Lesson 16.5 · 60 min

Non-linear Simultaneous Equations

Where a straight line meets a parabola, a circle or another curve: forming one quadratic equation, using the discriminant $b^{2} - 4 a c$ to count — or control — the points of intersection, and solving to find them, with the formula when it will not factorise.

In this lesson you will learn to…

Form a single quadratic equation by setting a line equal to a curve, or by substituting
Use the discriminant $b^{2} - 4 a c$ to decide whether there are 2, 1 or 0 points of intersection
Solve by factorising and substitute back to find both coordinates
Solve a line and a circle $x^{2} + y^{2} = r^{2}$ simultaneously, expanding brackets carefully
Recognise when a line is a tangent to a curve
Stretch: use the quadratic formula when the equation does not factorise, giving exact (surd) or 3 s.f. answers
Stretch: find the values of $k$ for which a line meets, touches or misses a curve
Stretch: solve with other curves, such as $x y = 12$ or $y = \frac{6}{x}$

Why we get a quadratic

Two linear equations make two straight lines, which cross at most once. Now one of the equations is a curve — a parabola $y = ax^2 + bx + c$ or a circle $x^2 + y^2 = r^2$ . The points where the line meets the curve satisfy both equations, and combining them gives a quadratic — which is why there can be two answers.

Try it

Move the line across the parabola

Gradient of the line, m1

y-intercept of the line2

Curve: y = x² − 4
Line: y = x + 2
Combined equation: x² − x − 6 = 0
Discriminant: (-1)² − 4(1)(-6) = 25
So: b² − 4ac > 0: two points of intersection
Points: (-2, 0) and (3, 5)

This is $y = x^{2} - 4$ and $y = x + 2$ — question 1 from class. Change the line and watch the discriminant. Can you make the line just touch the curve? (Try a gradient of 2 with $k = - 5$ , or a gradient of 0 with $k = - 4$ .) Can you make it miss completely?

The discriminant tells you how many

Once you have the combined equation $ax^2 + bx + c = 0$ , the discriminant $b^2 - 4ac$ predicts what will happen before you solve anything:

$b^2 - 4ac$	Points of intersection	Picture
positive	2	the line cuts through the curve
zero	1	the line just touches — a tangent
negative	0	the line misses the curve

Solving: a line and a parabola

When both equations start with $y =$ , set them equal. Collect everything on one side, check the discriminant, factorise, and then substitute back to find each $y$ .

A line and a circle

A circle with centre $(0, 0)$ and radius $r$ has equation $x^2 + y^2 = r^2$ . Make $y$ the subject of the linear equation, substitute it into the circle, and expand carefully — the squared bracket is where most marks are lost.

Stretch: beyond the class questions

The class questions all factorise neatly. Exam questions often do not — and they combine this topic with the discriminant work you already know.

When it will not factorise

If $b^2 - 4ac$ is positive but not a square number, there are still two points — they just are not whole numbers. Use the quadratic formula, and give the answer exactly (with surds) or to 3 significant figures, as the question asks.

Questions about $k$

Instead of a fixed line, you may be given a family of lines such as $y = 2x + k$ and asked which values of $k$ make it meet, touch or miss the curve. Keep $k$ in the combined equation and turn the discriminant into an inequality — exactly the “range of values of $k$ ” method from your discriminant work.

Other curves

Nothing in the method depends on the curve being a parabola or a circle. With $xy = 12$ or $y = \frac{6}{x}$ , substitute the line, multiply out, and you get a quadratic again.

Key rules and formulas

The method
1. Make $y$ the subject of the linear equation (if it is not already). 2. Set the two equations equal, or substitute, to get one equation in $x$ . 3. Rearrange to $a x^{2} + b x + c = 0$ . 4. Solve — usually by factorising. 5. Substitute each $x$ back into the linear equation to find $y$ . 6. Write the answers as coordinates.
The discriminant counts the points
$b^{2} - 4 a c > 0 : 2 points b^{2} - 4 a c = 0 : 1 point (tangent) b^{2} - 4 a c < 0 : none$
Work it out from the combined equation before you try to factorise. If it is negative, there is nothing to find. If it is a perfect square (1, 4, 9, 16, 25, …) the equation will factorise.
Lines and circles
For $x^{2} + y^{2} = r^{2}$ , substitute the line into the circle. Expand the squared bracket in full: $(2 x - 5)^{2} = (2 x - 5) (2 x - 5) = 4 x^{2} - 20 x + 25$ .
Give pairs of values
Each $x$ has its own $y$ . Write them as coordinates, $(3, 5)$ and $(- 2, 0)$ , not as four loose numbers.
Stretch: when it will not factorise
$x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$
If $b^{2} - 4 a c$ is positive but not a square number, use the formula. Simplify the surd for exact answers ( $24 = 26$ ), or round to 3 s.f. only at the very end.
Stretch: questions about $k$
Form the combined equation with $k$ in it, then write the discriminant as an inequality and solve for $k$ : meets at two points → $> 0$ ; tangent → $= 0$ ; misses → $< 0$ .

Worked examples

Worked example 1

A line and a parabola

Solve

y = x + 1

and

y = x^{2} - 5

simultaneously.

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

When $a$ is not 1

Solve

y = 3 x - 2

and

y = 2 x^{2} + x - 6

simultaneously.

Try it yourself first, then reveal the solution one step at a time.

Worked example 3

A line and a circle

Solve

x^{2} + y^{2} = 10

and

y = x + 2

simultaneously.

Try it yourself first, then reveal the solution one step at a time.

Worked example 4

A tangent

Show that the line

y = 2 x - 1

is a tangent to

y = x^{2}

, and find where it touches.

Try it yourself first, then reveal the solution one step at a time.

Worked example 5

Stretch: when does a line miss a circle?

Find the values of

k

for which the line

y = x + k

does not meet the circle

x^{2} + y^{2} = 2

Try it yourself first, then reveal the solution one step at a time.

Practice: non-linear simultaneous equations

20 questions

Questions 1–12 are at class level; 13–20 are stretch questions at exam level. Combine, rearrange to = 0, check b² − 4ac, then solve and substitute back into the line. Give the point with the smaller x first.

Question 1 of 20Type the answer
The line $y = 2 x + 3$ meets the curve $y = x^{2}$ . Form the combined equation and find its discriminant.
Your answer
Question 2 of 20Multiple choice
How many points of intersection do $y = x^{2} + 1$ and $y = 2 x$ have?
Question 3 of 20Multiple choice
How many points of intersection do $y = x - 3$ and $y = x^{2} + x$ have?
Question 4 of 20Fill the gaps
Solve $y = x + 6$ and $y = x^{2}$ simultaneously.
Points: ( Gap x1 , Gap y1 ) and ( Gap x2 , Gap y2 )
Question 5 of 20Fill the gaps
Solve $y = 2 x + 3$ and $y = x^{2} + 4 x - 5$ simultaneously.
Points: ( Gap x1 , Gap y1 ) and ( Gap x2 , Gap y2 )
Question 6 of 20Fill the gapsChallenge
Solve $y = 5 x - 3$ and $y = 2 x^{2} + 2 x - 5$ simultaneously.
Points: ( Gap x1 , Gap y1 ) and ( Gap x2 , Gap y2 )
Question 7 of 20Fill the gaps
Solve $x^{2} + y^{2} = 25$ and $y = x + 1$ simultaneously.
Points: ( Gap x1 , Gap y1 ) and ( Gap x2 , Gap y2 )
Question 8 of 20Step by stepChallenge
A line that touches a circle.
The line $2 x + y = 5$ meets the circle $x^{2} + y^{2} = 5$ .
1. Make $y$ the subject of the line and substitute. Which equation do you get after simplifying?
Question 9 of 20Step by step
A circle with an easy-to-miss bracket.
Solve $x^{2} + y^{2} = 9$ and $y = x - 3$ simultaneously.
1. Expand $(x - 3)^{2}$ .
Question 10 of 20Type the answerChallenge
The line $y = 2 x + k$ is a tangent to the curve $y = x^{2}$ . Find $k$ .
Your answer
Question 11 of 20Explore and match
Move the line until it is a tangent to $y = x^{2} - 4$ (exactly one point of intersection).
Move the controls until you match the question, then press Check.
Gradient of the line, m1
y-intercept of the line2
Curve
y = x² − 4
Line
y = x + 2
Combined equation
x² − x − 6 = 0
Discriminant
(-1)² − 4(1)(-6) = 25
So
b² − 4ac > 0: two points of intersection
Points
(-2, 0) and (3, 5)
Aim for
1 point of intersection
Question 12 of 20Step by stepChallenge
In context.
A ball follows the path $y = - x^{2} + 6 x$ . A straight ramp follows $y = x + 4$ . Find where the ball's path meets the ramp.
1. Which equation do you get?
Question 13 of 20Multiple choiceChallenge
Stretch. Solve $y = 3 x$ and $y = x^{2} + x - 1$ simultaneously. What are the $x$ -coordinates, exactly?
Question 14 of 20Fill the gapsChallenge
Stretch. Solve $y = 2 x - 1$ and $y = x^{2} + x - 4$ simultaneously. Give your answers correct to 3 significant figures, the point with the smaller $x$ first.
( Gap x1 , Gap y1 ) and ( Gap x2 , Gap y2 )
Question 15 of 20Fill the gapsChallenge
Stretch. The line $y = x + k$ meets the curve $y = x^{2} - 3 x$ at two points. Find the range of values of $k$ .
$k >$ Gap a
Question 16 of 20Fill the gapsChallenge
Stretch. The line $y = k x - 1$ is a tangent to the curve $y = x^{2}$ . Find the two possible values of $k$ .
$k =$ Gap a or $k =$ Gap b
Question 17 of 20Fill the gapsChallenge
Stretch. For which values of $k$ does the line $y = x + k$ miss the circle $x^{2} + y^{2} = 8$ ?
$k <$ Gap a or $k >$ Gap b
Question 18 of 20Fill the gapsChallenge
Stretch. Solve $y = \frac{6}{x}$ and $y = x + 1$ simultaneously. Give the point with the smaller $x$ first.
( Gap x1 , Gap y1 ) and ( Gap x2 , Gap y2 )
Question 19 of 20Step by stepChallenge
Stretch. Substituting for $x$ instead of $y$ .
Solve $x^{2} + y^{2} = 25$ and $x - 2 y = 5$ simultaneously.
1. Which is easier to make the subject of $x - 2 y = 5$ ?
Question 20 of 20Put in order
Stretch. Put the lines of this proof in order: Show that the line $y = 2 x - 3$ does not meet the curve $y = x^{2} + 1$ .
Put these in the right order. Drag them, or use the arrow buttons.
1. $x^{2} + 1 = 2 x - 3$
2. $b^{2} - 4 a c = (- 2)^{2} - 4 (1) (4) = - 12$
3. $x^{2} - 2 x + 4 = 0$
4. $- 12 < 0$ , so there are no real solutions: the line does not meet the curve.

In this lesson you will learn to…

Why we get a quadratic

Move the line across the parabola

The discriminant tells you how many

Solving: a line and a parabola

A line and a circle

Stretch: beyond the class questions

When it will not factorise

Questions about kkk

Other curves

Key rules and formulas

The method

The discriminant counts the points

Lines and circles

Give pairs of values

Stretch: when it will not factorise

Stretch: questions about k

Worked examples

Practice: non-linear simultaneous equations

Questions about $k$

Stretch: questions about $k$