We are given a system of two linear equations with two variables, $x$ and $y$. We are asked to solve this system using the elimination method. The system of equations is: $x + 2y = 6$ $x - 2y = 2$

1. Problem Description

We are given a system of two linear equations with two variables, $x$ and $y$. We are asked to solve this system using the elimination method. The system of equations is:

$x + 2y = 6$

$x - 2y = 2$

2. Solution Steps

We will use the elimination method to solve this system of equations.

Notice that the $y$ terms in the two equations have opposite signs. We can add the two equations to eliminate the $y$ variable.

Add the two equations:

$(x + 2y) + (x - 2y) = 6 + 2$

$x + x + 2y - 2y = 8$

$2x = 8$

Divide both sides by 2 to solve for $x$:

$\frac{2x}{2} = \frac{8}{2}$

$x = 4$

Now that we have found the value of $x$, we can substitute it into either of the original equations to solve for $y$. Let's use the first equation:

$x + 2y = 6$

Substitute $x = 4$:

$4 + 2y = 6$

Subtract 4 from both sides:

$2y = 6 - 4$

$2y = 2$

Divide both sides by 2 to solve for $y$:

$\frac{2y}{2} = \frac{2}{2}$

$y = 1$

So the solution to the system of equations is $x = 4$ and $y = 1$.

3. Final Answer

The solution to the system of equations is $(4, 1)$.