Introduction to simultaneous equations - KS3 Maths

Key points

Images of one smoothie and one banana. A question mark is written beneath each. — Image caption,
The unknown values in an equation could be the individual costs of a smoothie and a banana when the combined cost is known.

A good understanding of how to solve equations with 𝒙 on one side is useful when working with with one value.
A set of equations with more than one unknown value are called equations.
When there are two unknown values, two equations will be required in order to solve them.
The two equations cannot be solved on their own. Each one by itself has an number of solutions.

Video

Simultaneous equations can be useful when running a business.

Watch the video to listen to Kim, a textiles designer, talk about how simultaneous equations help her when deciding on how to deliver her products to customers.

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Equations with two unknown variables

One equation with two unknown values will have solutions. This means there is not enough information provided to find the exact solution to the equation.
- For example, if two different items are bought together and only the combined cost of them is known, then it is not possible to work out the individual cost of each item.
- There are a number of potential solutions to this problem, including solutions that are not values.

Example

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The combined cost of a smoothie and a banana is £6. Work out the cost of the banana.
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The two items could cost the same amount. The smoothie could cost £3 and the banana could also cost £3. This is not the only possible solution.
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Either item could cost more than the other, but still total the same amount when added together. For example, the smoothie could cost £4 and the banana could cost £2. Add the two amounts: £4 + £2 = £6
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The cost of each item may not be an integer value. The smoothie could be £2.50 and the banana could be £3.50. Take care when checking solutions that are not integer values. £2.50 + £3.50 = £6
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An equation with two unknown values will have an infinite number of solutions. The unknown values are the cost of the smoothie and the cost of the banana. When there are two unknown values, then two equations will be required to work them out.

Solving a simultaneous equation

One with two values will have an number of solutions.
If a second equation is introduced, then it is possible to solve the problem and find the value of both .
Two equations with the same solutions are called equations.
Comparing the two equations often leads to one variable being eliminated, meaning the value of the remaining variable can be calculated.
The other variable can be calculated using .

Example

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The total cost of a smoothie and a banana is £6. The cost of a strawberry smoothie and two bananas is £8.50. Work out how much each item costs.
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Compare the two sets of items. Look at what is the same and what is different.
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Each set of items contains a smoothie and a banana. The second set has one extra banana and costs £2.50 more than the first. Calculate £8.50 - £6
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A banana costs £2.50
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The first set of one smoothie and one banana costs £6. The cost of the banana is £2.50. Substitute the price of the banana into the equation. Smoothie + £2.50 = £6
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Calculate the price of the smoothie using subtraction. £6 - £2.50 = £3.50
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Check solutions work for both equations by substituting the values, so £3.50 + £2.50 = £6 and £3.50 + £2.50 + £2.50 = £8.50

Question

Two smoothies and two bananas cost £11

Two smoothies and three bananas cost £14

What is the price of a banana?

Images of two smoothies and two bananas. Written below: two s plus two b equals eleven. Written right: Total, eleven pounds. Beneath that: images of two smoothies and three bananas. Written below: two s plus three b equals fourteen. Written right: Total, fourteen pounds.

Representing the problem using a bar model

A can be used to represent a problem, instead of drawing or using images for each object.
Using images helps to identify the structure of the mathematical problem, however a bar model can be drawn more quickly, especially if the problem involves a larger number of items.
The representation of a bar model can be used in other mathematical contexts.

Example

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One smoothie and one banana cost £6.50. One smoothie and three bananas cost £13.50. Work out the price of a smoothie.
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This information can be represented as a bar model rather than using images for the different items. The first set of items is represented by the bar at the top. The second set of items is represented by the bottom bar. The bar model helps identify what is the same and what is different.
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Each set of items contains one smoothie and one banana. The difference is that the second set of items has two extra bananas.
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The extra two bananas cost £7 (£13.50 - £6.50 = £7)
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If two bananas cost £7, each banana costs £3.50. Adding the price of each banana to the bar model makes it simpler to work out the price of the smoothie.
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Using the first set of items, one smoothie and one banana cost £6.50. £6.50 - £3.30 = £3. Each smoothie costs £3
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It is important to check that solutions work for both equations by substituting the values. The first set included one smoothie and one banana. £3 + £3.50 = £6.50
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The second set included one smoothie and three bananas. £3 + £3.50 + £3.50 + £3.50 = £13.50. The solution works for both equations.

Question

Two cups of tea and two cups of coffee cost £10 in total.

Two cups of tea and one cup of coffee cost £7

What is the price of one cup of tea?

The problem represented as a bar model. The bar model has two rows. On the first row, four rectangles joined side by side, labelled tea, tea, coffee and coffee. Written above: a brace the length of the four rectangles, labelled total cost, ten pounds. On the second row, three rectangles joined side by side, labelled tea, tea and coffee. Written below: a brace the length of the three rectangles, labelled total cost, seven pounds.

Practise solving simultaneous equations

Practise solving simultaneous equations with this quiz. You may need a pen and paper to help you work out your answers.

Quiz

Real-life maths

Cyclist riding up a hill on a country road. — Image caption,
Simultaneous equations can be used by athletes to set different goals, such as how to maximise their time or speed.

Simultaneous equations can be used by athletes, for example when calculating the best routes for running or cycling training.

They can create a mathematical expression that takes into account the distance and average speed for different parts of the route they plan to take. Equations can be used to set different goals, such as maximising time for building endurance, or speed for best performance.