To solve the given system of linear equations:
[tex]\[
\left\{
\begin{array}{l}
3x + 2y = 7 \\
4x - 3y = -2
\end{array}
\right.
\][/tex]
we can use the method of substitution or elimination. In this case, let's use the elimination method.
1. Multiply each equation to make the coefficients of [tex]\(y\)[/tex] the same:
We will multiply the first equation by 3 and the second equation by 2 so that the coefficients of [tex]\(y\)[/tex] will match in terms of their magnitude.
[tex]\[
\begin{aligned}
3(3x + 2y) &= 3(7) && \text{(Multiply the first equation by 3)} \\
9x + 6y &= 21
\end{aligned}
\][/tex]
[tex]\[
\begin{aligned}
2(4x - 3y) &= 2(-2) && \text{(Multiply the second equation by 2)} \\
8x - 6y &= -4
\end{aligned}
\][/tex]
2. Add the two equations to eliminate [tex]\(y\)[/tex]:
[tex]\[
\begin{aligned}
(9x + 6y) + (8x - 6y) &= 21 + (-4) \\
9x + 8x &= 17 \\
17x &= 17 \\
x &= 1
\end{aligned}
\][/tex]
3. Substitute the value of [tex]\(x\)[/tex] back into one of the original equations to find [tex]\(y\)[/tex]:
Let's substitute [tex]\(x = 1\)[/tex] into the first original equation:
[tex]\[
\begin{aligned}
3(1) + 2y &= 7 \\
3 + 2y &= 7 \\
2y &= 7 - 3 \\
2y &= 4 \\
y &= 2
\end{aligned}
\][/tex]
Therefore, the solution to the system of equations is:
[tex]\[
\boxed{(1, 2)}
\][/tex]
