Step by step
The question
Question: solve 3x − y = 7 and 2x + 5y = 16 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 3x is in the way.
To remove 3x, we take away 3x.
We do it on both sides, so they stay equal.
3x and − 3x make zero.
Cross them out.
Write what is left.
We write the x part first.
Each part keeps its sign.
−y means −1 times y.
We want just y.
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 y is left.
The −1 divides the whole right side.
So it divides every part: −3x and 7.
Each part keeps its sign on top.
Minus over minus is plus.
−3x divided by −1 is 3x.
Plus over minus is minus.
7 divided by −1 is −7.
So y = 3x − 7.
We call it ③.
2 Put ③ into ②
In ③, y is now alone.
y is the same as 3x − 7.
So in ②, we write (3x − 7) in place of y.
The brackets keep it together.
Putting one thing in place of another is called substitution.
3 Find x
The 5 in front of the bracket multiplies every part inside.
So we write 5 times each part.
Careful!
Not only the first part!
−7 is multiplied by 5 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.
2x + 15x = 17x.
Now we have 17x − 35 = 16.
We want x by itself.
The − 35 is in the way.
To remove − 35, we add 35.
We do it on both sides, so they stay equal.
− 35 and + 35 make zero.
Cross them out.
16 plus 35 is 51.
Now we have 17x = 51.
17x means 17 times x.
We want just x.
To remove times 17, we divide by 17.
We do it on both sides, so they stay equal.
17 divided by 17 is 1.
So just x is left.
51 divided by 17 is 3.
So x = 3.
4 Find y
Now we put x = 3 into ③.
That is 9 − 7.
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 7, 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 y alone in ①.
Then: put ③ into ②.
Then: find x.
Then: find y.
Last: check.