Step by step
The question
Question: solve x + 3y = 10 and 4x + y = 18 by substitution.
We call them ① and ②.
We want the x and y that make both true.
1 Get y alone in ②
No letter is alone yet.
In ②, 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 ①
In ③, y is now alone.
y is the same as −4x + 18.
So in ①, we write (−4x + 18) 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!
18 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.
x − 12x = −11x.
Now we have −11x + 54 = 10.
We want x by itself.
The + 54 is in the way.
To remove + 54, we take away 54.
We do it on both sides, so they stay equal.
+ 54 and − 54 make zero.
Cross them out.
10 minus 54 is −44.
Now we have −11x = −44.
−11x means −11 times x.
We want just x.
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 x is left.
−44 divided by −11 is 4.
So x = 4.
4 Find y
Now we put x = 4 into ③.
That is −16 + 18.
So y = 2.
So x = 4 and y = 2, both at the same time.
We write it as (4, 2).
On a graph, it is the point where the two lines meet.
5 Check
Let's check in both equations.
We put x = 4 and y = 2 into ①.
We get 10, the same as the right side.
Now ②.
We get 18, the same as the right side.
Both work.
So the answer is right!
Let's look back.
First: get y alone in ②.
Then: put ③ into ①.
Then: find x.
Then: find y.
Last: check.