Question

select the correct answer.
the elimination method is ideal for solving this system of equations. by which number must you multiply the second equation to eliminate the y - variable, and what is the solution for this system?
x + 3y = 42
2x - y = 14
a. multiply the second equation by - 3. the solution is x = 12, y = 9.
b. multiply the second equation by - 2. the solution is x = 12, y = 10.
c. multiply the second equation by 2. the solution is x = 15, y = 9.
d. multiply the second equation by 3. the solution is x = 12, y = 10.

Explanation:

Step1: Determine multiplier for elimination

To eliminate \( y \), the coefficients of \( y \) in both equations should be opposites (or equal). The first equation has \( 3y \), the second has \( -y \). Multiply the second equation by \( 3 \) to get \( -3y \), but wait, let's check the options. Wait, first equation: \( x + 3y = 42 \), second: \( 2x - y = 14 \). If we multiply the second equation by \( 3 \), we get \( 6x - 3y = 42 \). Now add to first equation: \( (x + 3y) + (6x - 3y) = 42 + 42 \), so \( 7x = 84 \), \( x = 12 \). Then substitute \( x = 12 \) into second equation: \( 2(12) - y = 14 \), \( 24 - y = 14 \), \( y = 10 \)? Wait no, wait option D: multiply by 3, solution \( x=12, y=10 \). Wait let's recalculate. Wait first equation: \( x + 3y = 42 \), second: \( 2x - y = 14 \). Multiply second by 3: \( 6x - 3y = 42 \). Add to first: \( x + 3y + 6x - 3y = 42 + 42 \) → \( 7x = 84 \) → \( x = 12 \). Then plug \( x = 12 \) into second equation: \( 2(12) - y = 14 \) → \( 24 - y = 14 \) → \( y = 10 \). Wait but let's check option D: "Multiply the second equation by 3. The solution is \( x = 12, y = 10 \)." Yes. Wait but let's check other options. Option A: multiply by -3: second equation becomes \( -6x + 3y = -42 \). Add to first: \( x + 3y -6x + 3y = 42 -42 \) → \( -5x + 6y = 0 \), not helpful. Option B: multiply by -2: second equation \( -4x + 2y = -28 \). Add to first: \( x + 3y -4x + 2y = 42 -28 \) → \( -3x + 5y = 14 \), not helpful. Option C: multiply by 2: second equation \( 4x - 2y = 28 \). Add to first: \( x + 3y + 4x - 2y = 42 + 28 \) → \( 5x + y = 70 \), not helpful. So correct is D.

Step2: Verify solution

Substitute \( x = 12 \), \( y = 10 \) into first equation: \( 12 + 3(10) = 12 + 30 = 42 \), correct. Second equation: \( 2(12) - 10 = 24 - 10 = 14 \), correct.

Answer:

D. Multiply the second equation by 3. The solution is \( x = 12, y = 10 \).