To solve the system of equations using the elimination method, follow these detailed steps:
The given system of equations is:
[tex]\[
\begin{cases}
3x + 5y = -2 \\
-x + 2y = 8
\end{cases}
\][/tex]
1. Eliminate one variable:
To eliminate [tex]\(x\)[/tex], we want to make the coefficients of [tex]\(x\)[/tex] in both equations equal in magnitude but opposite in sign so that they cancel out when we add the equations. To do this, we can multiply the second equation by [tex]\(3\)[/tex]:
[tex]\[
3(-x + 2y) = 3(8)
\][/tex]
Which simplifies to:
[tex]\[
-3x + 6y = 24
\][/tex]
2. Add the two equations:
Now add the new equation [tex]\(-3x + 6y = 24\)[/tex] to the first original equation [tex]\(3x + 5y = -2\)[/tex]:
[tex]\[
(3x + 5y) + (-3x + 6y) = -2 + 24
\][/tex]
Simplifying this, we get:
[tex]\[
3x - 3x + 5y + 6y = -2 + 24
\][/tex]
[tex]\[
0x + 11y = 22
\][/tex]
[tex]\[
11y = 22
\][/tex]
3. Solve for [tex]\(y\)[/tex]:
Divide both sides of the equation [tex]\(11y = 22\)[/tex] by [tex]\(11\)[/tex] to isolate [tex]\(y\)[/tex]:
[tex]\[
y = \frac{22}{11}
\][/tex]
[tex]\[
y = 2
\][/tex]
4. Substitute [tex]\(y\)[/tex] back into one of the original equations to find [tex]\(x\)[/tex]:
Now substitute [tex]\(y = 2\)[/tex] back into the original first equation [tex]\(3x + 5y = -2\)[/tex]:
[tex]\[
3x + 5(2) = -2
\][/tex]
Simplifying this, we get:
[tex]\[
3x + 10 = -2
\][/tex]
[tex]\[
3x = -2 - 10
\][/tex]
[tex]\[
3x = -12
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
Divide both sides of the equation [tex]\(3x = -12\)[/tex] by [tex]\(3\)[/tex] to isolate [tex]\(x\)[/tex]:
[tex]\[
x = \frac{-12}{3}
\][/tex]
[tex]\[
x = -4
\][/tex]
Thus, the solution to the system of equations is:
[tex]\[
(x, y) = (-4, 2)
\][/tex]
