Step by step
The question
Question: solve x² + y² = 25 and y = 2x − 5.
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 − 5.
So in ①, we write (2x − 5) in place of y.
The brackets keep it together.
2 Open the brackets
(2x − 5) squared is (2x − 5) times (2x − 5).
Each part times each part, then like terms: 4x² − 20x + 25.
Careful!
Squaring a bracket is not squaring each part.
(2x − 5) squared has a middle part too: −20x.
So we get x² + 4x² − 20x + 25 = 25.
Like terms together: x² and 4x² make 5x².
We want 0 on one side.
We take away 25 on both sides.
Now we have 5x² − 20x = 0.
Every number can be divided by 5.
We divide both sides by 5.
0 divided by 5 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 − 4) = 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 − 4 = 0, we add 4 on both sides.
So x = 0 or x = 4.
5 Find y
Each x goes into the line ②: x = 0 gives y = −5.
And x = 4 gives y = 3.
So the two points are (0, −5) and (4, 3).
6 Check
Let's check with a picture.
We draw both on a grid.
① is a circle around (0, 0) with radius 5.
② is a line.
They meet at (0, −5) and (4, 3): our answers.
Let's check each point in both equations.
First (0, −5) in ①: x² + y² is 25.
True!
In ②: −5 again.
It works!
Now (4, 3) in ①: x² + y² is 25.
True!
In ②: 3 again.
It works!
Let's look back.
First: put ② into ①.
Then: open the brackets.
Then: factor.
Then: one bracket is 0.
Then: find y.
Last: check.