Step by step
The question
Question: solve 2x + 3y = 8 and 3x − 2y = −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 3y, in ② it is −2y.
We make them opposite: 6y and −6y.
Then adding removes y.
We multiply every part of ① by 2.
We do it on both sides, so they stay equal.
2 times 2x is 4x.
2 times 3y is 6y.
2 times 8 is 16.
We call it ③.
We multiply every part of ② by 3.
We do it on both sides, so they stay equal.
3 times 3x is 9x.
3 times −2y is −6y.
3 times −1 is −3.
We call it ④.
2 Add the equations
We write ③ and ④ under each other: x under x, y under y.
Look at y: 6y and −6y.
They are opposites.
Added, they make zero.
Why can we add equations?
Both sides of ④ are the same amount.
So we add the same amount to both sides of ③.
6y and −6y make zero.
Cross them out.
We add ③ and ④: the x parts together, and the numbers together.
4x + 9x = 13x.
16 + (−3) = 13.
Now we have 13x = 13.
3 Find x
13x means 13 times x.
We want just x.
To remove times 13, we divide by 13.
We do it on both sides, so they stay equal.
13 divided by 13 is 1.
So just x is left.
On the right too: 13 divided by 13 is 1.
So x = 1.
4 Find y
We put x = 1 into ①.
2 times 1 is 2.
Now we have 2 + 3y = 8.
We want y by itself.
The 2 is in the way.
To remove 2, we take away 2.
We do it on both sides, so they stay equal.
2 and − 2 make zero.
Cross them out.
8 minus 2 is 6.
Now we have 3y = 6.
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.
6 divided by 3 is 2.
So y = 2.
So x = 1 and y = 2, both at the same time.
We write it as (1, 2).
On a graph, it is the point where the two lines meet.
5 Check
Let's check in both equations.
We put x = 1 and y = 2 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: make the y numbers opposite.
Then: add the equations.
Then: find x.
Then: find y.
Last: check.