ANSWER KEYS - Teacher Use Only

Linear Equations | Grade 8

Student Concept Worksheet Answers
1
x = 6
2x + 5 = 17 → 2x = 12 → x = 6. Check: 2(6) + 5 = 17 ✓
2
x = 8
4(x - 3) = 20 → 4x - 12 = 20 → 4x = 32 → x = 8. Check: 4(8-3) = 4(5) = 20 ✓
3
x = 5
6x - 4 = 3x + 11 → 3x = 15 → x = 5. Check: 6(5)-4 = 26 and 3(5)+11 = 26 ✓
4
No Solution
5x + 2 = 5x + 7 → 2 = 7 (FALSE). No value of x can make 2 equal to 7.
Practice Worksheet Answers

Part A: Two-Step Equations

# Answer Work
1x = 53x + 7 = 22 → 3x = 15 → x = 5
2x = 85x - 9 = 31 → 5x = 40 → x = 8
3x = -62x + 15 = 3 → 2x = -12 → x = -6
4x = -34x - 7 = -19 → 4x = -12 → x = -3

Part B: Equations with Distributive Property

#AnswerWork
5x = 42(x+5) = 18 → 2x + 10 = 18 → 2x = 8 → x = 4
6x = 93(x-4) = 15 → 3x - 12 = 15 → 3x = 27 → x = 9
7x = 35(2x+1) = 35 → 10x + 5 = 35 → 10x = 30 → x = 3
8x = 64(x-2) + 3 = 19 → 4x - 8 + 3 = 19 → 4x - 5 = 19 → 4x = 24 → x = 6

Part C: Variables on Both Sides

#AnswerWork
9x = 57x + 3 = 4x + 18 → 3x = 15 → x = 5
10x = 59x - 5 = 6x + 10 → 3x = 15 → x = 5
11x = 52x + 14 = 5x - 1 → 15 = 3x → x = 5
12x = 48x - 12 = 3x + 8 → 5x = 20 → x = 4

Part D: Identifying Special Cases

13
Infinite Solutions
3x + 6 = 3x + 6 → 6 = 6 (TRUE for all x). Any number works!
14
No Solution
2x + 5 = 2x + 9 → 5 = 9 (FALSE). No number can make this true.
15
Infinite Solutions
4(x + 2) = 4x + 8 → 4x + 8 = 4x + 8 → 8 = 8 (TRUE for all x).
16
No Solution
6x - 3 = 3(2x + 1) → 6x - 3 = 6x + 3 → -3 = 3 (FALSE).

Part E: Word Problems

17
Equation: 45 + 12x = 93 - 6x | Answer: x = approximately 2.67 or 8/3 weeks
Sarah: 45 + 12x. Brother: 93 - 6x. Set equal: 45 + 12x = 93 - 6x → 18x = 48 → x = 48/18 = 8/3 ≈ 2.67 weeks.
18
Equation: 25 + 0.10x = 15 + 0.15x | Answer: 200 texts
25 + 0.10x = 15 + 0.15x → 10 = 0.05x → x = 200 texts.
19
Width: 7 cm | Length: 17 cm
Let w = width, length = 2w + 3. Perimeter: 2w + 2(2w+3) = 48 → 2w + 4w + 6 = 48 → 6w = 42 → w = 7. Length = 2(7) + 3 = 17.

Challenge Questions:

#20: 3(2x-4) - 2(x+5) = 2(x-11) → 6x - 12 - 2x - 10 = 2x - 22 → 4x - 22 = 2x - 22 → 2x = 0 → x = 0. Check: 3(0-4) - 2(0+5) = -12 - 10 = -22 and 2(0-11) = -22 ✓

#21: Example: 4x + 3 = 4x + 8. When solved: 3 = 8, which is false. Therefore, no solution exists. (Any equation where variables cancel leaving a false statement works.)

FAST Practice Quiz Answers
1
B) x = 8
4x - 7 = 2x + 9 → 2x = 16 → x = 8. Check: 4(8)-7 = 25 and 2(8)+9 = 25 ✓
2
A) x = 4
3(x + 5) = 27 → 3x + 15 = 27 → 3x = 12 → x = 4. Check: 3(4+5) = 3(9) = 27 ✓
3
B) 3x + 4 = 3x + 7
A. 2x + 5 = 2x + 5 → 5 = 5 (TRUE) - Infinite solutions
B. 3x + 4 = 3x + 7 → 4 = 7 (FALSE) - NO SOLUTION ✓
C. 5x - 2 = 3x + 6 → 2x = 8 → x = 4 - One solution
D. 4(x+1) = 4x + 4 → 4x + 4 = 4x + 4 - Infinite solutions
4
Select: 2x + 4 = 10, 5x - 6 = 9, 3(x-1) = 6, 6x - 3 = 2x + 9
2x + 4 = 10 → x = 3 ✓
5x - 6 = 9 → x = 3 ✓
4x + 1 = 11 → x = 2.5 ✗
3(x-1) = 6 → x = 3 ✓
6x - 3 = 2x + 9 → 4x = 12 → x = 3 ✓
5
D) Infinitely many solutions
5(x-2) = 5x - 10 → 5x - 10 = 5x - 10 → -10 = -10 (TRUE). Every value of x works.
6
B) 50 + 15w = 110 - 5w
Maria starts with $50 and ADDS $15 each week: 50 + 15w. Juan starts with $110 and SPENDS (subtracts) $5 each week: 110 - 5w. Set equal: 50 + 15w = 110 - 5w.
FAST Quiz Scoring Guide
Score Interpretation Recommended Action
6/6 Mastery Ready for systems of equations and more complex algebraic expressions.
4-5/6 Approaching Mastery Review specific error patterns (distribution errors, sign errors, or special case confusion).
2-3/6 Developing Reteach using balance scale model. Practice two-step equations before adding variables on both sides.
0-1/6 Needs Intervention Small group instruction. Start with one-step equations and build up. Use manipulatives (algebra tiles). Focus on understanding the concept of "keeping the equation balanced."
FAST Success Kit | Grade 8 | FL B.E.S.T. Standards: MA.8.AR.2.1, MA.8.AR.2.2 | Answer Key - Teacher Use Only