GCSE · Maths · AQA · Spec 8300 · Foundation

Simultaneous equations (linear/linear)

Two numbers add up to 8, and the second is one more than double the first. Loads of pairs pass one test. Only one pair passes both. Which is it?

Which point works for both equations?

x: 4. y: 4

The line with the dot on it is x + y = 8. Drag the dot along it and watch the co-ordinates: they always add up to 8. Now slide it to where the red line crosses.

The red line is y = 2x + 1: Every point on the red line fits y = 2x + 1, just as every point on the dot's line fits x + y = 8. The crossing is the only point on both lines, so it's the only pair that makes both equations true at once. What do you read there? Something close to (2.3, 5.7). Close, but a graph can't tell you whether that's exact.

Watch out: A point on one line only is not a solution. The dot starts at (4, 4), which is on x + y = 8 because 4 + 4 = 8. But on the red line, x = 4 gives y = 2 × 4 + 1 = 9, not 4. The answer has to work in both equations.

Maths · Algebra

Now find it exactly

Elimination: add or subtract the equations so that one letter disappears.

GoalSolve x + y = 8 and y = 2x + 1

x + y = 8

(1)

Equation (1), as it is.

Step 1 of 6

Equation (1), as it is.

Exam line: Match the numbers in front of one letter. Same signs: subtract. Different signs: add. Then substitute to find the other letter, and check.

Watch out: When you multiply an equation, multiply every term. 3x + 2y = 13 doubled is 6x + 4y = 26, not 6x + 4y = 13.

Spot the slip

Where does this answer go wrong?

Solve 2x + 3y = 17 (1) and 2x − y = 5 (2).

A student's answer — which line goes wrong?

Both equations have 2x, so subtract: (1) − (2)
3y − y = 2y and 17 − 5 = 12, so 2y = 12
y = 6
Substitute into (1): 2x + 3 × 6 = 17, so 2x = −1
x = −0.5
Check in (1): 2 × (−0.5) + 3 × 6 = −1 + 18 = 17 ✓

Your turn

Turn the words into equations

At a cinema, 2 adult tickets and 3 child tickets cost £38 altogether. 1 adult ticket and 2 child tickets cost £22 altogether. Work out the cost of one adult ticket and the cost of one child ticket. Show how you form and solve your equations. [5 marks]

Your answer

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

The mark scheme

Says what each letter stands for and writes 2a + 3c = 38 (a = cost of an adult ticket, c = cost of a child ticket, in pounds) (1 mark)
Writes the second equation, a + 2c = 22 (1 mark)
Makes the numbers in front of one letter match and removes it, e.g. doubles the second equation to 2a + 4c = 44 and subtracts the first (1 mark)
Finds c = 6 and a = 10 (1 mark)
Answers the question in words with units: an adult ticket costs £10 and a child ticket costs £6 (1 mark)

Model answer

Let a be the cost of an adult ticket and c the cost of a child ticket, in pounds. 2a + 3c = 38 (1) and a + 2c = 22 (2). Double (2): 2a + 4c = 44. Subtract (1): c = 6. Substitute into (2): a + 12 = 22, so a = 10. Check in (1): 2 × 10 + 3 × 6 = 20 + 18 = 38 ✓. An adult ticket costs £10 and a child ticket costs £6.

Say what each letter stands for, write one equation per fact, solve, then answer in the problem's own words and units. a = 10 isn't finished until it's 'an adult ticket costs £10'.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

The answer is the point that sits on both lines. A graph shows you roughly where it is. Algebra pins it down exactly.

What you need to know

A solution is a pair of values, one for each letter, that makes both equations true.
On a graph, the solution is the point where the two lines cross. Reading it off the grid gives an estimate.
Elimination: make the numbers in front of one letter the same size, then subtract if their signs match or add if they differ.
If you multiply an equation, multiply every term, including the number on the other side.
Substitute the value you find into an original equation to get the other value, then check the pair in both equations.
In a problem in words or about a shape, say what each letter stands for and give the answer in context, with units.

The big picture

Two equations in two unknowns are solved simultaneously when you find the pair of values that makes both true at the same time. Each linear equation is a straight line, and the solution is the point where the two lines cross. Reading that point off a graph gives an estimate. Elimination gives the exact answer: match the numbers in front of one letter, add or subtract the equations to remove it, solve, substitute back, then check the pair in both equations. In a problem in words or about a shape, write one equation for each fact and finish by saying what your answer means.

Key points

1Every point on a line fits that line's equation. Where two lines cross, only the crossing point fits both.

2A graph gets you close; algebra gets you exact.

3Once the numbers in front of a letter match: same signs subtract, different signs add.

4Taking away a negative is adding: 3y − (−y) = 4y.

5A pair that fails either equation is not the solution.

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Worked example

Problem

A rectangle has one pair of opposite sides of length (3x − y) cm and (x + 7) cm. The other pair of sides are (x + y) cm and 8 cm. Find x and y, and then the area of the rectangle.

⚠ Watch out

Stopping at a pair that works in only one equation. After a slip, your values usually still fit the equation you substituted into, so that check looks fine. Test the pair in the other equation too: if it fails there, it isn't the solution.

🧠

Memory hook

Two lines, one crossing. Same signs subtract, different signs add, and a pair isn't the answer until it passes both tests.

✓

Check yourself

Solve 2x + y = 11 and x − y = 1, then check both equations. Next, multiply first: solve 3x + 2y = 12 and x + 3y = 11.

Flashcards

(9)

What does it mean to solve two equations simultaneously?

Find the pair of values that makes both equations true at the same time.

Where is the solution on a graph of the two lines?

At the point where the lines cross. Read x straight down to the x-axis and y straight across to the y-axis.

Why is a solution read from a graph only an estimate?

The crossing can fall between gridlines, so you can only read it approximately. Algebra gives the exact values.

Elimination: what do you do before adding or subtracting?

Make the numbers in front of one letter the same size, multiplying a whole equation if you need to.

The matched terms have the same sign. Add or subtract?

Subtract. If the signs are different, add.

When you multiply an equation to match coefficients, what must you not forget?

Multiply every term, including the number on the right-hand side.

You've found one letter. How do you find the other?

Substitute it into one of the original equations and solve.

How do you check a solution pair?

Substitute both values into both original equations. Both must work.

In a word problem, what does a finished answer look like?

It says what each value means in the problem, with units, e.g. 'an adult ticket costs £10'.

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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Which point works for both equations?

Now find it exactly

Where does this answer go wrong?

Turn the words into equations

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