Step by step
The question
Question: solve x + 3y = 13 and 2x − 3y = 8.
There are four choices.
First we solve it, the normal way.
Then we pick.
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 Add the equations
Look at y: 3y and −3y.
They are opposites.
Added, they make zero.
3y and −3y make zero.
Cross them out.
We add ① and ②: the x parts together, and the numbers together.
x + 2x = 3x.
13 + 8 = 21.
Now we have 3x = 21.
Is x alone now?
Not yet: 3x means 3 times x.
How do we remove it?
2 Find 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.
They cancel each other.
So just x is left.
21 divided by 3 is 7.
So x = 7.
So after step 2: we divided by 3, and now we have x = 7.
3 Find y
We put x = 7 into ①.
We want y by itself.
The 7 is in the way.
Is y alone now?
Not yet: + 7 is still with y.
To remove 7, we take away 7.
We do it on both sides, so they stay equal.
7 and − 7 cancel each other.
They make zero.
13 minus 7 is 6.
Now we have 3y = 6.
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.
6 divided by 3 is 2.
So y = 2.
So x = 7 and y = 2, both at the same time.
We write it as (7, 2).
On a graph, it is the point where the two lines meet.
4 Check
Let's check in both equations.
We put x = 7 and y = 2 into ①.
We get 13, the same as the right side.
Now ②.
We get 8, the same as the right side.
Both work.
So the answer is right!
5 Pick the choice
Our answer is (x, y) = (7, 2).
That is choice B.
6 Why not the others
A is (2, 7).
Wrong: x and y swapped.
C is (7, 6).
Wrong: 3y = 6, but we still had to divide by 3.
D is (7, 20/3).
Wrong: added 7 instead of taking it away: 3y = 20.
Let's look back.
First: add the equations.
Then: find x.
Then: find y.
Then: check.
Then: pick the choice.
Last: why not the others.