Step by step
The question
Question: solve 5x + y = 9 and 2x − 3y = 7 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 5x is in the way.
To remove 5x, we take away 5x.
We do it on both sides, so they stay equal.
5x and − 5x 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 −5x + 9.
So in ②, we write (−5x + 9) in place of y.
The brackets keep it together.
Putting one thing in place of another is called substitution.
3 Find x
The −3 in front of the bracket multiplies every part inside.
So we write −3 times each part.
Careful!
Not only the first part!
9 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 x terms go together.
2x + 15x = 17x.
Now we have 17x − 27 = 7.
We want x by itself.
The − 27 is in the way.
To remove − 27, we add 27.
We do it on both sides, so they stay equal.
− 27 and + 27 make zero.
Cross them out.
7 plus 27 is 34.
Now we have 17x = 34.
17x means 17 times x.
We want just 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.
So just x is left.
34 divided by 17 is 2.
So x = 2.
4 Find y
Now we put x = 2 into ③.
That is −10 + 9.
So y = −1.
So x = 2 and y = −1, both at the same time.
We write it as (2, −1).
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 = −1 into ①.
We get 9, the same as the right side.
Now ②.
We get 7, 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.