Let's solve the system of linear equations step by step:
[tex]\[
\left\{
\begin{array}{l}
-x + 3y = -8 \\
x - 4y = 9
\end{array}
\right.
\][/tex]
Step 1: Isolate one variable in one of the equations. Let's isolate [tex]\(x\)[/tex] in the second equation:
[tex]\[
x - 4y = 9
\][/tex]
Add [tex]\(4y\)[/tex] to both sides:
[tex]\[
x = 4y + 9
\][/tex]
Step 2: Substitute this expression for [tex]\(x\)[/tex] into the first equation:
[tex]\[
-(4y + 9) + 3y = -8
\][/tex]
Distribute the negative sign:
[tex]\[
-4y - 9 + 3y = -8
\][/tex]
Combine like terms:
[tex]\[
-4y + 3y - 9 = -8
\][/tex]
[tex]\[
-y - 9 = -8
\][/tex]
Add 9 to both sides:
[tex]\[
-y = 1
\][/tex]
Multiply by -1 to solve for [tex]\(y\)[/tex]:
[tex]\[
y = -1
\][/tex]
Step 3: Substitute [tex]\(y = -1\)[/tex] back into the expression we found for [tex]\(x\)[/tex]:
[tex]\[
x = 4y + 9
\][/tex]
[tex]\[
x = 4(-1) + 9
\][/tex]
[tex]\[
x = -4 + 9
\][/tex]
[tex]\[
x = 5
\][/tex]
So, the solution to the system of equations is:
[tex]\[
x = 5, \quad y = -1
\][/tex]
Thus, the solution is [tex]\((x, y) = (5, -1)\)[/tex].
