Step by step
The question
Question: solve x + 4y = 3 and 2x − 3y = −16 by substitution.
We call them ① and ②.
We want the x and y that make both true.
1 Get x alone in ①
No letter is alone yet.
In ①, x has just 1 in front.
So we get x alone there first.
We want x by itself.
The + 4y is in the way.
To remove + 4y, we take away 4y.
We do it on both sides, so they stay equal.
+ 4y and − 4y make zero.
Cross them out.
Write what is left.
We write the y part first.
Each part keeps its sign.
x is by itself.
We call it ③.
2 Put ③ into ②
In ③, x is now alone.
x is the same as −4y + 3.
So in ②, we write (−4y + 3) in place of x.
The brackets keep it together.
Putting one thing in place of another is called substitution.
3 Find y
The 2 in front of the bracket multiplies every part inside.
So we write 2 times each part.
Careful!
Not only the first part!
3 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 y terms go together.
−8y − 3y = −11y.
Now we have −11y + 6 = −16.
We want y by itself.
The + 6 is in the way.
To remove + 6, we take away 6.
We do it on both sides, so they stay equal.
+ 6 and − 6 make zero.
Cross them out.
−16 minus 6 is −22.
Now we have −11y = −22.
−11y means −11 times y.
We want just y.
To remove times −11, we divide by −11.
We do it on both sides, so they stay equal.
−11 divided by −11 is 1.
So just y is left.
−22 divided by −11 is 2.
So y = 2.
4 Find x
Now we put y = 2 into ③.
That is −8 + 3.
So x = −5.
So x = −5 and y = 2, both at the same time.
We write it as (−5, 2).
On a graph, it is the point where the two lines meet.
5 Check
Let's check in both equations.
We put x = −5 and y = 2 into ①.
We get 3, the same as the right side.
Now ②.
We get −16, the same as the right side.
Both work.
So the answer is right!
Let's look back.
First: get x alone in ①.
Then: put ③ into ②.
Then: find y.
Then: find x.
Last: check.