  # Understanding Simultaneous Equations

### Understanding Simultaneous Equations by using teas and coffees

As a young subject leader some years ago, I used various methods to help students understand and solve simultaneous equations in GCSE mathematics.  I remember vividly a significant moment when my classes made enormous progress by placing algebraic concepts into a practical setting.   I can highly recommend the following idea to help ensure amazing success in a short space of time:

Asking students to work in pairs or in a one to one learning session on their own, consider the following three coffee houses:

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### Café America

Two teas and one coffee costs £1.90

One tea and one coffee costs £1.20

How much is a cup of tea?

How much is a cup of coffee?

### Café Brazilia

Three teas and two coffees cost £3.50

Three tea and four coffees cost £4.80

How much is a cup of tea?

How much is a cup of coffee?

### Café Cuba

Four teas and five coffees cost £3.50

Three teas and two coffees cost £2.10

How much is a cup of tea?

How much is a cup of coffee?

By allocating around seven to ten minutes to allow students to explore these three scenarios can be an illuminating experience and reinforces the need to show clear method work.  Sometimes, the correct answers are obtained by guesswork in one, two or all three coffee houses.  However, often students need an easy to understand strategy that provides a beacon of light to obtain the correct solutions showing clear and concise method work.

For each question, it is amazing to see huge progress by allowing X to represent the price of a cup of tea and Y to represent the price of a cup of coffee.  It is also helpful to write down equations in terms of pence rather than pounds.

### For Café America we obtain:

2X  + Y  =  190 (1)

X +  Y  =  120  (2)

It is very helpful to reinforce the importance of numbering each equation as this brings structure and also helps the person checking the work to understand what is happening.

By asking a ‘What If’ question, students can be guided towards a solution by asking ‘what if we ADD the two equations together?’ or ‘What if we SUBTRACT the two equations’?  – can we eliminate the X or the Y?

Clearly by subtracting equation (2) from equation (1) we easily obtain X = 70 pence.   This value for X can be substituted into either of the first two equations to obtain Y = 50 pence.    By contextualising X and Y into something practical such as a cup of tea and a cup of coffee brings great success and improved confidence very quickly.

### For Café Brazilia, students are quick to write down:

3X  + 2Y  =  360 (1)

3X  + 4Y  =  480 (2)

By subtracting equation (2) from equation (1) gives 2Y = 120 and hence Y = 60 pence.  This value again can be substituted back into either equation (1) or equation (2) to obtain X = 80 pence.

### For Café Cuba, students begin with:

4X  + 5Y =  350  (1)

3X  + 2Y  =  210 (2)

It is intriguing to realise that adding the two equations or subtracting the two equations will not eliminate the X or Y values consistent with the first two coffee houses so we must use a sophisticated technique of ‘equalising the coefficients’.  I have found it helpful on some occasions just to write down:

• +2  =  3

and then ask the student to multiply every number by a number of their choice – for example 10:

10 + 20 = 30

When students can see that the equals sign still holds true they can transfer the new learning to see that, for Café Cuba, we can multiply equation (1) by 3 and multiply equation (2) by 4 to produce:

12X  + 15Y  =  1050 (3)

12X  + 8Y   =   840   (4)

Students can quickly see that progress has been made as there are now an equal number of X terms.

By subtracting equation (4) from (3) gives 7Y =  210 and hence Y = 30 pence.  By substituting Y = 30 into equation (1) or equation (2) gives X = 50 pence.

Students are asked if there is an alternative strategy to find the price of a cup of tea and the price of a cup of coffee for Café Cuba.  Often, they can quickly see that another way to equalise coefficients is to multiply equation (1) by 2 and equation (2) by 5.   This allows the price of a cup of coffee to be obtained first and is equally correct and successful.

### Conclusion

I hold the view that this elegant strategy is the finest way in to solve simultaneous equations in an imaginative way.  There is still a need to practice lots of examples to consolidate understanding and improve confidence and awareness.  Innovative ‘ways in’ to teaching and learning mathematical topics is clearly helpful in the journey towards demystifying the universal language of mathematics.  I wish you luck if you wish to try this out in your teaching in the future.

Christopher is a highly-experienced Maths tutor and the inventor of the popular game Prometheus

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