Step by step
The question
Question: solve y = x² + 3 and y = x.
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 = x.
Both ① and ② say what y is.
So the two right sides are equal.
2 Get 0 on one side
We take x away from both sides.
Careful!
The answers are points, not just x numbers.
After x, we must find each y.
Now we have x² − x + 3 = 0.
3 Can it be 0?
It does not factor.
b² − 4ac, the part under the root in the formula, is −11.
Negative.
No real number has a negative square root.
So no x works: the line and the parabola never meet.
There is no solution.
We can see it on a picture.
We draw both on a grid.
① is a parabola.
② is a line.
Let's look back.
First: put ② into ①.
Then: get 0 on one side.
Last: can it be 0?.