Systems by elimination, step by step

Elimination adds or subtracts two equations so that one letter disappears.

The question

−6x + y = 2312x − y = −47

−6x + y = 23 12x − y = −47

−6x + y = 23

12x − y = −47

−6x + y = 23

12x − y = −47

−6x + y = 23

12x − y = −47

6x = −24

x=−4

y=−1

Two equations, two letters. Neither one can be solved on its own — so the first move, whichever method you use, is to get down to ONE equation with ONE letter in it.
Look at the y terms: y in the first and −y in the second. They are opposites, so adding the two equations makes them cancel.
Adding is allowed because both sides of each equation are equal — adding left to left and right to right keeps the balance, the same as adding a number to both sides. That gives 6x = −24.
One letter left. −24 ÷ 6 = −4, so x = −4.
Now put x back into either equation to get y — this is the half that gets forgotten. It gives y = −1, so the solution is (−4, −1).

How it works.

01
Two equations, two letters. Neither one can be solved on its own — so the first move, whichever method you use, is to get down to ONE equation with ONE letter in it.
02
Look at the y terms: y in the first and −y in the second. They are opposites, so adding the two equations makes them cancel.
03
Adding is allowed because both sides of each equation are equal — adding left to left and right to right keeps the balance, the same as adding a number to both sides. That gives 6x = −24.
04
One letter left. −24 ÷ 6 = −4, so x = −4.
05
Now put x back into either equation to get y — this is the half that gets forgotten. It gives y = −1, so the solution is (−4, −1).