Step by step
The question
Question: solve 2x + y = 7 and x − 2y = 6 by substitution.
We call them ① and ②.
We want the x and y that make both true.
1 Get y alone in ①
No letter is alone yet.
In ①, y has just 1 in front.
So we get y alone there first.
We want y by itself.
The 2x is in the way.
To remove 2x, we take away 2x.
We do it on both sides, so they stay equal.
2x and − 2x make zero.
Cross them out.
Write what is left.
We write the x part first.
Each part keeps its sign.
y is by itself.
We call it ③.
2 Put ③ into ②
In ③, y is now alone.
y is the same as −2x + 7.
So in ②, we write (−2x + 7) in place of y.
The brackets keep it together.
Putting one thing in place of another is called substitution.
3 Find x
The −2 in front of the bracket multiplies every part inside.
So we write −2 times each part.
Careful!
Not only the first part!
7 is multiplied by −2 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 x terms go together.
x + 4x = 5x.
Now we have 5x − 14 = 6.
We want x by itself.
The − 14 is in the way.
To remove − 14, we add 14.
We do it on both sides, so they stay equal.
− 14 and + 14 make zero.
Cross them out.
6 plus 14 is 20.
Now we have 5x = 20.
5x means 5 times x.
We want just x.
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 x is left.
20 divided by 5 is 4.
So x = 4.
4 Find y
Now we put x = 4 into ③.
That is −8 + 7.
So y = −1.
So x = 4 and y = −1, both at the same time.
We write it as (4, −1).
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 = −1 into ①.
We get 7, the same as the right side.
Now ②.
We get 6, the same as the right side.
Both work.
So the answer is right!
Let's look back.
First: get y alone in ①.
Then: put ③ into ②.
Then: find x.
Then: find y.
Last: check.