Difficulty: Medium
Correct Answer: 10
Explanation:
Introduction:
This algebra question asks you to solve a pair of simultaneous linear equations and then find the value of a particular linear combination of x and y. It tests your understanding of basic equation solving techniques such as elimination or substitution.
Given Data / Assumptions:
Concept / Approach:
We can solve the system for x and y using elimination or substitution. Once we have x and y, we can compute 2x + 2y directly. Another efficient approach is to combine the equations in a way that directly produces x + y, and then double it.
Step-by-Step Solution:
Step 1: Solve the system using elimination. Equation 1: 3x + y = 19. (1)
Equation 2: x + 3y = 1. (2)
Multiply Equation 2 by 3 to align coefficients of x: 3x + 9y = 3. (3) Now subtract Equation 1 from Equation 3: (3x + 9y) - (3x + y) = 3 - 19. 3x + 9y - 3x - y = -16. 8y = -16 ⇒ y = -2. Step 2: Substitute y back to find x. Use Equation 2: x + 3y = 1. x + 3 * (-2) = 1 ⇒ x - 6 = 1 ⇒ x = 7. Step 3: Find 2x + 2y. Compute x + y first: x + y = 7 + (-2) = 5. Then: 2x + 2y = 2(x + y) = 2 * 5 = 10.
Verification / Alternative Check:
Check x = 7, y = -2 in both original equations: 3x + y = 3 * 7 + (-2) = 21 - 2 = 19 (satisfies Equation 1). x + 3y = 7 + 3 * (-2) = 7 - 6 = 1 (satisfies Equation 2). Everything is consistent, so 2x + 2y = 10 is reliable.
Why Other Options Are Wrong:
Values like 20, 18, 11, or 8 would arise from incorrect algebra, such as sign mistakes, adding equations incorrectly, or miscomputing the combination 2x + 2y from x and y. Since the verified solution yields 10, all other answers are incorrect.
Common Pitfalls:
Typical errors include mixing up coefficients, forgetting to multiply both sides of an equation during elimination, or miscalculating when substituting values back. It is often simpler to find x + y first when the desired quantity is 2x + 2y, since it is just twice x + y.
Final Answer:
The value of 2x + 2y is 10.
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