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 \( -3y \), which will cancel with \( 3y \). Wait, wait, let's check the options. Wait, the first equation: \( x + 3y = 42 \), second: \( 2x - y = 14 \). To eliminate \( y \), we need the coefficient of \( y \) in the second equation to be \( -3 \) (so that \( 3y + (-3y) = 0 \))? Wait no, wait: if we multiply the second equation by \( 3 \), the second equation becomes \( 6x - 3y = 42 \). Then add to the first equation: \( (x + 3y) + (6x - 3y) = 42 + 42 \), so \( 7x = 84 \), so \( x = 12 \). Then substitute \( x = 12 \) into the second original equation: \( 2(12) - y = 14 \), \( 24 - y = 14 \), \( y = 10 \). Wait, let's check the options. Option D: Multiply the second equation by 3. The solution is x = 12, y = 10. Let's verify each option:

Option A: Multiply second equation by -3: \( -6x + 3y = -42 \). Add to first equation: \( x + 3y -6x + 3y = 42 -42 \), \( -5x + 6y = 0 \). Not helpful. So A is wrong.

Option B: Multiply by -2: \( -4x + 2y = -28 \). Add to first equation: \( x + 3y -4x + 2y = 42 -28 \), \( -3x + 5y = 14 \). Not helpful. B wrong.

Option C: Multiply by 2: \( 4x - 2y = 28 \). Add to first equation: \( x + 3y +4x -2y = 42 +28 \), \( 5x + y = 70 \). Not eliminating y. C wrong.

Option D: Multiply second equation by 3: \( 6x - 3y = 42 \). Add to first equation: \( x + 3y +6x -3y = 42 +42 \), \( 7x = 84 \), so \( x = 12 \). Then \( 2(12) - y = 14 \), \( 24 - y = 14 \), \( y = 10 \). So D is correct.

Step2: Verify the solution

Substitute \( x = 12 \) and \( y = 10 \) into both original equations:

First equation: \( 12 + 3(10) = 12 + 30 = 42 \), which matches.

Second equation: \( 2(12) - 10 = 24 - 10 = 14 \), which matches. So D is correct.

Answer:

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