To solve the system of equations:
[tex]\[
\left\{\begin{array}{l}
-2 x+3 y=13 \\
3 x+4 y=6
\end{array}\right.
\][/tex]
We'll use the method of elimination or substitution to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
1. Step 1: Write down the system of equations:
[tex]\[
\left\{\begin{array}{l}
-2x + 3y = 13 \\
3x + 4y = 6
\end{array}\right.
\][/tex]
2. Step 2: Choose one of the equations and solve for one variable. Let's solve the first equation for [tex]\(x\)[/tex]:
[tex]\[
-2x + 3y = 13
\][/tex]
We can solve for [tex]\(x\)[/tex] as follows:
[tex]\[
-2x = 13 - 3y \\
x = \frac{13 - 3y}{-2} \\
x = -\frac{13}{2} + \frac{3}{2}y
\][/tex]
3. Step 3: Substitute this expression for [tex]\(x\)[/tex] into the second equation:
[tex]\[
3\left(-\frac{13}{2} + \frac{3}{2}y\right) + 4y = 6
\][/tex]
Multiply through by 2 to clear the fraction:
[tex]\[
3\left(-13 + 3y\right) + 8y = 12 \\
-39 + 9y + 8y = 12 \\
-39 + 17y = 12
\][/tex]
Solve for [tex]\(y\)[/tex]:
[tex]\[
17y = 12 + 39 \\
17y = 51 \\
y = \frac{51}{17} \\
y = 3
\][/tex]
4. Step 4: Substitute the value of [tex]\(y\)[/tex] back into one of the original equations to solve for [tex]\(x\)[/tex]. Use the first equation:
[tex]\[
-2x + 3(3) = 13 \\
-2x + 9 = 13 \\
-2x = 13 - 9 \\
-2x = 4 \\
x = \frac{4}{-2} \\
x = -2
\][/tex]
So, the solution to the system of equations is:
[tex]\[
(x, y) = (-2, 3)
\][/tex]
Therefore, the correct choice among the provided options is:
[tex]$(-2, 3)$[/tex].
