Step by step
The question
Question: solve y = x + 2 and x² + y² = 34.
We call them ① and ②.
We want the points (x, y) that make both true.
② is a circle, and ① is a line.
Not both lines, so this is called a nonlinear system.
1 Put ① into ②
In ①, y is already alone: y = x + 2.
So in ②, we write (x + 2) in place of y.
The brackets keep it together.
2 Open the brackets
(x + 2) squared is (x + 2) times (x + 2).
Each part times each part, then like terms: x² + 4x + 4.
Careful!
Squaring a bracket is not squaring each part.
(x + 2) squared has a middle part too: 4x.
So we get x² + x² + 4x + 4 = 34.
Like terms together: x² and x² make 2x².
We want 0 on one side.
We take away 34 on both sides.
Now we have 2x² + 4x − 30 = 0.
Every number can be divided by 2.
We divide both sides by 2.
0 divided by 2 is still 0.
3 Factor
We need two numbers.
They multiply to −15, and they add to 2.
They multiply to a minus number, so one number is plus and one is minus.
They add to a plus number, so the bigger one is plus.
We list the pairs that make 15, smallest first.
We take the smaller from the bigger.
We need 2.
3 and 5 give 2.
Yes!
With the signs: −3 and 5.
So it is (x − 3)(x + 5) = 0.
4 One bracket is 0
Two numbers multiply to 0.
That only happens if one of them is 0.
So one bracket must be 0.
We make each bracket 0.
Two small equations.
In x − 3 = 0, we add 3 on both sides.
In x + 5 = 0, we take away 5 on both sides.
So x = 3 or x = −5.
5 Find y
Each x goes into the line ①: x = −5 gives y = −3.
And x = 3 gives y = 5.
So the two points are (−5, −3) and (3, 5).
6 Check
Let's check with a picture.
We draw both on a grid.
② is a circle around (0, 0).
① is a line.
They meet at (−5, −3) and (3, 5): our answers.
Let's check each point in both equations.
First (−5, −3) in ②: x² + y² is 34.
True!
In ①: −3 again.
It works!
Now (3, 5) in ②: x² + y² is 34.
True!
In ①: 5 again.
It works!
Let's look back.
First: put ① into ②.
Then: open the brackets.
Then: factor.
Then: one bracket is 0.
Then: find y.
Last: check.