Step by step
The question
Question: solve −x + y = 4 and 4x − y = 2 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 add 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 + 4.
So in ②, we write (x + 4) in place of y.
The brackets keep it together.
Putting one thing in place of another is called substitution.
3 Find x
A minus in front of a bracket changes the sign of every part inside.
Careful!
Not only the first sign!
The sign of 4 changes too.
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 − x = 3x.
Now we have 3x − 4 = 2.
We want x by itself.
The − 4 is in the way.
To remove − 4, we add 4.
We do it on both sides, so they stay equal.
− 4 and + 4 make zero.
Cross them out.
2 plus 4 is 6.
Now we have 3x = 6.
3x means 3 times x.
We want just x.
To remove times 3, we divide by 3.
We do it on both sides, so they stay equal.
3 divided by 3 is 1.
So just x is left.
6 divided by 3 is 2.
So x = 2.
4 Find y
Now we put x = 2 into ③.
So y = 6.
So x = 2 and y = 6, both at the same time.
We write it as (2, 6).
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 = 6 into ①.
We get 4, the same as the right side.
Now ②.
We get 2, 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.