Sure, let's solve the given system of equations using the addition method, also known as the elimination method.
The system of equations is:
[tex]\[
\begin{cases}
4x - 5y = 13 \\
x + 5y = -3
\end{cases}
\][/tex]
Step 1: Add the two equations.
First, let's align them:
[tex]\[
\begin{aligned}
1. & \quad 4x - 5y = 13 \\
2. & \quad x + 5y = -3
\end{aligned}
\][/tex]
Now add these two equations directly:
[tex]\[
4x - 5y + x + 5y = 13 + (-3)
\][/tex]
Simplify by combining like terms:
[tex]\[
5x = 10
\][/tex]
Step 2: Solve for [tex]\( x \)[/tex].
Divide both sides of the equation by 5:
[tex]\[
x = \frac{10}{5}
\][/tex]
So, we have:
[tex]\[
x = 2
\][/tex]
Step 3: Substitute [tex]\( x \)[/tex] back into one of the original equations to solve for [tex]\( y \)[/tex].
Let's substitute [tex]\( x = 2 \)[/tex] into the second equation:
[tex]\[
x + 5y = -3
\][/tex]
Substitute [tex]\( x = 2 \)[/tex]:
[tex]\[
2 + 5y = -3
\][/tex]
Step 4: Solve for [tex]\( y \)[/tex].
Isolate [tex]\( y \)[/tex] by subtracting 2 from both sides:
[tex]\[
5y = -3 - 2
\][/tex]
Simplify:
[tex]\[
5y = -5
\][/tex]
Divide both sides by 5:
[tex]\[
y = \frac{-5}{5}
\][/tex]
So, we have:
[tex]\[
y = -1
\][/tex]
Step 5: Write the solution as an ordered pair.
The solution to the system of equations is:
[tex]\[
(x, y) = (2, -1)
\][/tex]
Thus, [tex]\( x = 2 \)[/tex] and [tex]\( y = -1 \)[/tex] are the values that satisfy both equations in the system.
