Step by step
The question
Question: solve x − 2y = −2 and 3x + 2y = 34 by substitution.
We call them ① and ②.
We want the x and y that make both true.
1 Get x alone in ①
No letter is alone yet.
In ①, x has just 1 in front.
So we get x alone there first.
We want x by itself.
The − 2y is in the way.
To remove − 2y, we add 2y.
We do it on both sides, so they stay equal.
− 2y and + 2y make zero.
Cross them out.
Write what is left.
We write the y part first.
Each part keeps its sign.
x is by itself.
We call it ③.
2 Put ③ into ②
In ③, x is now alone.
x is the same as 2y − 2.
So in ②, we write (2y − 2) in place of x.
The brackets keep it together.
Putting one thing in place of another is called substitution.
3 Find y
The 3 in front of the bracket multiplies every part inside.
So we write 3 times each part.
Careful!
Not only the first part!
−2 is multiplied by 3 too.
Work out each multiplication.
Terms are the parts between the + and − signs.
Like terms are the same kind: same letters, same powers.
All the y terms go together.
6y + 2y = 8y.
Now we have 8y − 6 = 34.
We want y by itself.
The − 6 is in the way.
To remove − 6, we add 6.
We do it on both sides, so they stay equal.
− 6 and + 6 make zero.
Cross them out.
34 plus 6 is 40.
Now we have 8y = 40.
8y means 8 times y.
We want just y.
To remove times 8, we divide by 8.
We do it on both sides, so they stay equal.
8 divided by 8 is 1.
So just y is left.
40 divided by 8 is 5.
So y = 5.
4 Find x
Now we put y = 5 into ③.
That is 10 − 2.
So x = 8.
So x = 8 and y = 5, both at the same time.
We write it as (8, 5).
On a graph, it is the point where the two lines meet.
5 Check
Let's check in both equations.
We put x = 8 and y = 5 into ①.
We get −2, the same as the right side.
Now ②.
We get 34, the same as the right side.
Both work.
So the answer is right!
Let's look back.
First: get x alone in ①.
Then: put ③ into ②.
Then: find y.
Then: find x.
Last: check.