Step by step
The question
Question: solve x + y = 6 and 2x + 3y = 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 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 + 6.
So in ②, we write (−x + 6) in place of y.
The brackets keep it together.
Putting one thing in place of another is called substitution.
3 Find x
The 3 in front of the bracket multiplies every part inside.
So we write 3 times each part.
Careful!
Not only the first part!
6 is multiplied by 3 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 − 3x = −x.
Now we have −x + 18 = 16.
We want x by itself.
The + 18 is in the way.
To remove + 18, we take away 18.
We do it on both sides, so they stay equal.
+ 18 and − 18 make zero.
Cross them out.
16 minus 18 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.
So y = 4.
So x = 2 and y = 4, both at the same time.
We write it as (2, 4).
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 = 4 into ①.
We get 6, 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.