Step by step
The question
Question: solve 4x + y = 2 and 3x + 2y = −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 4x is in the way.
To remove 4x, we take away 4x.
We do it on both sides, so they stay equal.
4x and − 4x 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 −4x + 2.
So in ②, we write (−4x + 2) 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!
2 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 − 8x = −5x.
Now we have −5x + 4 = −1.
We want x by itself.
The + 4 is in the way.
To remove + 4, we take away 4.
We do it on both sides, so they stay equal.
+ 4 and − 4 make zero.
Cross them out.
−1 minus 4 is −5.
Now we have −5x = −5.
−5x means −5 times x.
We want just x.
To remove times −5, we divide by −5.
We do it on both sides, so they stay equal.
−5 divided by −5 is 1.
So just x is left.
On the right too: −5 divided by −5 is 1.
So x = 1.
4 Find y
Now we put x = 1 into ③.
That is −4 + 2.
So y = −2.
So x = 1 and y = −2, both at the same time.
We write it as (1, −2).
On a graph, it is the point where the two lines meet.
5 Check
Let's check in both equations.
We put x = 1 and y = −2 into ①.
We get 2, 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.