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