How to solve system of linear equations by elimination

Say you have the system:

2x + 7y = 4

3x + 5y = - 5

To solve this system using elimination, you want to cancel out either the x terms or the y terms. In this equation, it makes most sense to get rid of the x terms because they can easily be calculated as opposites. So, what we need to do is multiply each term in the first equation by 3 and multiply each term in the second equation by - 2:

6x + 21y = 12 (2 * 3 = 6; 7 * 3 = 21; 4 * 3 = 12)

-6x - 10y = 10 (3 * - 2 = - 6; 5 * - 2 = - 10; - 5 * - 2 = 10)

With that, the x terms automatically cancel out and we're left with:

21y = 12

-10y = 10

From here, we can add both equations together and get:

11y = 22

y = 2 (divide both sides by 11)

After we have one variable, we can plug it right back into either of the first two original equations; ours were 2x + 7y = 4 and 3x + 5y = - 5.

So we'll take the first one and put 2 in the place of y to solve for x:

2x + 7 (2) = 4

2x + 14 = 4 (multiply 7 and 2)

2x = - 10 (subtract 14 from both sides of the equation)

x = - 5 (divide both sides by 2)

We have that y = 2 and x = - 5; there is only one more step - check your work!

Plug both values back into both original equations to check your work:

2 (-5) + 7 (2) = 4 = > - 10 + 14 = 4 Correct!

3 (-5) + 5 (2) = - 5 = > - 15 + 10 = - 5 Correct!

I hope this was comprehensive enough. Let me know if you have any more questions.