Step by step
The question
Question: solve 2x + y = 5 and 4x − 3y = 0 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 + 5.
So in ②, we write (−2x + 5) 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!
5 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.
4x + 6x = 10x.
Now we have 10x − 15 = 0.
We want x by itself.
The − 15 is in the way.
To remove − 15, we add 15.
We do it on both sides, so they stay equal.
− 15 and + 15 make zero.
Cross them out.
0 plus 15 is 15.
Now we have 10x = 15.
10x means 10 times x.
We want just x.
To remove times 10, we divide by 10.
We do it on both sides, so they stay equal.
10 divided by 10 is 1.
So just x is left.
We can make 15/10 simpler.
It is the same as 3/2.
So x = 3/2.
4 Find y
Now we put x = 3/2 into ③.
That is −3 + 5.
So y = 2.
So x = 3/2 and y = 2, both at the same time.
We write it as (3/2, 2).
On a graph, it is the point where the two lines meet.
5 Check
Let's check in both equations.
We put x = 3/2 and y = 2 into ①.
We get 5, the same as the right side.
Now ②.
We get 0, 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.