Step by step
The question
Question: solve y − 2x = 1 and 4x − 2y = 6 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 − 2x is in the way.
To remove − 2x, we add 2x.
We do it on both sides, so they stay equal.
− 2x and + 2x 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 2x + 1.
So in ②, we write (2x + 1) in place of y.
The brackets keep it together.
Putting one thing in place of another is called substitution.
3 Find x
The −2 in front of the bracket multiplies every part inside.
So we write −2 times each part.
Careful!
Not only the first part!
1 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 x terms go together.
4x − 4x = 0.
Now we have −2 = 6.
Both letters are gone, and −2 = 6 is never true.
The two lines never meet.
So there is no solution.
Let's look back.
First: get y alone in ①.
Then: put ③ into ②.
Last: find x.