ANSWER KEYS - Teacher Use Only

Systems of Equations | Grade 8

Student Concept Worksheet Answers
1
YES - (2, 5) IS a solution
Check equation 1: y = 3x - 1 → 5 = 3(2) - 1 = 6 - 1 = 5 ✓
Check equation 2: y = x + 3 → 5 = 2 + 3 = 5 ✓
Both equations are satisfied!
2
Solution: (2, 6)
Substitute y = x + 4 into 2x + y = 10: 2x + (x + 4) = 10 → 3x + 4 = 10 → 3x = 6 → x = 2. Then y = 2 + 4 = 6.
3
Solution: (2, 1) - Read from graph intersection
The lines intersect at the point (2, 1) based on the graph coordinates.
4
NO SOLUTION - Lines are parallel
Both equations have slope 2 but different y-intercepts (3 and -1). Parallel lines never intersect.
Practice Worksheet Answers

Part A: Is It a Solution?

#AnswerWork
1YESEq1: 5 = 2(3)-1 = 5 ✓ | Eq2: 5 = 3+2 = 5 ✓
2YESEq1: 4+(-2) = 2 ✓ | Eq2: 2(4)-(-2) = 10 ✓
3NOEq1: 4 = 3(1)+1 = 4 ✓ | Eq2: 2(1)+4 = 6 ≠ 5 ✗

Part B: Solve by Substitution

4
(3, 6)
2x + (x + 3) = 12 → 3x + 3 = 12 → x = 3, y = 6
5
(3, 1)
x + (2x - 5) = 4 → 3x - 5 = 4 → x = 3, y = 2(3) - 5 = 1
6
(6, 2)
2(3y) + y = 14 → 7y = 14 → y = 2, x = 3(2) = 6
7
(2, 6)
3x + (-x + 8) = 12 → 2x + 8 = 12 → x = 2, y = -2 + 8 = 6

Part C: How Many Solutions?

#AnswerExplanation
8No solutionSame slope (3), different y-intercepts (2 and -4). Parallel lines.
9One solutionDifferent slopes (2 and -1). Lines must intersect.
10Infinite solutions2y = 8x - 6 → y = 4x - 3. Same line as the first equation!

Part D: Word Problems

11
5 tickets (both cost $60)
Theater A: y = 12x. Theater B: y = 8x + 20. Set equal: 12x = 8x + 20 → 4x = 20 → x = 5. Cost: $60.
12
Numbers are 8 and 12
x + y = 20 and x = y + 4. Substitute: (y + 4) + y = 20 → 2y = 16 → y = 8, x = 12.
13
200 texts (both cost $40)
Plan 1: y = 30 + 0.05x. Plan 2: y = 20 + 0.10x. 30 + 0.05x = 20 + 0.10x → 10 = 0.05x → x = 200.

Challenge Questions:

#14: Let c = chickens, w = cows. System: c + w = 30 (heads) and 2c + 4w = 74 (legs). Solve: From eq1, c = 30 - w. Substitute: 2(30-w) + 4w = 74 → 60 - 2w + 4w = 74 → 2w = 14 → w = 7 cows. c = 30 - 7 = 23 chickens and 7 cows.

#15: Answers vary. Example: y = x + 3 and y = 2x - 3. Check: For (5,2): 2 = 5+3? No, that's 8. Let's try: y = 2x - 8 and y = x - 3. Check: 2 = 2(5)-8 = 2 ✓ and 2 = 5-3 = 2 ✓.

FAST Practice Quiz Answers
1
A) Yes, because it satisfies both equations
y = 3x - 1: 5 = 3(2) - 1 = 5 ✓ | x + y = 7: 2 + 5 = 7 ✓
2
A) (2, 4)
x + 2 = -x + 6 → 2x = 4 → x = 2. Then y = 2 + 2 = 4. Solution: (2, 4).
3
C) (1, 2)
Based on the graph, the intersection point is at x = 1 and y = 2.
4
Select: (2, 5) only
Solve: y = 2x + 1 and y = x + 3 → 2x + 1 = x + 3 → x = 2, y = 5.
(0, 1): y = 0+3 = 3 ≠ 1 ✗
(2, 5): y = 2(2)+1 = 5 ✓ and y = 2+3 = 5 ✓
(1, 3): y = 2(1)+1 = 3 ✓ but y = 1+3 = 4 ≠ 3 ✗
(3, 6): y = 2(3)+1 = 7 ≠ 6 ✗
5
B) No solution (the lines are parallel)
Both have slope 4 but different y-intercepts (-2 and +5). Parallel lines never intersect.
6
B) y = 20 + 5x and y = 50
Plan A: $20 base + $5 per class = 20 + 5x. Plan B: flat $50 = 50. Option B correctly represents this.
FAST Quiz Scoring Guide
Score Interpretation Recommended Action
6/6 Mastery Ready for more complex systems and real-world applications.
4-5/6 Approaching Mastery Review specific areas: graphing, substitution, or special cases.
2-3/6 Developing Reteach using graphing first. Emphasize checking solutions in BOTH equations.
0-1/6 Needs Intervention Review solving single linear equations first. Use visual graphs extensively. Focus on understanding what "solution" means.
FAST Success Kit | Grade 8 | FL B.E.S.T. Standards: MA.8.AR.4.1, MA.8.AR.4.2, MA.8.AR.4.3 | Answer Key - Teacher Use Only