Step by step
The question
Question: solve x + y = 1 and 3x − y = 11 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 + 1.
So in ②, we write (−x + 1) 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 1 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.
3x + x = 4x.
Now we have 4x − 1 = 11.
We want x by itself.
The − 1 is in the way.
To remove − 1, we add 1.
We do it on both sides, so they stay equal.
− 1 and + 1 make zero.
Cross them out.
11 plus 1 is 12.
Now we have 4x = 12.
4x means 4 times x.
We want just x.
To remove times 4, we divide by 4.
We do it on both sides, so they stay equal.
4 divided by 4 is 1.
So just x is left.
12 divided by 4 is 3.
So x = 3.
4 Find y
Now we put x = 3 into ③.
So y = −2.
So x = 3 and y = −2, both at the same time.
We write it as (3, −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 and y = −2 into ①.
We get 1, the same as the right side.
Now ②.
We get 11, 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.