Step by step
The question
Question: solve 2x + y = 1 and x + y = −1.
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 y numbers opposite
We want to remove y.
In ① it is y, in ② it is y.
We make them opposite: y and −y.
Then adding removes y.
We multiply every part of ② by −1.
We do it on both sides, so they stay equal.
−1 times x is −x.
−1 times y is −y.
−1 times −1 is 1.
We call it ③.
2 Add the equations
We write ① and ③ under each other: x under x, y under y.
Look at y: y and −y.
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.
y and −y make zero.
Cross them out.
We add ① and ③: the x parts together, and the numbers together.
2x − x = x.
1 + 1 = 2.
Now we have x = 2.
3 Find x
So x = 2.
4 Find y
We put x = 2 into ①.
2 times 2 is 4.
Now we have 4 + y = 1.
We want y by itself.
The 4 is in the way.
To remove 4, we take away 4.
We do it on both sides, so they stay equal.
4 and − 4 make zero.
Cross them out.
1 minus 4 is −3.
So y = −3.
So x = 2 and y = −3, both at the same time.
We write it as (2, −3).
On a graph, it is the point where the two lines meet.
5 Check
Let's check in both equations.
We put x = 2 and y = −3 into ①.
We get 1, the same as the right side.
Now ②.
We get −1, the same as the right side.
Both work.
So the answer is right!
Let's look back.
First: make the y numbers opposite.
Then: add the equations.
Then: find x.
Then: find y.
Last: check.