Step by step
The question
Question: solve x − y = 2 and 3x + 2y = 21 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 − y is in the way.
To remove − y, we add y.
We do it on both sides, so they stay equal.
− y and + y 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 y + 2.
So in ②, we write (y + 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.
3y + 2y = 5y.
Now we have 5y + 6 = 21.
We want y by itself.
The + 6 is in the way.
To remove + 6, we take away 6.
We do it on both sides, so they stay equal.
+ 6 and − 6 make zero.
Cross them out.
21 minus 6 is 15.
Now we have 5y = 15.
5y means 5 times y.
We want just y.
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 y is left.
15 divided by 5 is 3.
So y = 3.
4 Find x
Now we put y = 3 into ③.
So x = 5.
So x = 5 and y = 3, both at the same time.
We write it as (5, 3).
On a graph, it is the point where the two lines meet.
5 Check
Let's check in both equations.
We put x = 5 and y = 3 into ①.
We get 2, the same as the right side.
Now ②.
We get 21, 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.