Step by step
The question
Question: solve 2x + 2y = 10 and x + y = 5.
We call them ① and ②.
We want the x and y that make both true.
We will add the equations to remove one letter.
This way is called elimination.
1 Make the x numbers opposite
We want to remove x.
In ① it is 2x, in ② it is x.
We make them opposite: 2x and −2x.
Then adding removes x.
We multiply every part of ② by −2.
We do it on both sides, so they stay equal.
−2 times x is −2x.
−2 times y is −2y.
−2 times 5 is −10.
We call it ③.
2 Add the equations
We write ① and ③ under each other: x under x, y under y.
Look at x: 2x and −2x.
They are opposites.
Added, they make zero.
Why can we add equations?
Both sides of ③ are the same amount.
So we add the same amount to both sides of ①.
2x and −2x make zero.
Cross them out.
We add ① and ③: the y parts together, and the numbers together.
2y − 2y = 0.
10 + (−10) = 0.
We get 0 = 0.
Both letters are gone, and 0 = 0 is always true.
The two equations are the same line.
So every point on the line works.
Let's look back.
First: make the x numbers opposite.
Last: add the equations.