through (-3, 2) and (1, -6)____________
6. Through (0, 5) and (4, 5) ____________
7. Through (2, -1) and (2, 8) ____________
Section C: Slope-Intercept Form
Identify slope and y-intercept:
8. y = 4x - 7 ____________
9. y = -(2/3)x + 1 ____________
10. y = 5x ____________
Convert to slope-intercept form:
11. 5x + y = 10 ____________
12. 2x - 3y = 9 ____________
13. x - 2y = 6 ____________
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HOMESCHOOL MATHEMATICS: GRADE 7
Section D: Graphing
Graph each equation:
14. y = x + 2 ____________
15. y = -2x + 4 ____________
16. y = (1/3)x - 1 ____________
17. 3x + y = 6 (use intercepts) ____________
Section E: Writing Equations
Write the equation:
18. Slope 5, y-intercept -2 ____________
19. Slope -1, through (3, 2) ____________
20. Through (0, 4) and (2, 10) ____________
21. Through (-1, 5) and (3, -3) ____________
22. Horizontal line through (2, -4) ____________
23. Vertical line through (-3, 5) ____________
Section F: Parallel and Perpendicular
24. Write an equation parallel to y = 3x - 2 through (2, 1). ____________
25. Write an equation perpendicular to y = 5x + 1 through (0, 3). ____________
26. Are y = 4x - 1 and 8x - 2y = 6 parallel, perpendicular, or neither? ____________
27. Are y = 3x + 2 and y = -(1/3)x - 1 parallel, perpendicular, or neither? ____________
Section G: Applications
28. A plumber charges $50 plus $40 per hour. Write an equation for the total cost C in terms of hours h. What is the slope, and what does it represent? ____________
29. A candle is 12 inches tall and burns 2 inches per hour. Write an equation for height H in terms of time t. When will the candle burn out? ____________
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30. Two points on a linear graph are (2, 150) and (5, 240), representing distance in miles after hours of driving. Find the equation and interpret the slope. ____________
೦೦ Challenge Problems
1. Find the equation of a line perpendicular to y = (3/4)x - 2 that passes through the point where the line crosses the x-axis.____________
2. A line passes through (1, k) and (k, 5) with slope 2. Find k. ____________
3. Find the area of the triangle formed by the line 2x + 3y = 12 and the coordinate axes. ____________
4. Line A passes through (0, 3) and (4, 0). Line B passes through (0, -1) and (2, 5). Find where the lines intersect. ____________
5. Write the equation of a line that passes through (3, 4) and is equidistant from (1, 2) and (5, 6). ____________
Chapter 3 Summary
Key Formulas
Slope formula: m = (y₂ - y₁)/(x₂ - x₁)
Slope-intercept form: y = mx + b
Point-slope form: y - y₁ = m(x - x₁)
Slope Relationships
Relationship Slopes
Parallel Equal (m₁ = m₂)
Perpendicular Negative reciprocals (m₁ × m₂ = -1) Graphing Methods
1. Slope and y-intercept: Plot b, then use slope 2. Table of values: Calculate several (x, y) pairs
3. Intercepts: Find where line crosses each axis
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HOMESCHOOL MATHEMATICS: GRADE 7
Key Concepts
Slope represents rate of change
Y-intercept represents starting value
Horizontal lines: m = 0
Vertical lines: m = undefined
Linear equations are the foundation of algebra and essential for understanding more complex relationships.
Chapter 4·SYSTEMS OF EQUATIONS What You'll
Learn
Understanding systems of linear equations
Solving systems by graphing
Identifying types of solutions (one, none, infinitely many)
Interpreting solutions in context
Introduction to algebraic methods
Why This Matters Many real-world problems involve multiple conditions that must be satisfied simultaneously. When will two phone plans cost the same? Where do two roads intersect? How many of each item should a store stock? Systems of equations help solve problems with multiple unknowns and multiple constraints.
LESSON 4.1: What Is a System of Equations?
Definition
A system of equations is a set of two or more equations with the same variables.
Example: y = 2x + 1 y = -x + 7
We want to find values of x and y that make BOTH equations true simultaneously.
Global Sovereign University Press Page: 49
Solutions to Systems
A solution to a system is an ordered pair (x, y) that satisfies ALL equations in the system.
Example 1: Is (2, 5) a solution to
the system? y = 2x + 1
y = -x + 7
Check equation 1: 5 = 2(2) + 1 = 5 ✓
Check equation 2: 5 = -(2) + 7 = 5 ✓
Yes, (2, 5) is the solution.
Example 2: Is (3, 4) a solution to
the system? y = x + 1
y = 2x - 1
Check equation 1: 4 = 3 + 1 = 4 ✓
Check equation 2: 4 = 2(3) - 1 = 5 ✗
No, (3, 4) is NOT a solution (fails the second equation).
Example 3: Is (1, 3) a solution to
the system? 2x + y = 5
x - y = -2
Check equation 1: 2(1) + 3 = 5 ✓
Check equation 2: 1 - 3 = -2 ✓
Yes, (1, 3) is the solution.
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HOMESCHOOL MATHEMATICS: GRADE 7 ✏️ Try This
Is the given point a solution to the system?
1. (4, 1): y = x - 3 and y = -x + 5
2. (2, 3): y = 2x - 1 and y = x + 2
3. (-1, 2): 3x + y = -1 and x - 2y = -5
LESSON 4.2: Solving Systems by Graphing
The Method
1. Graph both equations on the same coordinate plane
2. Find the point of intersection
3. Check the solution in both equations
Example 4: Solve by graphing:
y = x + 1
y = -x + 5
Graph y = x + 1: Y-intercept: 1, Slope: 1 Points: (0, 1), (1, 2), (2, 3)
Graph y = -x + 5: Y-intercept: 5, Slope: -1 Points: (0, 5), (1, 4), (2, 3) The lines intersect at (2, 3).
Check: y = x + 1: 3 = 2 + 1 ✓ y = -x + 5: 3 = -2 + 5 ✓
Example 5: Solve by graphing:
y = 2x - 1 ____________
y = (1/2)x + 2 ____________
Graph y = 2x - 1: Y-intercept: -1, Slope: 2 Points: (0, -1), (1, 1), (2, 3)
Graph y = (1/2)x + 2: Y-intercept: 2, Slope: 1/2 Points: (0, 2), (2, 3), (4, 4)
The lines intersect at (2, 3). Check: y = 2x - 1: 3 = 2(2) - 1 = 3 ✓ y = (1/2)x + 2: 3 = (1/2)(2) + 2 = 3 ✓
Global Sovereign University Press Page: 51
Example 6: Solve by graphing:
x + y = 4 ____________________________________
2x - y = 2 ____________________________________
First, convert to slope-intercept form:
Equation 1: y = -x + 4 (slope -1, y-int 4) ____________________________________
Equation 2: y = 2x - 2 (slope 2, y-int -2) ____________________________________
Graph both lines.
Intersection: (2, 2)
Check: x + y = 4: 2 + 2 = 4 ✓ 2x - y = 2: 2(2) - 2 = 2 ✓
✏️ Try This
Solve by graphing:
1. y = x + 2 and y = -x + 4 ____________________________________
2. y = 3x - 1 and y = x + 3 ____________________________________
3. x + y = 5 and x - y = 1 ____________________________________
LESSON 12.3: Types of Solutions
Three Possibilities
Graph Lines Solutions Slopes
Intersecting Different slopes
Parallel: Same slope, different y-intercepts
Same line overlaps completely
Infinite solutions Same slope, same y-intercept
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Cross at one point One solution
Never cross No solution
HOMESCHOOL MATHEMATICS: GRADE 7
One Solution (Most Common)
The lines intersect at exactly one point.
Example 7:
y = 2x + 1 (slope 2)
y = -x + 4 (slope -1)
Different slopes → lines must intersect → One solution
No Solution (Parallel Lines)
The lines never intersect.
Example 8: Solve:
y = 3x + 2 ____________
y = 3x - 1 ____________
Both have slope 3, but different y-intercepts (2 and -1).
These are parallel lines → No solution
The system is called inconsistent.
Example 9: Determine if there's a solution:
2x + y = 5____________
4x + 2y = 7 ____________
Rewrite in slope-intercept form:
y = -2x + 5 ____________
y = -2x + 3.5 ____________
Same slope (-2), different y-intercepts → Parallel, no solution
Infinite Solutions (Same Line)
The equations describe the same line.
Global Sovereign University Press Page: 53
Example 10: Solve:
y = 2x + 3 ____________
4x - 2y = -6 ____________
Rewrite the second equation:
-2y = -4x - 6 ____________
y = 2x + 3 ____________
Both equations are the same line!
Every point on the line is a solution → Infinitely many solutions The system is called dependent.
Example 11: Determine the number of solutions:
3x + 6y = 12 ____________
x + 2y = 4 ____________
Rewrite both:
y = -(1/2)x + 2 ____________
Same equation → Infinitely many solutions
Notice: The first equation is 3 times the second equation.
Identifying Without Graphing
Compare slopes and y-intercepts: Example 12: How many solutions?
y = 4x - 5 (slope 4, y-int -5) ____________
y = 4x + 2 (slope 4, y-int 2) ____________
Same slope, different y-intercepts → Parallel → No solution
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HOMESCHOOL MATHEMATICS: GRADE 7
Example 13: How many solutions?
y = -2x + 7 (slope -2) ____________
y = 3x - 1 (slope 3) ____________
Different slopes → Lines intersect → One solution
✏️ Try This
How many solutions? (Don't solve—just determine the number.)
1. y = 5x + 3 and y = 5x - 2 ____________
2. y = -x + 4 and y = 2x + 1 ____________
3. 2x + y = 8 and 4x + 2y = 16 ____________
4. y = 3x + 1 and y = 3x + 1 ____________
LESSON 4.4: Applications of Systems
Setting Up Systems from Word Problems
Example 14: Movie tickets cost $12 for adults and $8 for children. A group buys 9 tickets totaling $84. How many of each type? ____________
Define variables: Let a = number of adult tickets Let c = number of child tickets
Write equations: Total tickets: a + c = 9 Total cost: 12a + 8c = 84
Solve by graphing (or substitution—we'll preview this):
From equation 1: c = 9 - a
Substitute: 12a + 8(9 - a) = 84
12a + 72 - 8a = 84
4a = 12
a = 3
Then c = 9 - 3 = 6
3 adult tickets and 6 child tickets Check: 3 + 6 = 9 ✓ and 12(3) + 8(6) = 36 + 48 = 84 ✓
Global Sovereign University Press Page: 55
Example 15: Two phone plans:
Plan A: $30/month + $0.10 per text
Plan B: $40/month + $0.05 per text
When do they cost the same? ____________
Let t = number of texts
Let C = total cost
Plan A: C = 30 + 0.10t
Plan B: C = 40 + 0.05t
Set equal (find intersection):
30 + 0.10t = 40 + 0.05t
0.05t = 10
t = 200
At 200 texts, both plans cost $50.
For fewer than 200 texts, Plan A is cheaper.
For more than 200 texts, Plan B is cheaper.
Example 16: A rectangle's perimeter is 36 cm. Its length is 3 cm more than its width. Find the dimensions. Let l = length, w = width
Perimeter: 2l + 2w = 36
Relationship: l = w + 3
Substitute:
2(w + 3) + 2w = 36
2w + 6 + 2w = 36
4w = 30
w = 7.5
Then l = 7.5 + 3 = 10.5
Length = 10.5 cm, Width = 7.5 cm
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HOMESCHOOL MATHEMATICS: GRADE 7
✏️ Try This
1. Concert tickets: $15 for students, $25 for adults. 200 tickets were sold for $3,500. How many of each? ____________
2. A number is 5 more than twice another number. Their sum is 26. Find both numbers. ____________
3. Gym A: $100 joining fee + $30/month. Gym B: $50 joining fee + $40/month. When are costs equal? ____________
LESSON 4.5: Preview of Algebraic Methods
Why Use Algebraic Methods?
Graphing has limitations:
Hard to read exact solutions that aren't integers
Time-consuming for complex equations
Not precise enough for many applications
Algebraic methods (substitution and elimination) give exact answers.
Substitution Method (Preview)
If one variable is isolated, substitute it into the other equation.
Example 17: Solve:
y = 3x - 5
2x + y = 15
Step 1: y is already isolated in equation 1.
Step 2: Substitute into equation 2: 2x + (3x - 5) = 15 5x - 5 = 15 5x = 20 x = 4
Step 3: Find y: y = 3(4) - 5 = 7
Solution: (4, 7)
Global Sovereign University Press Page: 57
Example 18: Solve:
x = 2y + 1 ____________
3x - 4y = 7 ____________
Substitute x = 2y + 1 into equation 2:
3(2y + 1) - 4y = 7 ____________
6y + 3 - 4y = 7 ____________
2y = 4
y = 2
Find x: x = 2(2) + 1 = 5 ____________
Solution: (5, 2) Elimination Method (Preview)
Add or subtract equations to eliminate one variable
Example 19: Solve:
x + y = 10 ____________
x - y = 4 ____________
Add the equations: (x + y) + (x - y) = 10 + 4 2x = 14 x = 7
Substitute to find y: 7 + y = 10 y = 3
Solution: (7, 3)
Example 20: Solve:
2x + 3y = 12
2x + y = 8
Subtract to eliminate x: (2x + 3y) - (2x + y) = 12 - 8 2y = 4 y = 2
Substitute: 2x + 2 = 8, 2x = 6 x = 3
Solution: (3, 2)
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HOMESCHOOL MATHEMATICS: GRADE 7
✏️ Try This
Solve using substitution:
1. y = x + 3 and 2x + y = 12 ____________
2. x = 4y and x + y = 15 ____________
Solve using elimination:
3. x + y = 9 and x - y = 3 ____________
4. 3x + 2y = 19 and 3x - 2y = 5 ____________
Chapter 4 Practice Problems
Section A: Checking Solutions
Is the point a solution?
1. (3, 5): y = 2x - 1 and y = -x + 8 ____________
2. (4, 2): y = x - 2 and y = 3x - 10____________
3. (-2, 1): x + y = -1 and 2x - y = -5 ____________
Section B: Solving by Graphing
Solve by graphing:
4. y = x + 3 and y = -x + 1 ____________
5. y = 2x and y = -x + 6 ____________
6. y = (1/2)x + 1 and y = -x + 4 ____________
7. x + y = 6 and x - y = 2 ____________
Section C: Types of Solutions
Determine: one solution, no solution, or infinitely many?
8. y = 4x + 1 and y = 4x - 3 ____________
9. y = -2x + 5 and y = 3x - 5 ____________
10. 2x + 4y = 8 and x + 2y = 4 ____________
11. y = 5x - 2 and 10x - 2y = 4 ____________
Global Sovereign University Press Page: 59
12. 3x - y = 6 and y = 3x + 2 ____________
Section D: Applications
13. Admission: $10 adults, $6 children. Total 120 people, $960 collected. How many adults?
14. A number is 12 more than another. Their sum is 48. Find both.
15. Two cars leave a city. Car A goes 60 mph, and Car B goes 50 mph (starts 1 hour later, same direction). When does B catch A?
16. A 30-foot rope is cut into two pieces. One piece is 6 feet longer than the other. Find each length. ____________
17. Plan A: $25 + $0.15/mile. Plan B: $40 + $0.10/mile. When are they equal? ____________
Section E: Algebraic Methods (Preview)
Solve using substitution:
18. y = 2x + 4 and x + y = 10 ____________
19. x = y - 5 and 2x + 3y = 15 ____________
20. y = -x + 7 and 3x + 2y = 11 ____________
Solve using elimination:
21. x + y = 14 and x - y = 6 ____________
22. 2x + y = 11 and x + y = 7____________
23. 4x + 3y = 25 and 2x + 3y = 17 ____________
೦೦ Challenge Problems
1. Find values of k so that the system has no solution:
y = 3x + 5 ____________
y = kx + 2 ____________
2. A grocer mixes nuts costing $4/lb with nuts costing $7/lb to make 12 pounds of mixture worth $5.50/lb. How many pounds of each?____________
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HOMESCHOOL MATHEMATICS: GRADE 7 3. A boat travels 30 km downstream in 2 hours and 30 km upstream in 3 hours. Find the boat's speed in still water and the current's speed. ____________
4. Three consecutive integers sum to 72. Write this as a system of equations and solve. ____________
5. Find the equation of a line that passes through the intersection of y = 2x - 1 and y = -x + 5, and is perpendicular to y = 3x + 4. ____________
Chapter 4 Summary
Key Concepts
System of equations: Two or more equations with the same variables
Solution: An ordered pair that satisfies ALL equations
Types of Systems
Type Lines Solutions: How to Identify
Independent Different slopes
Inconsistent: Same slope, different y-intercepts
Dependent: Same line Infinite Same slope AND y-intercept
Solving Methods
1. Graphing: Plot both lines, find intersection
2. Substitution: Solve one equation for a variable, substitute
3. Elimination: Add/subtract equations to eliminate a variable
Problem-Solving Steps
1. Define variables
2. Write equations from given information
3. Solve the system
4. Check the solution
5. Answer in context
Global Sovereign University Press Page: 61
Intersect One
Parallel None
Systems of equations are essential tools for solving real-world problems with multiple constraints.
Chapter 5·REVIEW AND APPLICATIONS
What You'll Learn
Connecting concepts across all chapters
Solving multi-step problems that combine skills
Applying mathematical concepts to real-world situations
Building problem-solving strategies
Why This Matters Mathematics isn't learned in isolation—real problems require combining multiple skills. This chapter brings together everything you've learned this year, showing how different concepts connect and how to tackle complex problems systematically.
LESSON 5.1: Number Sense and Operations Review
Rational Number Operations
Example 1: Simplify: -3/4 + 2/3 × (-6/5)
Follow the order of operations (multiply first):
= -3/4 + (2/3 × -6/5)
= -3/4 + (-12/15)
= -3/4 + (-4/5)
Find common denominator (20): = -15/20 + (-16/20) = -31/20 or -1 11/20
Example 2: A submarine is at -250 feet. It rises 75 feet, then dives 120 feet. What is its final depth?
-250 + 75 + (-120) = -250 + 75 - 120 = -175 - 120 = -295 feet
Proportional Relationships
Example 3: A car travels 195 miles on 6.5 gallons of gas. At this rate, how far can it travel on 10 gallons? Find unit rate: 195 ÷ 6.5 = 30 miles per gallon
Distance on 10 gallons: 30 × 10 = 300 miles
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HOMESCHOOL MATHEMATICS: GRADE 7
Percent Applications
Example 4: A $450 television is on sale for 25% off. Sales tax is 8%. What is the final price?
Discount: $450 × 0.25 = $112.50 Sale price: $450 - $112.50 = $337.50 Tax: $337.50 × 0.08 = $27 Final price: $337.50 + $27 = $364.50
Example 5: You invest $2,000 at 4.5% simple interest. How much will you have after 3 years?
I = Prt = $2,000 × 0.045 × 3 = $270 Total = $2,000 + $270 = $2,270
✏️ Practice Problems
1. Simplify: -5/6 - 1/3 + (-1/2) ____________
2. A recipe for 8 servings uses 2.5 cups of flour. How much flour for 12 servings? ____________
3. A price increased from $80 to $92. What was the percent increase? ____________
4. Find the simple interest on $3,500 at 6% for 2.5 years. ____________
LESSON 5.2: Expressions, Equations, and Inequalities Review
Simplifying Expressions
Example 6: Simplify: 4(2x - 3) - 3(x + 5)
= 8x - 12 - 3x - 15 = 8x - 3x - 12 - 15 = 5x - 27
Solving Multi-Step Equations
Example 7: Solve: 5(x - 2) = 3(x + 4) - 2
5x - 10 = 3x + 12 - 2 5x - 10 = 3x + 10 2x = 20 x = 10
Example 8: Solve: (2x + 1)/3 = (x - 1)/2
Cross multiply: 2(2x + 1) = 3(x - 1) 4x + 2 = 3x - 3 x = -5
Global Sovereign University Press Page: 63
Solving Inequalities
Example 9: Solve and graph: -3x + 7 < 16 -3x < 9
x > -3 (flip sign when dividing by negative)
Graph: Open circle at -3, arrow pointing right.
Absolute Value
Example 10: Solve: |2x - 5| = 9
Case 1: 2x - 5 = 9 → 2x = 14 → x = 7
Case 2: 2x - 5 = -9 → 2x = -4 → x = -2
Solutions: x = 7, x = -2
Word Problems with Equations
Example 11: The sum of three consecutive even integers is 150. Find them.
Let n = first integer
n + (n + 2) + (n + 4) = 150
3n + 6 = 150
3n = 144
n = 48
The integers are 48, 50, and 52.
✏️ Practice Problems
5. Simplify: 2(3a + 4) - 5(a - 2) ________________________
6. Solve: 4x - 3(x + 2) = 2(x - 5) ________________________
7. Solve: -2(x + 1) ≥ 4x - 8 ________________________
8. Solve: |x + 3| = 7 ________________________
9. The perimeter of a rectangle is 84 cm. The length is 6 cm more than the width. Find the dimensions. ________________________
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HOMESCHOOL MATHEMATICS: GRADE 7
LESSON 5.3: Geometry Review
Angle Relationships
Example 12: Two parallel lines are cut by a transversal. One angle measures 65°. Find all eight angles. If one angle is 65°:
Corresponding angles: 65°
Alternate interior angles: 65°
Same-side interior angles: 180° - 65° = 115°
Angles: 65°, 65°, 65°, 65°, 115°, 115°, 115°, 115°
Triangle Properties
Example 13: In a triangle, one angle is 20° more than the second, and the third is twice the second. Find all angles.
Let x = second angle
Then x + 20 = first angle
And 2x = third angle
x + (x + 20) + 2x = 180
4x + 20 = 180
4x = 160
x = 40
Angles: 60°, 40°, and 80°
Circle Measurements
Example 14: A circular pool has a diameter of 24 feet. Find the circumference and area.
Circumference: C = πd = 3.14 × 24 = 75.36 feet Area: A = πr² = 3.14 × 12² = 3.14 × 144 = 452.16
square feet
Global Sovereign University Press Page: 65
Cylinder Volume and Surface Area
Example 15: A cylindrical can has a radius of 4 cm and a height of 10 cm. Find the volume and total surface area.
Volume: V = πr²h = 3.14 × 16 × 10 = 502.4 cm³
Example 15: A cylindrical can has a radius of 4 cm and a height of 10 cm. Find the volume and total surface area. Volume: V = πr²h = 3.14 × 16 × 10 = 502.4 cm³
Surface Area: SA = 2πr² + 2πrh = 2(3.14)(16) + 2(3.14)(4)(10) = 100.48 + 251.2 = 351.68 cm²
Similar Figures
Example 16: Two similar triangles have perimeters of 15 cm and 25 cm. If the area of the smaller is 18 cm², find the area of the larger.
Scale factor (linear): 25/15 = 5/3
Area scale factor: (5/3)² = 25/9
Larger area: 18 × (25/9) = 450/9 = 50 cm²
✏️ Practice Problems
10. An exterior angle of a triangle is 120°. One remote interior angle is 50°. Find the other remote interior angle and the third angle of the triangle. ____________
11. Find the area and circumference of a circle with radius 7 cm (use π ≈ 22/7). ____________
12. A cylinder has a diameter of 10 m and a height of 8 m. Find the volume. ____________
13. Can sides 5, 9, and 15 form a triangle? Explain. ____________
14. Two similar rectangles have widths of 6 and 9. If the smaller has area 48, find the area of the larger. ____________
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HOMESCHOOL MATHEMATICS: GRADE 7
LESSON 5.4: Statistics and Probability Review
Statistical Analysis
Example 17: Compare two data sets:
Class A scores: 72, 78, 80, 85, 85
Class B scores: 65, 75, 80, 85, 95
Class A: Mean: (72 + 78 + 80 + 85 + 85) ÷ 5 = 400 ÷ 5 = 80 Median: 80 (middle value)
Class B: Mean: (65 + 75 + 80 + 85 + 95) ÷ 5 = 400 ÷ 5 = 80
Class A: Mean: (72 + 78 + 80 + 85 + 85) ÷ 5 = 400 ÷ 5 = 80 Median: 80 (middle value)
Class B: Mean: (65 + 75 + 80 + 85 + 95) ÷ 5 = 400 ÷ 5 = 80 Median: 80
Both have the same mean and median, but Class B has more spread (range: 30 vs. 13). Making Inferences
Example 18: In a sample of 250 students, 175 prefer pizza for lunch. The school has 1,200 students. Estimate how many prefer pizza.
Sample proportion: 175/250 = 0.70
Estimate: 0.70 × 1,200 = 840 students
Probability
Example 19: A bag contains 5 red, 3 blue, and 2 green marbles. You draw one, replace it, and draw again. Find P(red, then blue).
P(red) = 5/10 = 1/2
P(blue) = 3/10
P(red, then blue) = (1/2) × (3/10) = 3/20 = 0.15 = 15%
Example 20: Two dice are rolled. Find P(sum = 8).
Outcomes with a sum of 8: (2,6), (3,5), (4,4), (5,3), (6,2) = 5 outcomes
Total outcomes: 36 P(sum = 8) = 5/36 ≈ 0.139 or about 13.9%
Global Sovereign University Press Page: 67
✏️ Practice Problems
15. Find the mean and MAD of: 10, 14, 16, 18, 22 ____________
16. In a sample of 80 light bulbs, 4 are defective. Predict defects in 2,000 bulbs. ____________
17. A coin is flipped and a die is rolled. Find P(heads and number > 4). ____________
18. From a deck, find P(heart or face card). ____________
16. In a sample of 80 light bulbs, 4 are defective. Predict defects in 2,000 bulbs. ____________
17. A coin is flipped and a die is rolled. Find P(heads and number > ____________ 4).
18. From a deck, find P(heart or face card). ____________
LESSON 5.5: Linear Relationships Review
Slope and Equations
Example 21: Find the equation of the line through (-2, 5)
and (4, -1). Slope: m = (-1 - 5)/(4 - (-2)) = -6/6 = -1
Using point (4, -1):
y - (-1) = -1(x - 4)
y + 1 = -x + 4
y = -x + 3
Equation: y = -x + 3
Graphing
Example 22: Graph the line 3x - 2y = 6.
Find intercepts:
X-intercept (y = 0): 3x = 6, x = 2 → Point (2, 0)
Y-intercept (x = 0): -2y = 6, y = -3 → Point (0, -3)
Plot (2, 0) and (0, -3), and connect with a line.
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HOMESCHOOL MATHEMATICS: GRADE 7
Parallel and Perpendicular
Example 23: Write an equation perpendicular to y = 3x - 2 through (6, 4).
Original slope: 3
Perpendicular slope: -1/3
y - 4 = -1/3(x - 6)
y - 4 = -1/3x + 2
y = -1/3x + 6
Equation: y = -1/3x + 6 Systems of Equations
Example 24: A school sells adult and student tickets. Adult tickets cost $8, and student tickets cost $5. They sell 400 tickets for $2,600. How many of each?
Let a = adult, s = student
a + s = 400
8a + 5s = 2,600
From equation 1: s = 400 - a
Substitute: 8a + 5(400 - a) = 2,600
8a + 2,000 - 5a = 2,600
3a = 600
a = 200
s = 400 - 200 = 200
200 adult tickets and 200 student tickets
✏️ Practice Problems
19. Find the slope and y-intercept of: 4x + 2y = 10 ________________________
20. Write the equation of a line parallel to y = -2x + 5 through (3, 1). ________________________
21. Solve the system: y = 2x - 1 and y = -x + 8 ________________________
22. A rental car costs $35/day plus $0.20/mile. You have $100 for one day. How many miles can you drive? ________________________
Global Sovereign University Press Page: 69
LESSON 5.6: Multi-Step Problem Solving
Strategy for Complex Problems
1. Understand: What are you trying to find?
2. Plan: What concepts and formulas apply?
3. Solve: Show all steps clearly
4. Check: Does your answer make sense?
Example 25: A cylindrical water tank has a radius of 3 meters and a height of 10 meters. Water is pumped in at 2 cubic meters per minute. How long to fill the tank?
Step 1: Find the volume of the tank. V = πr²h = 3.14 × 9 × 10 = 282.6 m³
Step 2: Find time to fill. Time = Volume ÷ Rate = 282.6 ÷ 2 = 141.3 minutes (about 2 hours 21 minutes) Example 26: A store marks up items 40% and then offers a 25% discount. What is the overall percent change from cost?
Let cost = $100 (choose an easy number)
After 40% markup: $100 × 1.40 = $140
After 25% discount: $140 × 0.75 = $105
Overall change: ($105 - $100)/$100 = 5% increase
The overall change is a 5% increase.
Example 27: Two trains leave the same station at the same time, traveling in opposite directions. One travels at 60 mph, the other at 75 mph. When will they be 405 miles apart?
Combined rate: 60 + 75 = 135 mph (moving apart) Time = Distance ÷ Rate = 405 ÷ 135 = 3 hours
Example 28: A rectangular garden is surrounded by a path 2 meters wide. The garden is 12 m by 8 m. Find the area of the path.
Outer rectangle: (12 + 4) × (8 + 4) = 16 × 12 = 192 m² Inner rectangle (garden): 12 × 8 = 96 m² Path area: 192 - 96 = 96 m²
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✏️ Practice Problems
23. A cone has radius 6 cm and height 10 cm. Find the volume. (V = 1/3 πr²h) ____________
24. A price is increased by 20% and then decreased by 20%. Is the final price the same as the original? Calculate. ____________
25. Two cars start 300 miles apart and drive toward each other at 55 mph and 45 mph. When do they meet? ____________
26. A circular pond has a diameter of 20 m. A path 3 m wide surrounds it. Find the area of the path. ____________
COMPREHENSIVE REVIEW PROBLEMS
Mixed Practice
Solve or simplify:
1. -2/3 × (9/4 - 3/2) ____________
2. What is 15% of 240? ____________
3. 35 is what percent of 140? ____________
4. Simplify: 3(2x - 4) - 2(x + 5)____________
5. Solve: 4x - 7 = 2x + 9 ____________
6. Solve: |3x - 6| = 12 ____________
7. Find the slope through (2, -3) and (-4, 9) ____________
8. Write the equation through (3, 5) with slope -2 ____________
9. A cylinder has r = 5 cm and h = 12 cm. Find V and SA. ____________
10. P(sum = 6) when rolling two dice ____________
Word Problems
11. A population grew from 25,000 to 28,500. Find the percent increase. ____________
12. You invest $4,000 at 5.5% simple interest. Balance after 4 years? ____________
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13. A map scale is 1:250,000. Two cities are 8 cm apart on the map. Actual distance in km? ____________
14. Similar triangles have corresponding sides 5 and 8. The smaller has an area of 30. Find the larger area. ____________
15. In a sample of 150, 45 prefer option A. Predict preference in a population of 2,000. ____________
16. Adult tickets: $12, Child: $7. Sold 350 tickets for $3,200. How many adult tickets? ____________
17. A candle is 15 inches tall and burns at 2 inches per hour. Write an equation for height h after t hours. When will it burn out? ____________
18. Find three consecutive odd integers whose sum is 117. ____________
೦೦ Challenge Problems
1. A rectangular prism has dimensions that are consecutive integers. The surface area is 148 cm². Find the dimensions. ____________
2. Two numbers have a sum of 36 and a product of 320. Find the numbers. ____________
3. A boat travels 24 km upstream in 3 hours and returns in 2 hours. Find the boat's speed in still water and the current speed. ____________
4. A circle is inscribed in a square with side 10 cm. Find the area between the square and circle. ____________
5. Three people can paint a house in 6 days. How long would it take 5 people (working at the same rate)? ____________
Chapter 5 Summary
Key Formulas Reference
Percent Change: (New - Original)/Original × 100 Simple Interest: I = Prt
Circle: C = 2πr, A = πr²
Cylinder: V = πr²h, SA = 2πr² + 2πrh
Slope: m = (y₂ - y₁)/(x₂ - x₁)
Slope-Intercept: y = mx + b
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Scale: Drawing/Actual = Scale Factor
Area scaling: k² (for scale factor k)
Volume scaling: k³
Problem-Solving Strategies
1. Read carefully—identify what's given and what's asked
2. Draw a diagram if helpful
3. Define variables clearly
4. Write equations that model the situation
5. Solve step-by-step
6. Check—does your answer make sense?
7. State your answer in context
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This chapter demonstrates how mathematical concepts work together to solve complex real-world problems.
Chapter 6·PREPARING FOR ALGEBRA
What You'll Learn
Understanding functions and function notation
Exploring exponent rules
Working with square roots and cube roots
Introduction to the Pythagorean theorem
Preview of Algebra 1 concepts
Why This Matters This chapter bridges your 7th grade mathematics to Algebra 1. The concepts here—functions, exponents, roots, and the Pythagorean theorem—are foundational tools you'll use throughout high school math and beyond. Mastering these now will give you a strong start in algebra.
LESSON 6.1: Introduction to Functions
What Is a Function?
A function is a rule that assigns exactly one output to each input.
Think of a function like a machine:
Input goes in
The machine applies a rule
Exactly one output comes out
Function Notation
f(x) is read as "f of x" and represents the output when x is the input.
If f(x) = 2x + 3:
f(1) means "plug in 1 for x" = 2(1) + 3 = 5
f(5) means "plug in 5 for x" = 2(5) + 3 = 13
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Example 1: If f(x) = 3x - 4, find:
a) f(2) = 3(2) - 4 = 6 - 4 = 2
b) f(-1) = 3(-1) - 4 = -3 - 4 = -7
c) f(0) = 3(0) - 4 = 0 - 4 = -4
Example 2: If g(x) = x² + 1, find:
a) g(3) = 3² + 1 = 9 + 1 = 10
b) g(-2) = (-2)² + 1 = 4 + 1 = 5
c) g(0) = 0² + 1 = 1
Identifying Functions
A relationship is a function if each input has exactly ONE output.
Example 3: Is this a function?
x 1 2 3 4
y 5 7 9 11 Each x has exactly one y. Yes, this is a function.
Example 4: Is this a function?
x 1 2 1 3
y 4 5 6 7
The input x = 1 has two different outputs (4 and 6). No, not a function.
Vertical Line Test
On a graph, a relation is a function if no vertical line intersects the graph more than once.
Example 5: Is y = x² a function? Draw any vertical line—it hits the parabola at most once. Yes, it's a function.
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Example 6: Is x² + y² = 25 (a circle) a function?
A vertical line can hit the circle twice. No, not a function.
Domain and Range
Domain: All possible input values (x-values)
Range: All possible output values (y-values)
Example 7: Find the domain and range of: {(1, 4), (2, 5), (3, 6), (4, 7)}
Domain: {1, 2, 3, 4}
Range: {4, 5, 6, 7}
✏️ Try This
1. If f(x) = 4x - 1, find f(3) and f(-2). ____________
2. If g(x) = x² - 2x, find g(4) and g(-1). ____________
3. Is this a function? {(2, 3), (4, 5), (2, 7), (6, 9)} ____________
4. Find domain and range: {(-1, 2), (0, 4), (1, 6), (2, 8)} ____________
LESSON 6.2: Linear vs. Nonlinear Functions
Linear Functions
A linear function has:
Constant rate of change (slope) Graph is a straight line
Equation form: f(x) = mx + b
Example 8: Is f(x) = 3x + 2 linear?
Yes—it's in the form f(x) = mx + b with m = 3, b = 2.
Linear
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Nonlinear Functions
A nonlinear function does NOT have a constant rate of change.
Type Example Shape
Quadratic Parabola Exponential Curve
Absolute value f(x) = x
Example 9: Is this table linear or nonlinear?
x 0 1 2 3
y 1 3 9 27
Check rate of change:
3 - 1 = 2
9 - 3 = 6
27 - 9 = 18
Rate of change is NOT constant.
Nonlinear
(This is y = 3ˣ, an exponential function.)
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f(x) = x²
f(x) = 2ˣ
Example 10: Is this linear or nonlinear? ____________
x 0 2 4 6
y 5 11 17 23
Rate of change:
11 - 5 = 6
17 - 11 = 6
23 - 17 = 6
Constant rate of change = 6 ÷ 2 = 3. Linear (f(x) = 3x + 5)
✏️ Try This
Linear or nonlinear?
1. f(x) = x² + 3 ____________
2. f(x) = -2x + 7 ____________
LESSON 14.3: Exponent Rules
Review of Exponents
a^n means multiply a by itself n times.
3⁴ = 3 × 3 × 3 × 3 = 81
Product Rule
a^m × a^n = a^(m+n)
When multiplying with the same base, ADD exponents.
Example 11: Simplify: x³ × x⁵
x³ × x⁵ = x^(3+5) = x⁸
Example 12: Simplify: 2⁴ × 2³
2⁴ × 2³ = 2^(4+3) = 2⁷ = 128
Quotient Rule
a^m ÷ a^n = a^(m-n)
When dividing with the same base, SUBTRACT exponents.
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Example 13: Simplify: y⁸ ÷ y³
y⁸ ÷ y³ = y^(8-3) = y⁵ Example 14: Simplify: 5⁶/5²
5⁶/5² = 5^(6-2) = 5⁴ = 625
Power Rule
(a^m)^n = a^(m×n)
When raising a power to a power, MULTIPLY
exponents. Example 15: Simplify: (x²)⁴
(x²)⁴ = x^(2×4) = x⁸
Example 16: Simplify: (3²)³
(3²)³ = 3^(2×3) = 3⁶ = 729
Zero Exponent
a⁰ = 1 (for any a ≠ 0)
Example 17: Simplify: 7⁰ = 1
Example 18: Simplify: (-5)⁰ = 1
Negative Exponents
a^(-n) = 1/a^n
A negative exponent means "take the reciprocal."
Example 19: Simplify: 2^(-3)
2^(-3) = 1/2³ = 1/8 Example 20: Simplify: x^(-4)
x^(-4) = 1/x⁴
Example 21: Simplify: (1/3)^(-2)
(1/3)^(-2) = (3/1)² = 3² = 9
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✏️ Try This
Simplify:
1. x⁴ × x⁶ ____________
2. y⁹ ÷ y⁴ ____________
3. (a³)² ____________
4. 5⁰ ____________
5. 3^(-2) ____________
6. (2³ × 2²)/2⁴ ____________
LESSON 6.4: Square Roots and Cube Roots
Square Roots
The square root of a number n is a value that, when multiplied by itself, equals n.
√n means "what number times itself equals n?"
Example 22: Find √49
√49 = 7 because 7 × 7 = 49
Example 23: Find √121
√121 = 11 because 11 × 11 = 121
Perfect Squares
Memorize these:
n 1 4 9 16 25 36 49 64 81 100 121 144
√n 1 2 3 4 5 6 7 8 9 10 11 12
Estimating Square Roots
Example 24: Estimate √50
√49 = 7 and √64 = 8
50 is between 49 and 64, closer to 49.
√50 ≈ 7.1 (actual: 7.071...)
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Example 25: Between which two integers is √75? √64 = 8 and √81 = 9
64 < 75 < 81
√75 is between 8 and 9 (closer to 9, actually ≈ 8.66)
Cube Roots
The cube root of n is a value that, when cubed, equals n.
∛n means "what number cubed equals n?"
Example 26: Find ∛27
∛27 = 3 because 3³ = 27
Example 27: Find ∛-8
∛-8 = -2 because (-2)³ = -8
Note: Cube roots can be negative!
Perfect Cubes
n 1 8 27 64 125 216
∛n 1 2 3 4 5 6
✏️ Try This
1. √81 ____________
2. √144 ____________
3. Estimate √40 to one decimal place ____________
4. Between which integers is √90? ____________
5. ∛64 ____________
6. ∛-125 ____________
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LESSON 6.5: The Pythagorean Theorem
The Theorem
In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. a² + b² = c² Where:
a and b are the legs (shorter sides)
c is the hypotenuse (longest side, opposite the right angle)
Finding the Hypotenuse
Example 28: A right triangle has legs 3 and 4. Find the hypotenuse.
a² + b² = c² 3² + 4² = c² 9 + 16 = c² 25 = c² c = √25 = 5
Example 29: Find the hypotenuse of a right triangle with legs 5 and 12.
5² + 12² = c² 25 + 144 = c² 169 = c² c = √169 = 13
Finding a Leg
Example 30: A right triangle has hypotenuse 10 and one leg 6. Find the other leg.
a² + b² = c² a² + 6² = 10² a² + 36 = 100 a² = 64 a = √64 = 8
Example 31: Find the missing leg if c = 15 and one leg = 9.
a² + 9² = 15² a² + 81 = 225 a² = 144 a = √144 = 12
Pythagorean Triples
Some right triangles have all integer sides. These sets are called
Pythagorean triples. Common triples:
3, 4, 5
5, 12, 13
8, 15, 17
7, 24, 25
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Multiples also work:
6, 8, 10 (double of 3, 4, 5)
9, 12, 15 (triple of 3, 4, 5)
Real-World Applications
Example 32: A ladder leans against a wall. The base is 5 feet from the wall, and the ladder reaches 12 feet up. How long is the ladder?
This forms a right triangle. 5² + 12² = c² 25 + 144 = c² 169 = c² c = 13 feet
Example 33: A TV screen is 40 inches wide and 30 inches tall. What is the diagonal measurement? 40² + 30² = d² 1600 + 900 = d² 2500 = d² d = √2500 = 50 inches
Converse of the Pythagorean Theorem
If a² + b² = c² for a triangle's sides, then the triangle is a right triangle.
Example 34: Is a triangle with sides 6, 8, and 10 a right triangle?
Check: 6² + 8² = 36 + 64 = 100 = 10²
Yes, it is a right triangle.
Example 35: Is a triangle with sides 7, 8, and 12 a right triangle?
Check: 7² + 8² = 49 + 64 = 113
12² = 144
113 ≠ 144, so it is not a right triangle.
✏️ Try This
1. Find the hypotenuse: legs 6 and 8 ____________
2. Find the hypotenuse: legs 9 and 12 ____________
3. Find the missing leg: hypotenuse 17, one leg 8 ____________
4. Is a triangle with sides 5, 6, 8 a right triangle? ____________
5. A rectangular field is 80 m by 60 m. What is the diagonal distance across? ____________
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LESSON 6.6: Preview of Algebra 1
What's Coming in Algebra 1
Solving more complex equations:
Literal equations (solving for any variable)
Systems of equations (all methods)
Quadratic equations Working with polynomials:
Adding, subtracting, multiplying polynomials
Factoring expressions
Polynomial long division
Graphing:
Quadratic functions (parabolas)
Exponential functions
Transformations of functions
Radicals and rational expressions:
Simplifying square roots
Operations with radicals
Working with rational expressions
Skills You're Ready For
Based on 7th grade, you can already:
✓ Solve multi-step equations
✓ Work with variables on both sides
✓ Graph linear equations
✓ Understand slope and y-intercept
✓ Work with exponents and roots
✓ Apply the Pythagorean theorem
✓ Understand functions
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A Glimpse Ahead
Example 36: (Algebra 1 preview) Solve by factoring: x² - 5x + 6 = 0 Factor: (x - 2)(x - 3) = 0
Set each factor equal to 0:
x - 2 = 0 → x = 2
x - 3 = 0 → x = 3
Solutions: x = 2 and x = 3
Example 37: (Algebra 1 preview) Simplify: √50 √50 = √(25 × 2) = √25 × √2 = 5√2 ✏️ Try This
Simplify (preview):
1. √72 ____________
2. √32 ____________
Chapter 6 Practice Problems
Section A: Functions
1. If f(x) = 5x - 3, find f(4). ____________
2. If g(x) = x² + 2x, find g(-3). ____________
3. If h(x) = |x - 4|, find h(1) and h(7). ____________
4. Is {(1, 2), (2, 4), (3, 2), (4, 5)} a function? ____________
5. Find domain and range: {(0, 3), (1, 5), (2, 7), (3, 9)} ____________
Section B: Linear vs. Nonlinear
Identify as linear or nonlinear:
6. f(x) = -4x + 9 ____________
7. f(x) = x³ ____________
Section C: Exponent Rules
Simplify:
8. a⁵ × a³ ____________
9. b⁷ ÷ b² ____________
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10. (c⁴)³ ____________
11. 4⁰ + 4¹ ____________
12. 2^(-4) ____________
13. (x³ × x⁵)/(x²)____________ Section D: Square and Cube Roots
14. √196 ____________
15. √169 ____________
16. Estimate √55 (between which integers? approximate value?) ____________
17. ∛1000 ____________
18. ∛-64 ____________
19. √(144/49) ____________
Section E: Pythagorean Theorem
20. Find c: a = 9, b = 12 ____________
21. Find a: b = 24, c = 25 ____________
22. Is 9, 40, 41 a Pythagorean triple? ____________
23. A 20-foot ladder reaches 16 feet up a wall. How far is its base from the wall? ____________
24. Find the diagonal of a rectangle 15 cm by 8 cm. ____________
Section F: Mixed Review
25. Write the equation of a line with slope 3 through (2, 7). ____________
26. Solve: 3(x - 4) = 2(x + 1) - 5 ____________
27. Find the surface area of a cylinder with r = 4, h = 10. ____________
28. P(sum = 9) when rolling two dice ____________
29. In similar triangles, sides are 5, 12, 13 and 10, 24, x. Find x. ____________
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೦೦ Challenge Problems
1. If f(x) = 2x + 1, find x when f(x) = 15. ____________
2. Simplify: (3² × 3⁴)/(3³ × 3²) ____________
3. The diagonal of a square is 10 cm. Find the side length. ____________
4. A rectangular box has dimensions 3 × 4 × 5. Find the length of the space diagonal (corner to opposite corner through the box). ____________
5. If f(x) = x² - 3x + 2, find all values of x where f(x) = 0. (Hint: Factor)____________
Chapter 6 Summary
Functions
Function: Each input has exactly one output
Notation: f(x) means output when input is x
Domain: All possible x-values
Range: All possible y-values
Vertical line test: If any vertical line hits the graph more than once, it's not a function
Exponent Rules
Rule Formula Example
Product x³ × x⁴ = x⁷ Quotient x⁵ ÷ x² = x³ Power (x²)³ = x⁶
Zero 5⁰ = 1
Negative a⁻ⁿ = 1/aⁿ 2⁻³ = 1/8
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aᵐ × aⁿ = aᵐ⁺ⁿ
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
(aᵐ)ⁿ = aᵐⁿ
a⁰ = 1
Roots
Square root: √n × √n = n
Cube root: ∛n × ∛n × ∛n = n
Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100...
Perfect cubes: 1, 8, 27, 64, 125, 216...
Pythagorean Theorem
a² + b² = c²
Use to find missing side of right triangle
Converse: If a² + b² = c², then it's a right triangle
Common triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25
Congratulations!
You've completed 7th Grade Mathematics! You now have:
✓ Strong number sense with rational numbers
✓ Proportional reasoning skills
✓ Equation and inequality solving abilities ✓ Geometric understanding of angles, triangles, and circles
✓ Statistical literacy and probability foundations
✓ Linear relationship expertise
✓ Introduction to functions and advanced algebra concepts
You are ready for Algebra 1!
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