Step by step
The question
Question: solve x² + y² = 4 and y = x + 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 = x + 5.
So in ①, we write (x + 5) in place of y.
The brackets keep it together.
2 Open the brackets
(x + 5) squared is (x + 5) times (x + 5).
Each part times each part, then like terms: x² + 10x + 25.
Careful!
Squaring a bracket is not squaring each part.
(x + 5) squared has a middle part too: 10x.
So we get x² + x² + 10x + 25 = 4.
Like terms together: x² and x² make 2x².
We want 0 on one side.
We take away 4 on both sides.
Now we have 2x² + 10x + 21 = 0.
3 Can it be 0?
It does not factor.
b² − 4ac, the part under the root in the formula, is −68.
Negative.
No real number has a negative square root.
So no x works: the line and the circle never meet.
There is no solution.
We can see it on a picture.
We draw both on a grid.
① is a circle around (0, 0) with radius 2.
② is a line.
Let's look back.
First: put ② into ①.
Then: open the brackets.
Last: can it be 0?.