To solve the system of equations using the elimination method, follow these detailed steps:
Given the system of equations:
[tex]\[
\begin{array}{r}
-9x - 4y = 1 \quad \text{(1)}\\
3x + 3y = 3 \quad \text{(2)}
\end{array}
\][/tex]
Step 1: Eliminate one variable by making the coefficients of one of the variables equal in both equations.
Let's aim to eliminate [tex]\( y \)[/tex]. To do this, we need to make the coefficients of [tex]\( y \)[/tex] in both equations equal. We can multiply:
- Equation (1) by 3
- Equation (2) by 4
[tex]\[
\begin{array}{r}
3 \times (-9x - 4y) = 3 \times 1 \\
4 \times (3x + 3y) = 4 \times 3
\end{array}
\][/tex]
This gives us:
[tex]\[
\begin{array}{r}
-27x - 12y = 3 \quad \text{(3)} \\
12x + 12y = 12 \quad \text{(4)}
\end{array}
\][/tex]
Step 2: Add the transformed equations to eliminate [tex]\( y \)[/tex].
[tex]\[
(-27x - 12y) + (12x + 12y) = 3 + 12
\][/tex]
This simplifies to:
[tex]\[
-27x + 12x = 15
\][/tex]
[tex]\[
-15x = 15
\][/tex]
Step 3: Solve for [tex]\( x \)[/tex].
[tex]\[
x = \frac{15}{-15} = -1
\][/tex]
Step 4: Substitute [tex]\( x \)[/tex] back into one of the original equations to solve for [tex]\( y \)[/tex].
Using Equation (2):
[tex]\[
3x + 3y = 3
\][/tex]
Substitute [tex]\( x = -1 \)[/tex]:
[tex]\[
3(-1) + 3y = 3
\][/tex]
[tex]\[
-3 + 3y = 3
\][/tex]
[tex]\[
3y = 6
\][/tex]
[tex]\[
y = \frac{6}{3} = 2
\][/tex]
So, the solution to the system of equations is:
[tex]\[
(x, y) = (-1, 2)
\][/tex]
Therefore, the final answer is:
[tex]\[
(-1, 2)
\][/tex]
