Sure! Let's solve the system of equations using the addition (elimination) method step by step.
We have the following system of equations:
[tex]\[
\begin{cases}
4x + 3y = -18 \quad \text{(1)} \\
x + 9y = -21 \quad \text{(2)}
\end{cases}
\][/tex]
Step 1: Make the coefficients of one of the variables equal in both equations.
To eliminate [tex]\(x\)[/tex], we need the coefficients of [tex]\(x\)[/tex] to be equal. We can do this by multiplying the second equation by 4.
[tex]\[
4(x + 9y = -21) \implies 4x + 36y = -84 \quad \text{(3)}
\][/tex]
Step 2: Subtract the modified second equation from the first equation to eliminate [tex]\(x\)[/tex].
Now we have:
[tex]\[
\begin{cases}
4x + 3y = -18 \quad \text{(1)} \\
4x + 36y = -84 \quad \text{(3)}
\end{cases}
\][/tex]
Subtract equation (1) from equation (3):
[tex]\[
(4x + 36y) - (4x + 3y) = -84 - (-18)
\][/tex]
Simplify the left-hand side:
[tex]\[
4x + 36y - 4x - 3y = -84 + 18
\][/tex]
[tex]\[
33y = -66
\][/tex]
Step 3: Solve for [tex]\(y\)[/tex].
Divide both sides by 33:
[tex]\[
y = \frac{-66}{33}
\][/tex]
[tex]\[
y = -2
\][/tex]
Step 4: Substitute [tex]\(y = -2\)[/tex] back into one of the original equations to solve for [tex]\(x\)[/tex].
Using the second equation [tex]\(x + 9y = -21\)[/tex]:
[tex]\[
x + 9(-2) = -21
\][/tex]
[tex]\[
x - 18 = -21
\][/tex]
[tex]\[
x = -21 + 18
\][/tex]
[tex]\[
x = -3
\][/tex]
Step 5: Write the solution.
The solution to the system is:
[tex]\[
(x, y) = (-3, -2)
\][/tex]
So, the solution is:
[tex]\[
\boxed{(-3, -2)}
\][/tex]
