To solve the system of equations using the elimination method, follow these steps:
Given the system:
[tex]\[ 5x + 2y = 3 \][/tex]
[tex]\[ 4x - 8y = 12 \][/tex]
1. Align the equations:
[tex]\[
\begin{array}{l}
5x + 2y = 3 \\
4x - 8y = 12
\end{array}
\][/tex]
2. Multiply the first equation by a factor that will allow the coefficients of [tex]\( y \)[/tex] in both equations to become additive inverses. In this case, we can multiply the first equation by 4 so that the coefficient of [tex]\( y \)[/tex] in the second equation (-8) is the opposite of [tex]\( 8y \)[/tex] in the modified first equation.
[tex]\[
4(5x + 2y) = 4(3)
\][/tex]
[tex]\[
20x + 8y = 12
\][/tex]
Now our system of equations is:
[tex]\[
\begin{array}{l}
20x + 8y = 12 \\
4x - 8y = 12
\end{array}
\][/tex]
3. Add the two equations together to eliminate [tex]\( y \)[/tex]:
[tex]\[
\begin{array}{l}
(20x + 8y) + (4x - 8y) = 12 + 12 \\
20x + 8y + 4x - 8y = 24 \\
24x = 24
\end{array}
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[
24x = 24
\][/tex]
[tex]\[
x = \frac{24}{24}
\][/tex]
[tex]\[
x = 1
\][/tex]
5. Substitute [tex]\( x = 1 \)[/tex] back into one of the original equations to solve for [tex]\( y \)[/tex]:
Using the first equation:
[tex]\[
5x + 2y = 3
\][/tex]
Substitute [tex]\( x = 1 \)[/tex]:
[tex]\[
5(1) + 2y = 3
\][/tex]
[tex]\[
5 + 2y = 3
\][/tex]
[tex]\[
2y = 3 - 5
\][/tex]
[tex]\[
2y = -2
\][/tex]
[tex]\[
y = \frac{-2}{2}
\][/tex]
[tex]\[
y = -1
\][/tex]
Therefore, the solution to the system of equations is [tex]\( (x, y) = (1, -1) \)[/tex].
So the correct answer is [tex]\( (1, -1) \)[/tex].
