What is the first step in solving a linear and a quadratic equation together?

rearrange the linear equation and substitute it into the other. One letter is eliminated.

How many pairs of solutions are there usually?

two. A line usually cuts a curve of this kind twice.

What are the solutions of y = x and x² + y² = 8?

(2, 2) and (−2, −2). 2x² = 8 gives x = 2 or −2, and y = x.

Where does the line y = x + 2 meet the curve y = x²?

(2, 4) and (−1, 1). x² − x − 2 = 0 gives x = 2 or −1.

The quadratic formed has two equal roots. What does that mean?

The line is a tangent to the curve. They meet at one point only.

Simultaneous equations: one linear, one not · Cambridge IGCSE Additional Mathematics

In plain words

A straight line and a curve usually meet at two points. Finding them means solving both equations at once, and there is only one sensible way: use the line to get rid of a letter.

5 things to know

Rearrange the linear equation to give x or y.
Substitute into the other equation. This leaves a quadratic in one letter.
Solve the quadratic. There are usually two values.
Put each value into the linear equation to find the other letter. Give the answers as pairs.
If the quadratic has equal roots, the line is a tangent to the curve.

Worked example

Solve y = x + 1 and x² + y² = 25.

Substitute for y: x² + (x + 1)² = 25.
Expand: 2x² + 2x + 1 = 25, so x² + x − 12 = 0.
Factorise: (x − 3)(x + 4) = 0, so x = 3 or x = −4.
From y = x + 1: y = 4 or y = −3. The solutions are x = 3, y = 4 and x = −4, y = −3.

Tips and tricks

Substitute back into the linear equation, not the curve. It is easier, and it cannot give extra false answers.
Pair the answers up. Writing "x = 3, −4 and y = 4, −3" without saying which goes with which can lose a mark.

5 questions, about 2 minutes.

It lands in your notebook with its questions as flashcards.

Simultaneous equations: one linear, one not: 5 questions and answers

These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.

What is the first step in solving a linear and a quadratic equation together?
- square both equations
- rearrange the linear equation and substitute it into the other (the answer)
- add the equations
- differentiate
One letter is eliminated.
How many pairs of solutions are there usually?
- one
- two (the answer)
- three
- none
A line usually cuts a curve of this kind twice.
What are the solutions of y = x and x² + y² = 8?
- (2, 2) only
- (2, 2) and (−2, −2) (the answer)
- (4, 4) and (−4, −4)
- (2, −2) and (−2, 2)
2x² = 8 gives x = 2 or −2, and y = x.
Where does the line y = x + 2 meet the curve y = x²?
- (2, 4) only
- (2, 4) and (−1, 1) (the answer)
- (−2, 4) and (1, 1)
- (0, 2) and (2, 0)
x² − x − 2 = 0 gives x = 2 or −1.
The quadratic formed has two equal roots. What does that mean?
- The line misses the curve.
- The line is a tangent to the curve. (the answer)
- The line cuts the curve twice.
- There is an error.
They meet at one point only.

5 questions

Tap an answer and you’ll see straight away whether it’s right, and why.

Worksheet

2 questions, 8 marks. Write your answers on paper, then check them.

Simultaneous equations: one linear, one not

Cambridge IGCSE Additional Mathematics 0606 · 8 marks · papermunch.org

Name ______________________________ Date ______________

Solve y = 2x − 1 and y = x² − 4.[4]
Show answerHide answer
x = 3, y = 5 and x = −1, y = −3. x² − 2x − 3 = 0.
I got this one
Solve x + y = 5 and xy = 6.[4]
Show answerHide answer
x = 2, y = 3 and x = 3, y = 2. x(5 − x) = 6 gives x² − 5x + 6 = 0.
I got this one

Answers: Simultaneous equations: one linear, one not

1. x = 3, y = 5 and x = −1, y = −3. x² − 2x − 3 = 0.
2. x = 2, y = 3 and x = 3, y = 2. x(5 − x) = 6 gives x² − 5x + 6 = 0.