Step by step
The question
Question: solve 4x + 3y = 17 and 3x − 2y = 0.
We call them ① and ②.
We want the x and y that make both true.
No letter is alone here.
So we add the equations to remove one letter.
This way is called elimination.
1 Make the y parts opposite
We want to remove one letter.
In ① it is 3y, in ② it is −2y.
Both letters take the same work, so we can pick either.
We pick y.
We need a number that 3 and 2 both go into.
An easy way: multiply them.
3 times 2 is 6.
We make them opposite: 6y and −6y.
Opposites are the same size, one plus and one minus.
Then adding removes y.
3y times 2 is 6y.
−2y times 3 is −6y.
We multiply every part of ① by 2.
We do it on both sides, so they stay equal.
2 times 4x is 8x.
2 times 3y is 6y.
2 times 17 is 34.
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 0 is 0.
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.
6y and −6y make zero.
Cross them out.
We add ③ and ④: the x parts together, and the numbers together.
8x + 9x = 17x.
34 + 0 = 34.
Now we have 17x = 34.
Is x alone now?
Not yet: 17x means 17 times x.
How do we remove it?
3 Find x
To remove times 17, we divide by 17.
We do it on both sides, so they stay equal.
17 divided by 17 is 1.
They cancel each other.
So just x is left.
34 divided by 17 is 2.
So x = 2.
So after step 3: we divided by 17, and now we have x = 2.
4 Find y
We put x = 2 into ①.
4 times 2 is 8.
Now we have 8 + 3y = 17.
We want y by itself.
The 8 is in the way.
Is y alone now?
Not yet: + 8 is still with y.
To remove 8, we take away 8.
We do it on both sides, so they stay equal.
8 and − 8 cancel each other.
They make zero.
17 minus 8 is 9.
Now we have 3y = 9.
3y means 3 times y.
Is y alone now?
Not yet: 3y means 3 times 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.
They cancel each other.
So just y is left.
9 divided by 3 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 17, the same as the right side.
Now ②.
We get 0, the same as the right side.
Both work.
So the answer is right!
Let's look back.
First: make the y parts opposite.
Then: add the equations.
Then: find x.
Then: find y.
Last: check.