Step by step
The question
Question: solve 3x + 2y = 16 and 2x + 3y = 14.
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 3x, in ② it is 2x.
We make them opposite: 6x and −6x.
Then adding removes x.
We multiply every part of ① by 2.
We do it on both sides, so they stay equal.
2 times 3x is 6x.
2 times 2y is 4y.
2 times 16 is 32.
We call it ③.
We multiply every part of ② by −3.
We do it on both sides, so they stay equal.
−3 times 2x is −6x.
−3 times 3y is −9y.
−3 times 14 is −42.
We call it ④.
2 Add the equations
We write ③ and ④ under each other: x under x, y under y.
Look at x: 6x and −6x.
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.
6x and −6x make zero.
Cross them out.
We add ③ and ④: the y parts together, and the numbers together.
4y − 9y = −5y.
32 + (−42) = −10.
Now we have −5y = −10.
3 Find y
−5y means −5 times y.
We want just y.
To remove times −5, we divide by −5.
We do it on both sides, so they stay equal.
−5 divided by −5 is 1.
So just y is left.
−10 divided by −5 is 2.
So y = 2.
4 Find x
We put y = 2 into ①.
2 times 2 is 4.
Now we have 3x + 4 = 16.
We want x 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.
16 minus 4 is 12.
Now we have 3x = 12.
3x means 3 times x.
We want just x.
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 x is left.
12 divided by 3 is 4.
So x = 4.
So x = 4 and y = 2, both at the same time.
We write it as (4, 2).
On a graph, it is the point where the two lines meet.
5 Check
Let's check in both equations.
We put x = 4 and y = 2 into ①.
We get 16, the same as the right side.
Now ②.
We get 14, the same as the right side.
Both work.
So the answer is right!
Let's look back.
First: make the x numbers opposite.
Then: add the equations.
Then: find y.
Then: find x.
Last: check.