To solve the system of linear equations given by:
[tex]\[
\begin{cases}
2x + y = -5 \\
2x - 5y = 13
\end{cases}
\][/tex]
we will use the method of elimination to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
1. Step 1: Align the two equations:
[tex]\[
\begin{cases}
2x + y = -5 \quad \text{(Equation 1)} \\
2x - 5y = 13 \quad \text{(Equation 2)}
\end{cases}
\][/tex]
2. Step 2: Eliminate one variable:
Since both equations have the term [tex]\(2x\)[/tex], we can eliminate [tex]\(x\)[/tex] by subtracting Equation 1 from Equation 2.
Subtract Equation 1 from Equation 2:
[tex]\[
(2x - 5y) - (2x + y) = 13 - (-5)
\][/tex]
Simplifying the left-hand side and the right-hand side:
[tex]\[
2x - 5y - 2x - y = 13 + 5
\][/tex]
[tex]\[
-6y = 18
\][/tex]
3. Step 3: Solve for [tex]\(y\)[/tex]:
Divide both sides by [tex]\(-6\)[/tex]:
[tex]\[
y = \frac{18}{-6}
\][/tex]
[tex]\[
y = -3
\][/tex]
4. Step 4: Substitute [tex]\(y\)[/tex] back into one of the original equations to find [tex]\(x\)[/tex]:
We can use Equation 1:
[tex]\[
2x + y = -5
\][/tex]
Substitute [tex]\(y = -3\)[/tex]:
[tex]\[
2x + (-3) = -5
\][/tex]
Simplify:
[tex]\[
2x - 3 = -5
\][/tex]
Add 3 to both sides:
[tex]\[
2x = -5 + 3
\][/tex]
[tex]\[
2x = -2
\][/tex]
5. Step 5: Solve for [tex]\(x\)[/tex]:
Divide both sides by 2:
[tex]\[
x = \frac{-2}{2}
\][/tex]
[tex]\[
x = -1
\][/tex]
Therefore, the solution to the system of equations is:
[tex]\[
x = -1 \quad \text{and} \quad y = -3
\][/tex]
So, the ordered pair [tex]\((x, y)\)[/tex] is [tex]\((-1, -3)\)[/tex].
