Step by step
The question
Question: solve x + 2y = 8 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 Add the equations
Look at x: x and −x.
They are opposites.
Added, they make zero.
Why can we add equations?
Both sides of ② are the same amount.
So we add the same amount to both sides of ①.
x and −x make zero.
Cross them out.
We add ① and ②: the y parts together, and the numbers together.
2y + y = 3y.
8 + 1 = 9.
Now we have 3y = 9.
2 Find y
3y means 3 times y.
We want just y.
To remove times 3, we divide by 3.
We do it on both sides, so they stay equal.
3 divided by 3 is 1.
So just y is left.
9 divided by 3 is 3.
So y = 3.
3 Find x
We put y = 3 into ①.
2 times 3 is 6.
Now we have x + 6 = 8.
We want x by itself.
The + 6 is in the way.
To remove + 6, we take away 6.
We do it on both sides, so they stay equal.
+ 6 and − 6 make zero.
Cross them out.
8 minus 6 is 2.
So x = 2.
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.
4 Check
Let's check in both equations.
We put x = 2 and y = 3 into ①.
We get 8, 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: add the equations.
Then: find y.
Then: find x.
Last: check.