To solve the system of equations
[tex]\[ 3x - 2y = 4 \][/tex]
[tex]\[ 2x + 3y = -6 \][/tex]
we need to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both equations simultaneously. Here is the step-by-step solution:
1. Multiply the first equation by 3 and the second equation by 2 to align the coefficients of [tex]\(x\)[/tex]:
[tex]\[ 3(3x - 2y) = 3(4) \][/tex]
[tex]\[ 9x - 6y = 12 \][/tex]
[tex]\[ 2(2x + 3y) = 2(-6) \][/tex]
[tex]\[ 4x + 6y = -12 \][/tex]
2. Add the new equations together to eliminate [tex]\(y\)[/tex]:
[tex]\[ (9x - 6y) + (4x + 6y) = 12 + (-12) \][/tex]
[tex]\[ 9x + 4x - 6y + 6y = 0 \][/tex]
[tex]\[ 13x = 0 \][/tex]
3. Solve for [tex]\(x\)[/tex]:
[tex]\[ x = 0 \][/tex]
4. Substitute [tex]\(x = 0\)[/tex] back into one of the original equations to solve for [tex]\(y\)[/tex]. We'll use the first equation [tex]\(3x - 2y = 4\)[/tex]:
[tex]\[ 3(0) - 2y = 4 \][/tex]
[tex]\[ -2y = 4 \][/tex]
5. Solve for [tex]\(y\)[/tex]:
[tex]\[ y = -2 \][/tex]
Thus, the value of [tex]\(y\)[/tex] is
[tex]\[
\boxed{-2}
\][/tex]
Therefore, the correct answer is:
(c) -2
