To solve the system of equations using elimination, follow these steps:
Given equations:
[tex]\[
\begin{aligned}
(1) & \quad -x + y = 1 \\
(2) & \quad -6x + 2y = -34
\end{aligned}
\][/tex]
1. Multiply the first equation by 2 to align the coefficients of [tex]\( y \)[/tex] in both equations:
[tex]\[
2(-x + y) = 2 \cdot 1 \implies -2x + 2y = 2
\][/tex]
Now we have the system:
[tex]\[
\begin{aligned}
(1') & \quad -2x + 2y = 2 \\
(2) & \quad -6x + 2y = -34
\end{aligned}
\][/tex]
2. Subtract equation [tex]\((1')\)[/tex] from equation [tex]\((2)\)[/tex]:
[tex]\[
(-6x + 2y) - (-2x + 2y) = -34 - 2
\][/tex]
[tex]\[
-6x + 2y + 2x - 2y = -34 - 2
\][/tex]
[tex]\[
-4x = -36
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{-36}{-4} = 9
\][/tex]
4. Substitute [tex]\( x = 9 \)[/tex] back into the first equation to find [tex]\( y \)[/tex]:
[tex]\[
-x + y = 1 \implies -9 + y = 1 \implies y = 1 + 9 \implies y = 10
\][/tex]
Thus, the solution to the system of equations is:
[tex]\[
(x, y) = (9, 10)
\][/tex]
