Step by step
The question
Question: solve x + y = 3 and 2x − 5y = −1 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 x is in the way.
To remove x, we take away x.
We do it on both sides, so they stay equal.
x and − x 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 −x + 3.
So in ②, we write (−x + 3) in place of y.
The brackets keep it together.
Putting one thing in place of another is called substitution.
3 Find x
The −5 in front of the bracket multiplies every part inside.
So we write −5 times each part.
Careful!
Not only the first part!
3 is multiplied by −5 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 + 5x = 7x.
Now we have 7x − 15 = −1.
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.
−1 plus 15 is 14.
Now we have 7x = 14.
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.
14 divided by 7 is 2.
So x = 2.
4 Find y
Now we put x = 2 into ③.
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 3, the same as the right side.
Now ②.
We get −1, 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.