Step by step
The question
Question: solve 3x + y = 5 and 2x + 4y = −10 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 is by itself.
We call it ③.
2 Put ③ into ②
In ③, y is now alone.
y is the same as −3x + 5.
So in ②, we write (−3x + 5) in place of y.
The brackets keep it together.
Putting one thing in place of another is called substitution.
3 Find x
The 4 in front of the bracket multiplies every part inside.
So we write 4 times each part.
Careful!
Not only the first part!
5 is multiplied by 4 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 − 12x = −10x.
Now we have −10x + 20 = −10.
We want x by itself.
The + 20 is in the way.
To remove + 20, we take away 20.
We do it on both sides, so they stay equal.
+ 20 and − 20 make zero.
Cross them out.
−10 minus 20 is −30.
Now we have −10x = −30.
−10x means −10 times x.
We want just x.
To remove times −10, we divide by −10.
We do it on both sides, so they stay equal.
−10 divided by −10 is 1.
So just x is left.
−30 divided by −10 is 3.
So x = 3.
4 Find y
Now we put x = 3 into ③.
That is −9 + 5.
So y = −4.
So x = 3 and y = −4, both at the same time.
We write it as (3, −4).
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 = −4 into ①.
We get 5, the same as the right side.
Now ②.
We get −10, 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.