To solve the system of equations using the elimination method, we start with the given equations:
- \( x + 2y = 13 \) (Equation 1)
- \( -x + y = 5 \) (Equation 2)
Our goal is to eliminate one of the variables by adding or subtracting the equations.
Let's first manipulate Equation 2 to have its \( x \) term in positive form, so we multiply it by -1:
\[ x - y = -5 \quad \text{(Equation 2 multiplied by -1)} \]
Now we have:
- \( x + 2y = 13 \)
- \( x - y = -5 \)
Next, we can align these two equations:
\[ \begin{align*} 1: & \quad x + 2y = 13 \ 2: & \quad x - y = -5 \ \end{align*} \]
Now, we can eliminate \( x \) by subtracting Equation 2 from Equation 1:
\[ (x + 2y) - (x - y) = 13 - (-5) \]
This simplifies to:
\[ x + 2y - x + y = 13 + 5 \]
\[ 3y = 18 \]
Now, divide both sides by 3:
\[ y = 6 \]
Now that we have \( y \), we can substitute it back into one of the original equations to find \( x \). We'll use Equation 1:
\[ x + 2(6) = 13 \]
This simplifies to:
\[ x + 12 = 13 \]
Subtracting 12 from both sides gives:
\[ x = 1 \]
Thus, the solution to the system of equations is:
\[ x = 1, \quad y = 6 \]
So, the final solution is \( (1, 6) \).