Step by step
The question
Question: solve y = x² + 1 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 x² − 2x + 1 = 0.
3 Factor
We need two numbers.
They multiply to 1, and they add to −2.
They multiply to a plus number, so the two numbers have the same sign.
They add to a minus number, so both are minus.
We list the pairs that make 1, smallest first.
We add each pair.
We need 2.
1 and 1 give 2.
Yes!
With the signs: −1 and −1.
So it is (x − 1)(x − 1) = 0.
4 One bracket is 0
Two numbers multiply to 0.
That only happens if one of them is 0.
So one bracket must be 0.
We make each bracket 0.
The two brackets are the same.
In x − 1 = 0, we add 1 on both sides.
Both brackets give the same answer.
So there is only one answer: x = 1.
5 Find y
Each x goes into the line ②: x = 1 gives y = 2.
Only one x, so only one point: (1, 2).
The line just touches the parabola.
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 (1, 2): our answer.
Let's check each point in both equations.
First (1, 2) in ①: the right side is 2, the same as y.
In ②: 2 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.