Question

Solve these equations with the elimination method.

X+2y=13
-x+y=5

Answer 1

9x+y=2
-4x-y=-17

Answer 2

To solve the system of equations using the elimination method, we start with the given equations:

1. \( x + 2y = 13 \) (Equation 1)
2. \( -x + y = 5 \) (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:

1. \( x + 2y = 13 \)
2. \( x - y = -5 \)

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 \( (1, 6) \).