Sure, let's solve the system of linear equations step-by-step:
[tex]\[
\begin{cases}
-7x + 8y = 9 \\
5x - 4y = -3
\end{cases}
\][/tex]
### Step 1: Express one of the variables in terms of the other
Let's solve the second equation for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[
5x - 4y = -3
\][/tex]
First, isolate the term with [tex]\( y \)[/tex]:
[tex]\[
-4y = -5x - 3
\][/tex]
Divide both sides by [tex]\( -4 \)[/tex]:
[tex]\[
y = \frac{5x + 3}{4}
\][/tex]
### Step 2: Substitute [tex]\( y \)[/tex] into the other equation
Substitute [tex]\( y = \frac{5x + 3}{4} \)[/tex] into the first equation:
[tex]\[
-7x + 8\left(\frac{5x + 3}{4}\right) = 9
\][/tex]
Simplify inside the parentheses:
[tex]\[
-7x + 10x + 6 = 9
\][/tex]
Combine like terms:
[tex]\[
3x + 6 = 9
\][/tex]
Subtract 6 from both sides:
[tex]\[
3x = 3
\][/tex]
Divide both sides by 3:
[tex]\[
x = 1
\][/tex]
### Step 3: Find the value of [tex]\( y \)[/tex]
Now that we have [tex]\( x = 1 \)[/tex], substitute this value back into the expression for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{5(1) + 3}{4}
\][/tex]
Simplify:
[tex]\[
y = \frac{5 + 3}{4} = \frac{8}{4} = 2
\][/tex]
### Step 4: Verify the solution
Substitute [tex]\( x = 1 \)[/tex] and [tex]\( y = 2 \)[/tex] back into the original equations to ensure they are true.
For the first equation:
[tex]\[
-7(1) + 8(2) = -7 + 16 = 9
\][/tex]
This is correct.
For the second equation:
[tex]\[
5(1) - 4(2) = 5 - 8 = -3
\][/tex]
This is also correct.
### Conclusion
The solution to the system of equations is:
[tex]\[
\boxed{x = 1, y = 2}
\][/tex]
