Learn7357Paper 3 · Pure and StatisticsPure content (sections A–J)Simultaneous equations and lines meeting curves

Pure content (sections A–J) · Subtopic 5 of 66

Simultaneous Equations and Lines Meeting Curves

Solve one linear and one quadratic equation together by substitution, find the points where a line meets a curve, and use the discriminant to decide whether a line crosses, touches or misses a curve.

Cambridge 9709 Paper 1Edexcel 9MA0 Paper 1Edexcel 9MA0 Paper 2AQA 7357 Paper 1AQA 7357 Paper 2AQA 7357 Paper 3≈ 40 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

Solve a linear and a quadratic equation simultaneously by substitution.
Give the answers as matching pairs, and use them to find lengths and midpoints.
Use the discriminant of the resulting quadratic to decide how many times a line meets a curve.
Find the condition for a line to be a tangent to a curve, and the point where it touches.

Pause and think

A straight line and a U-shaped parabola. What are all the possible numbers of meeting points?

Section 2 of 14

Core idea

Start hereWhere does the ramp meet the hill?

A skateboard ramp follows the straight line $y=x+2$ and the ground beside it curves as $y=x^2$ . Where do they meet? A meeting point lies on both, so its coordinates satisfy both equations at once. Solving two equations together is called solving them simultaneously. The answer: $(2,4)$ and $(-1,1)$ .

The big idea

To solve a linear and a quadratic equation simultaneously, substitute the linear one into the quadratic. This gives a single quadratic equation in one variable. Its roots are the $x$ -coordinates of the intersection points; put them into the linear equation to get the $y$ -coordinates. The discriminant tells you how many intersections there are.

1. Two linear equations (revision)

Two unknowns need two equations. Elimination: add or subtract the equations to remove one variable. Substitution: make one variable the subject and replace it in the other. Both work because if two things are equal, you can swap one for the other, or add equal amounts to both sides.

Worked example $2x+3y=12$ and $x-y=1$

Solve $2x+3y=12$ and $x-y=1$ .

Goal: The pair $(x,y)$ satisfying both.
Strategy: Substitution: the second equation easily gives $x=y+1$ .

$x=y+1$
Add $y$ to both sides of $x-y=1$ .
$2(y+1)+3y=12$
Replace $x$ in the first equation.
$5y+2=12\Rightarrow y=2$
Expand, collect, solve.
$x=2+1=3$
Back-substitute.

Answer

x=3

y=2

✓ Check

2(3)+3(2)=12

✓;

3-2=1

✓.

2. A linear and a quadratic

y = x² and y = x + 2 cross twice

The two crossing points are the two solutions of the pair of equations.

Worked exampleThe ramp: $y=x+2$ and $y=x^2$

Find where $y=x+2$ meets $y=x^2$ .

Goal: All intersection points $(x,y)$ .
Strategy: Both equal $y$ , so set them equal; solve; back-substitute into the line.

$x^2=x+2$
At a meeting point the $y$ -values are equal.
$x^2-x-2=0$
Rearrange to $=0$ .
$(x-2)(x+1)=0\Rightarrow x=2\text{ or }x=-1$
Factorise.
$x=2:\ y=4;\qquad x=-1:\ y=1$
Use the linear equation $y=x+2$ (simplest).

Answer

(2,4)

and

(-1,1)

✓ CheckBoth satisfy

y=x^2

too:

2^2=4

✓,

(-1)^2=1

✓.

Worked exampleA harder pair: $x+2y=5$ and $x^2+y^2=10$

Solve $x+2y=5$ and $x^2+y^2=10$ .

Goal: All solutions.
Strategy: Make $x$ the subject of the linear equation (avoids fractions), substitute into the circle.

$x=5-2y$
Rearrange the linear equation.
$(5-2y)^2+y^2=10$
Substitute.
$25-20y+4y^2+y^2=10$
Expand $(5-2y)^2$ carefully: three terms.
$5y^2-20y+15=0\Rightarrow y^2-4y+3=0$
Collect; divide by $5$ .
$(y-1)(y-3)=0\Rightarrow y=1\text{ or }3$
Factorise.
$y=1:\ x=3;\qquad y=3:\ x=-1$
Back-substitute into $x=5-2y$ .

Answer

(3,1)

and

(-1,3)

✓ Check

9+1=10

✓ and

1+9=10

✓;

3+2=5

✓ and

-1+6=5

✓.

Common misconceptionPairing the answers wrongly Each

x

goes with its own

y

. Write answers as coordinate pairs, and substitute into the linear equation; substituting into the quadratic can give extra, false

y

-values (

y^2=1

gives

y=\pm1

3. How many intersections?

The line y = x + c moving down past y = x²

c = 3.00

Substituting gives

x^2-x-c=0

, with discriminant

1+4c

. Two points while

c>-\tfrac14

, one (a tangent) at

c=-\tfrac14

, none below that.

y = x² + 1 and lines y = mx through the origin

gradient m = 0.00

x^2+1=mx\Rightarrow x^2-mx+1=0

\Delta=m^2-4

. Shallow lines (

m<2

) miss;

m=2

touches at

(1,2)

; steeper lines cut twice.

Worked exampleTangent condition

Find $c$ such that $y=x+c$ is a tangent to $y=x^2$ , and find the point of contact.

Goal: $c$ with exactly one meeting point.
Strategy: Substitute; set $\Delta=0$ .

$x^2=x+c\Rightarrow x^2-x-c=0$
Equate and rearrange.
$\Delta=1+4c=0\Rightarrow c=-\tfrac14$
$a=1$ , $b=-1$ , $c_{\text{coef}}=-c$ .
$x^2-x+\tfrac14=\left(x-\tfrac12\right)^2=0\Rightarrow x=\tfrac12$
Repeated root.
$y=\tfrac12-\tfrac14=\tfrac14$
Use the line.

Answer

c=-\frac14

; contact at

\left(\frac12,\frac14\right)

✓ Check

\left(\frac12\right)^2=\frac14

✓: the point is on the parabola.

Linear + quadratic: the method

Drawing diagram…

Always finish with coordinate pairs.

Pause and think

Show that $y=2x-5$ does not meet $y=x^2$ .

Summary

Substitute the linear equation into the quadratic.
Solve; back-substitute into the linear one; pair answers.
$\Delta$ counts intersections; $\Delta=0$ is a tangent.

Simultaneous equations: Equations true at the same time.
Intersection: A point lying on both graphs.
Back-substitute: Put a found value into an earlier equation to get the other unknown.

HistoryTwo thousand years of simultaneous equations. The ancient Chinese book The Nine Chapters on the Mathematical Art, compiled over several centuries up to about the 1st century CE, solves systems of equations about grain and livestock with a method almost identical to the row operations used in computers today.

In the real worldYour phone does this to find you. A GPS receiver knows its distance from several satellites. Each distance puts it on a sphere around a satellite, and its position is where those spheres meet. Finding that point means solving simultaneous quadratic equations, many times a second.

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