Step by step
The question
Question: solve x − 3y = 1 and 2x + y = 9 by substitution.
We call them ① and ②.
We want the x and y that make both true.
1 Get y alone in ②
No letter is alone yet.
In ②, 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 ①
In ③, y is now alone.
y is the same as −2x + 9.
So in ①, we write (−2x + 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.
x + 6x = 7x.
Now we have 7x − 27 = 1.
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.
1 plus 27 is 28.
Now we have 7x = 28.
7x means 7 times x.
We want just x.
To remove times 7, we divide by 7.
We do it on both sides, so they stay equal.
7 divided by 7 is 1.
So just x is left.
28 divided by 7 is 4.
So x = 4.
4 Find y
Now we put x = 4 into ③.
That is −8 + 9.
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 1, the same as the right side.
Now ②.
We get 9, the same as the right side.
Both work.
So the answer is right!
Let's look back.
First: get y alone in ②.
Then: put ③ into ①.
Then: find x.
Then: find y.
Last: check.