To solve the given system of linear equations:
[tex]\[
\begin{cases}
9x + 6y = 12 \\
-18x - 8y = -4
\end{cases}
\][/tex]
We'll follow a systematic method to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
### Step 1: Simplify the Equations
First, let's simplify the equations if possible. Both equations appear to be in their simplest forms, so we'll proceed as is.
### Step 2: Express One Variable in Terms of Another
To make it easier, we can solve one of the equations for one variable. Let's solve the first equation for [tex]\( y \)[/tex]:
[tex]\[
9x + 6y = 12 \implies 6y = 12 - 9x \implies y = \frac{12 - 9x}{6} \implies y = 2 - \frac{3}{2}x.
\][/tex]
Now we have:
[tex]\[
y = 2 - \frac{3}{2}x.
\][/tex]
### Step 3: Substitute Into the Second Equation
Substitute [tex]\( y = 2 - \frac{3}{2}x \)[/tex] into the second equation:
[tex]\[
-18x - 8y = -4 \implies -18x - 8(2 - \frac{3}{2}x) = -4.
\][/tex]
Simplify the expression inside the parentheses:
[tex]\[
-18x - 8(2) + 8 \cdot \frac{3}{2}x = -4 \implies -18x - 16 + 12x = -4.
\][/tex]
Combine like terms:
[tex]\[
-6x - 16 = -4.
\][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
-6x = -4 + 16 \implies -6x = 12 \implies x = -2.
\][/tex]
So we have found [tex]\( x = -2 \)[/tex].
### Step 4: Solve for the Other Variable
Now, substitute [tex]\( x = -2 \)[/tex] back into the expression for [tex]\( y \)[/tex]:
[tex]\[
y = 2 - \frac{3}{2}(-2) \implies y = 2 + 3 \implies y = 5.
\][/tex]
Thus, we have found [tex]\( y = 5 \)[/tex].
### Conclusion
The solution to the system of equations is:
[tex]\[
\boxed{x = -2, \; y = 5}
\][/tex]
