To solve the given system of equations using substitution, we'll proceed step-by-step.
The system of equations is:
[tex]\[
\begin{cases}
5x + y = -2 \\
x + 2y = 5
\end{cases}
\][/tex]
Step 1: Solve one of the equations for one variable in terms of the other.
Let's solve the second equation [tex]\( x + 2y = 5 \)[/tex] for [tex]\( x \)[/tex]:
[tex]\[
x + 2y = 5 \implies x = 5 - 2y
\][/tex]
Step 2: Substitute this expression into the other equation.
We'll substitute [tex]\( x = 5 - 2y \)[/tex] into the first equation [tex]\( 5x + y = -2 \)[/tex]:
[tex]\[
5(5 - 2y) + y = -2
\][/tex]
Simplify the equation:
[tex]\[
25 - 10y + y = -2 \implies 25 - 9y = -2
\][/tex]
Step 3: Solve for [tex]\( y \)[/tex].
Move the constant term to the other side of the equation:
[tex]\[
25 - 9y = -2 \implies -9y = -2 - 25 \implies -9y = -27
\][/tex]
Divide both sides by [tex]\(-9\)[/tex]:
[tex]\[
y = \frac{-27}{-9} = 3
\][/tex]
Step 4: Substitute [tex]\( y \)[/tex] back into the expression we found for [tex]\( x \)[/tex].
Using [tex]\( x = 5 - 2y \)[/tex] and substituting [tex]\( y = 3 \)[/tex]:
[tex]\[
x = 5 - 2(3) \implies x = 5 - 6 \implies x = -1
\][/tex]
Conclusion:
The solution to the system of equations, written as an ordered pair [tex]\((x, y)\)[/tex], is:
[tex]\[
(x, y) = (-1, 3)
\][/tex]
Answer: [tex]\(\boxed{(-1, 3)}\)[/tex]
