Step by step
The question
Question: solve 2x − y = 3 and x + 3y = 12 by substitution.
We call them ① and ②.
We want the x and y that make both true.
1 Get x alone in ②
No letter is alone yet.
In ②, x has just 1 in front.
So we get x alone there first.
We want x by itself.
The + 3y is in the way.
To remove + 3y, we take away 3y.
We do it on both sides, so they stay equal.
+ 3y and − 3y 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 ①
In ③, x is now alone.
x is the same as −3y + 12.
So in ①, we write (−3y + 12) in place of x.
The brackets keep it together.
Putting one thing in place of another is called substitution.
3 Find y
The 2 in front of the bracket multiplies every part inside.
So we write 2 times each part.
Careful!
Not only the first part!
12 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 y terms go together.
−6y − y = −7y.
Now we have −7y + 24 = 3.
We want y by itself.
The + 24 is in the way.
To remove + 24, we take away 24.
We do it on both sides, so they stay equal.
+ 24 and − 24 make zero.
Cross them out.
3 minus 24 is −21.
Now we have −7y = −21.
−7y means −7 times y.
We want just y.
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 y is left.
−21 divided by −7 is 3.
So y = 3.
4 Find x
Now we put y = 3 into ③.
That is −9 + 12.
So x = 3.
So x = 3 and y = 3, both at the same time.
We write it as (3, 3).
On a graph, it is the point where the two lines meet.
5 Check
Let's check in both equations.
We put x = 3 and y = 3 into ①.
We get 3, the same as the right side.
Now ②.
We get 12, the same as the right side.
Both work.
So the answer is right!
Let's look back.
First: get x alone in ②.
Then: put ③ into ①.
Then: find y.
Then: find x.
Last: check.