Solving linear systems using the elimination method is also known as using the addition/subtraction method. The goal is to "eliminate" one of the variables in order to solve the system of equations.

`1/2x+2y=27` 

`x+1/3y =10` 

Simplifying we get:

`x/2 +2y=27` 

`x+y/3=10` 

Now let's look at the first equation. Multiply each term in the equation by 2.

`2 (x/2 +2y) = 2(27)`

`x+4y = 54`

Now let's go back to the second equation. Multiply each term in the equation by 3.

`3(x+ y/3) =3(10)`

`3x+y =30`

Now the two equations will be easier to work with.

`x+4y=54`

`3x+y=30`

Let's eliminate the x variable to solve for y first. To do this multiply the first equation by -3.

`-3(x+4y=54)` >>> `-3x-12y=-162`

`3x+y=30`

Now we have:

`-3x-12y=-162`

`3x +y =30`

The 3x and -3x cancel out. Combine -12y with 1y which is -11y. Finally -162 combined with 30 is -132.

`-11y = -132`

Divide both sides  by -11 to get y alone.

`y=12`

Almost finished. Since you now know what y is, simply plug this value into one of the original equations to solve for x.

`1/2 x +2(12) =27`

`x/2 +24 =27`

`x/2 =3`

`2 (x/2) = 2 (3)`

`x=6`

The solution to the system of equations is

x=6, y=12