Step by step
The question
Question: solve x² + y² = 20 and y = 2x.
We call them ① and ②.
We want the points (x, y) that make both true.
① is a circle, 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.
So in ①, we write (2x) in place of y.
The brackets keep it together.
2 Open the brackets
(2x) squared is 2x times 2x: 4x².
So we get x² + 4x² = 20.
Like terms together: x² and 4x² make 5x².
We want 0 on one side.
We take away 20 on both sides.
Now we have 5x² − 20 = 0.
Every number can be divided by 5.
We divide both sides by 5.
0 divided by 5 is still 0.
3 Get x² alone
There is no x term, so we get x² by itself.
To remove the − 4, we add 4 on both sides.
0 + 4 = 4.
Now we have x² = 4.
4 Square root of both sides
x squared is 4.
So x is the square root of 4, or minus it.
Minus times minus is plus, so the negative one works too.
2 times 2 is 4.
So x = 2 or x = −2.
5 Find y
Each x goes into the line ②: x = −2 gives y = −4.
And x = 2 gives y = 4.
So the two points are (−2, −4) and (2, 4).
6 Check
Let's check with a picture.
We draw both on a grid.
① is a circle around (0, 0).
② is a line.
They meet at (−2, −4) and (2, 4): our answers.
Let's check each point in both equations.
First (−2, −4) in ①: x² + y² is 20.
True!
In ②: −4 again.
It works!
Now (2, 4) in ①: x² + y² is 20.
True!
In ②: 4 again.
It works!
Let's look back.
First: put ② into ①.
Then: open the brackets.
Then: get x² alone.
Then: square root of both sides.
Then: find y.
Last: check.