Quiz Answers Explained

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📚 Part 1: Multiple Choice (Answers with brief explanations)

1. Solve by elimination: 2x + 3y = 12 and 4x + 3y = 18. What is the solution (x, y)?

(3, 2)

(−3, −2)

(2, 3)

No solution

Answer: (3, 2). Explanation: Subtract the first equation from the second to get 2x = 6, so x = 3. Substitute to find y = 2.

2. Consider 3x − 2y = 7 and 6x − 4y = 14. What is true about this system?

One unique solution

No solution

Infinitely many solutions

Exactly two solutions

Answer: Infinitely many solutions. Explanation: The second equation is 2× the first, so they are the same line.

3. Solve by elimination: x + 2y = 5 and 2x + 3y = 8. What is the solution (x, y)?

(1, 2)

(2, 1)

(3, −1)

(0, 2.5)

Answer: (1, 2). Explanation: Multiply the first equation by -2 and add to the second to eliminate x: -y = -2 → y = 2; then x = 1.

4. To eliminate y from 5x + 4y = 1 and 3x − 2y = 7, which step is most efficient?

Multiply the second equation by 2

Multiply the first equation by 3

Subtract the equations as written

Multiply the second equation by 4

Answer: Multiply the second equation by 2. Explanation: 2*(3x − 2y) = 6x − 4y, which will cancel the +4y when added to the first after adjusting for x if needed (or allows elimination of y by addition/subtraction).

✏️ Part 2: Short Answers — Elimination Steps & Explanations

I can: solve systems using the elimination method and explain each step.

Success criteria: Identify a variable to eliminate, multiply equations if needed, add/subtract correctly, solve for one variable, then substitute to find the other.

Differentiation: Provide hints, allow a calculator, or work with a partner.

Extension: Create a real-world word problem that leads to a system solved by elimination or solve a system with fractional coefficients.

Dyslexia-friendly options: Use a colored overlay, read equations aloud, break steps into short lines.

5. Solve using elimination: 2x + 3y = 12 and 4x + 3y = 18

Step 1: Subtract equation 1 from equation 2 to eliminate y: 4x + 3y − (2x + 3y) = 18 − 12 → 2x = 6 Step 2: x = 3 Step 3: Substitute into 2x + 3y = 12 → 6 + 3y = 12 → 3y = 6 → y = 2

Answer: (3, 2)

6. Solve using elimination: 3x + 2y = 18 and 6x − 4y = 12

Step 1: Multiply the first equation by -2 to align y: -2*(3x + 2y) = -36 → -6x − 4y = -36 Step 2: Add to the second equation: (-6x − 4y) + (6x − 4y) = -36 + 12 → -8y = -24 Step 3: y = 3. Substitute into 3x + 2(3) = 18 → 3x + 6 = 18 → x = 4

Answer: (4, 3)

7. Solve using elimination: 5x + 2y = 1 and 10x + 4y = 2. State whether the system has one solution, no solution, or infinitely many solutions.

Observation: The second equation is exactly 2× the first (10x + 4y = 2 is 2*(5x + 2y = 1)). Conclusion: The equations represent the same line, so there are infinitely many solutions.

Answer: Infinitely many solutions

8. Solve using elimination: 4x + 6y = 7 and 2x + 3y = 4. Explain your conclusion (unique, none, or infinite).

Check multiples: 2*(2x + 3y = 4) → 4x + 6y = 8, which conflicts with 4x + 6y = 7. Since the same left side would need to equal two different constants, the system is inconsistent. Conclusion: No solution (parallel lines).

Answer: No solution