Solve these equations with the elimination method.

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

2 answers

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 \( (1, 6) \).

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