Step by step
The question
Question: solve y = 2x² − 4x and y = 2x.
We call them ① and ②.
We want the points (x, y) that make both true.
① is a parabola, and ② is a line.
Not both lines, so this is called a nonlinear system.
1 Put ② into ①
In ②, y is already alone: y = 2x.
Both ① and ② say what y is.
So the two right sides are equal.
2 Get 0 on one side
We take 2x away from both sides.
Careful!
The answers are points, not just x numbers.
After x, we must find each y.
Now we have 2x² − 6x = 0.
Every number can be divided by 2.
We divide both sides by 2.
0 divided by 2 is still 0.
3 Factor
There is no plain number.
Both parts have x in them.
So we take x out.
So it is x(x − 3) = 0.
4 One bracket is 0
x times the bracket is 0.
That only happens if one of them is 0.
We make each part 0.
Two small equations.
x = 0 is done already.
In x − 3 = 0, we add 3 on both sides.
So x = 0 or x = 3.
5 Find y
Each x goes into the line ②: x = 0 gives y = 0.
And x = 3 gives y = 6.
So the two points are (0, 0) and (3, 6).
6 Check
Let's check with a picture.
We draw both on a grid.
① is a parabola.
② is a line.
They meet at (0, 0) and (3, 6): our answers.
Let's check each point in both equations.
First (0, 0) in ①: the right side is 0, the same as y.
In ②: 0 again.
It works!
Now (3, 6) in ①: the right side is 6, the same as y.
In ②: 6 again.
It works!
Let's look back.
First: put ② into ①.
Then: get 0 on one side.
Then: factor.
Then: one bracket is 0.
Then: find y.
Last: check.