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