To solve the given system of linear equations:
[tex]\[
\begin{cases}
6x - 54y = -42 \\
-8x - 3y = 56 \\
\end{cases}
\][/tex]
we will follow a detailed, step-by-step approach.
### Step 1: Simplify the Equations (if possible)
First, let's see if the coefficients can be simplified to make the equations easier to work with.
For the first equation:
[tex]\[
6x - 54y = -42
\][/tex]
Divide every term by 6:
[tex]\[
x - 9y = -7
\][/tex]
So, the first equation simplifies to:
[tex]\[
x - 9y = -7 \quad \text{(Equation 1)}
\][/tex]
For the second equation:
[tex]\[
-8x - 3y = 56
\][/tex]
This equation does not simplify any further. Thus, we have:
[tex]\[
\begin{cases}
x - 9y = -7 \\
-8x - 3y = 56 \\
\end{cases}
\][/tex]
### Step 2: Solve One Equation for One Variable
From Equation 1:
[tex]\[
x - 9y = -7
\][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
x = 9y - 7 \quad \text{(Equation 3)}
\][/tex]
### Step 3: Substitute the Expression from Step 2 into the Other Equation
Now substitute Equation 3 into the second original equation:
[tex]\[
-8x - 3y = 56
\][/tex]
Substituting [tex]\( x = 9y - 7 \)[/tex]:
[tex]\[
-8(9y - 7) - 3y = 56
\][/tex]
Simplify and solve for [tex]\( y \)[/tex]:
[tex]\[
-72y + 56 - 3y = 56
\][/tex]
Combine like terms:
[tex]\[
-75y + 56 = 56
\][/tex]
Subtract 56 from both sides:
[tex]\[
-75y = 0
\][/tex]
Solve for [tex]\( y \)[/tex]:
[tex]\[
y = 0
\][/tex]
### Step 4: Substitute Back to Find the Other Variable
Substitute [tex]\( y = 0 \)[/tex] back into Equation 3:
[tex]\[
x = 9(0) - 7
\][/tex]
Simplify:
[tex]\[
x = -7
\][/tex]
### Conclusion
The solution to the system of equations is:
[tex]\[
(x, y) = (-7, 0)
\][/tex]
