GCSE · Maths · AQA · Spec 8300 · Higher

Simultaneous equations (linear/quadratic) (Higher)

Two straight lines that aren't parallel cross just once. A line and a curve can cross twice — so now you're hunting for two answers, each with two parts.

Where does the line cross the curve?

x: 0.5. y: -2.75

Drag the point along the curve. Stop where it sits on the straight line, read its coordinates, then find the other crossing.

The straight line is y = x − 1, the curve is y = x² − 3: At a crossing point, the same x and y fit on the line AND on the curve, so that pair makes both equations true. That's what a solution is.

Watch out: A solution is a pair, not just an x-value. Two crossings means two answers, and each x comes with its own y.

Maths · Algebra

Now make it exact: substitution

Same line, same curve as the graph. Step through and watch the two crossing points appear.

GoalSolve simultaneously: y = x² − 3 and x − y = 1

y = x² − 3 and x − y = 1

The curve is the quadratic equation; the line is the linear one. Start with the line — it's the easy one to rearrange.

Step 1 of 9

The curve is the quadratic equation; the line is the linear one. Start with the line — it's the easy one to rearrange.

Watch out: Find each y from the linear equation. It gives every x exactly one y, and the arithmetic is simpler. Then use the other equation as your check.

Two lines vs a line and a curve

Linear and linearvsLinear and quadratic

Start with the top row: it's the difference that trips people up.

Focus

How many solution pairs?

Linear and linear

Usually one pair, where the two lines cross. Parallel lines never meet, so they give none.

Linear and quadratic

Up to two pairs: two where the line cuts the curve twice, fewer where it just touches or misses.

The insight

Expect two answers, but not always. If your quadratic has only one (repeated) solution, the line just touches the curve; if it has no solutions, they never meet.

What the graphs look like

Linear and linear

Two straight lines

Linear and quadratic

A straight line and a curve

Method to reach for

Linear and linear

Elimination (match a coefficient, then add or subtract the equations) or substitution

Linear and quadratic

Substitution: rearrange the linear equation and put it into the quadratic

The one-letter equation you solve

Linear and linear

A linear equation, such as 5x = 20

Linear and quadratic

A quadratic equal to zero, such as x² − 4x + 3 = 0

Spot the line that loses the mark

Solve simultaneously: y = x² + 2x − 5 and y = x + 1

A student's answer — which line goes wrong?

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

Set it up, solve it, make sense of it

Write it, then mark it

Two squares have sides of x cm and y cm. The sides differ by 2 cm, and the two areas add up to 34 cm². Set up and solve a pair of simultaneous equations to find the side of each square. [5 marks]

Your answer

0 words · your answer stays on this page and is not sent anywhere.

The mark scheme

A linear equation from the sides: x − y = 2, or x = y + 2 (or y − x = 2 if y is taken as the larger side) (1 mark)
A quadratic equation from the areas: x² + y² = 34 (1 mark)
Substitutes to get one equation in one letter, equal to zero: e.g. (y + 2)² + y² = 34, so y² + 2y − 15 = 0 (1 mark)
Solves it: y = 3 or y = −5 (or x = 5 or x = −3) (1 mark)
Rejects the negative pair because a length can't be negative, and states the sides are 5 cm and 3 cm (1 mark)

Model answer

Let the larger square have side x cm. The sides differ by 2, so x − y = 2, which gives x = y + 2. The areas add to 34, so x² + y² = 34. Substitute: (y + 2)² + y² = 34, so y² + 4y + 4 + y² = 34, which gives 2y² + 4y − 30 = 0, or y² + 2y − 15 = 0. Factorise: (y + 5)(y − 3) = 0, so y = 3 or y = −5. If y = 3, x = 5. If y = −5, x = −3 — impossible, because a side can't have a negative length. So the squares have sides 5 cm and 3 cm. Check: 5² + 3² = 25 + 9 = 34 ✓.

Algebra gives you every pair that fits the equations; the context decides which pairs are real answers.

Exam line: Credit comes at three stages: setting up both equations, solving them, and interpreting the result. Rejecting an impossible answer earns its own credit.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

one crossing point = one (x, y) pair

A line and a curve can meet twice, once or not at all. Find every pair, and give each x its own y.

What you need to know

What a solution to a pair of simultaneous equations is, and why a line and a curve can give two.
How to find approximate solutions by reading where two graphs cross.
How to solve a linear and a quadratic equation exactly, by substitution.
How to set up two equations from a problem, solve them and interpret the answer.

The big picture

When one equation is linear and the other is quadratic, their graphs are a straight line and a curve, and a line can cut a curve twice. Each crossing point is one solution: an x with its own y. A graph lets you estimate those pairs; substitution finds them exactly. Rearrange the linear equation, substitute it into the quadratic, solve the quadratic, then put each x back into the linear equation to get its partner y.

Key points

1A solution of two simultaneous equations is a matched pair: an x and a y that make BOTH equations true.

2On a graph, the solutions are the points where the graphs cross. A line and a curve can meet twice, once or not at all.

3Reading crossing points from a graph gives approximate solutions; algebra gives exact ones.

4Method: make x or y the subject of the linear equation, substitute into the quadratic, rearrange to equal zero and solve.

5Substitute each x back into the linear equation to find its own y, and write the answers as pairs.

6In a problem set in context, check that each pair makes sense, and reject any that can't happen, such as a negative length.

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Worked example

Problem

Solve simultaneously: y = x² and y = 2x − 1. What does your answer tell you about the graphs?

⚠ Watch out

Stopping once the quadratic is solved. Two x-values are only half the answer: each one needs its own y, found from the linear equation, and the answer is written as two pairs.

🧠

Memory hook

Line → curve → line. Take y from the LINE, drop it into the CURVE, solve, then go back to the LINE for each partner y.

✓

Check yourself

Cover the page and solve y = x + 3 and y = x² + 1. You should get two pairs. Test each one in BOTH equations: if both come out true, you've got it.

Flashcards

(9)

What does one solution of a pair of simultaneous equations look like?

A pair of values — an x and its matching y — that makes both equations true at the same time.

On a graph, where are the solutions of two simultaneous equations?

Where the two graphs cross. The coordinates of each crossing point are one solution pair.

How many solution pairs can a linear and a quadratic equation have?

Two, one or none: the line can cut the curve twice, just touch it, or miss it completely.

How many solution pairs do two straight lines that aren't parallel have?

Exactly one — two lines that aren't parallel cross once.

Why are solutions read from a graph only approximate?

You're judging a crossing point by eye against a grid, so the values are only as accurate as the drawing. Algebra gives the exact values.

Which method solves a linear and a quadratic equation together?

Substitution: make x or y the subject of the linear equation, then substitute it into the quadratic.

After substituting, what form do you rearrange the equation into before solving?

A quadratic equal to zero, with every term on one side.

You've found x = 4 and x = −1. What's left to do?

Substitute each x into the linear equation to find its own y, then write the answer as two pairs.

In a problem set in context, when do you reject a solution pair?

When it can't happen in the situation — for example a negative length, or a negative number of people.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Where does the line cross the curve?

Now make it exact: substitution

Spot the line that loses the mark

Write it, then mark it

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