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Simultaneous equations are a set of at least two equations containing at least two unknowns. They are called simultaneous because all the equations must be solved at the same time. Contrary to popular belief, they are not just there to torture high school students and to calculate the price of apples and pears. Here are some everyday uses for simultaneous equations - and some you might not have thought of.

## Find the best mobile phone deal

So you want a new smart phone. There are dozens of different contracts available and you to need to decide which one is the best plan for you. To keep things simple, let's say that you have narrowed your choice down to two. Plan A charges a rate of £15 per month and 30p per minute of call time, after your 600 free minutes have expired. Plan B charges the higher rate of £20 per month, but only 20p per minute call time, and also gives 600 minutes free. Simultaneous equations can be used to calculate which plan is cheaper, depending on the number of phone calls you make in a month.

Let x be the number of minutes of calls you make in a month and y be the total cost of each plan. Under Plan A y= 15 + 0.30x. Under Plan B y= 20 + 0.20x. So:

15 + 0.30x = 20 + 0.2x 5 = 0.10x x = 50

Including the 600 free minutes, both plans cost the same with 650 minutes of calls per month. If I were to make more calls than that, say 675 minutes, the total monthly cost under Plan A would be y = 15 + 0.30(75), or £37.50. Under Plan B, the cost would be y = 20 + 0.20(75), or £35. So at 675 minutes, Plan B is cheaper.

## The food problem

You're in a coffee shop with your friends. You go to the counter to order. Friend A wants 1 sandwich, 2 biscotti and a coffee and hands over five pounds. Friend B, who's ordering for himself and his girlfriend, wants 1 sandwich, 3 biscotti and a 2 coffees and gives you the exact money of £8.75 - they've had this order before. Friend C wants a sandwich and a coffee and hands over the exact money of £4.50. You buy yourself 1 sandwich, 1 biscotti and 1 coffee and pay a total of £22.50 for everything.

Letting S be a sandwich, B be a biscotti and C be a coffee, the equations you get are:

Equation (1) 4S + 6B + 4C = 22.50 Equation (2) 1S + 3B + 2C = 8.75 Equation (3) S + C = 4.50

To solve it, multiply equation (2) by 2 to get 2S + 6B + 4C = 17.50, then take this away from equation (1) to get 2S = 5 or S = 2.50. From this you can deduce (using equation (3)) that a coffee costs £2.00 and a biscotti costs 75p. Or more importantly, that friend A owes you a pound.

## In the air

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Air traffic control use the graph method of solving simultaneous equations to find the intersection point of two flight paths, and use this to redirect planes before any accidents occur. Guided missile systems do the same thing but in reverse - here the "solution" to the simultaneous equation is the point of impact.

## Getting to the bottom of warranties

Practically every electrical or electronic product you buy these days comes with the offer of an extended warranty - your new tablet, for example. Company A wants to sell it to you for £400 with a five year warranty. Company B is selling the same tablet for £250 with a 1 year warranty. Company C has the tablet on offer at £200 with no warranty. You want to equalise the cost of the warranty to find the base price of the tablet. If W is the warranty and T is the tablet the equations you have are 5W + T = 400 and W + T = 250. From this we deduce that 4W = 150 or W is £37.50. Tablets A and B cost £212.50, so C is the cheaper unit.

## Your mortgage

If the Bank of England were lowering interest rates, a simultaneous equation could help you work out the exact point at which is is better to move from a fixed interest rate mortgage to a variable rate one.

## Find the best satellite or cable package

Basic TV packages often come bundled together with add-ons, such as the sports package or kids' channels, and this makes them difficult to compare. Company A, for example, is offering you a basic package of £15 per month plus 12 months of free sport, if you sign up for 12 months. Company B is offering you a basic package for £12 per month, with six months of free sport, but only if you sign up for 24 months. Which is the better basic deal?

If B is the basic package and S is the sport package, our equations are:

12B + 12S = 180 (£15 per month times 12 months) 24B + 6S = 288 (£12 per moth times 24 months)

If you halve the second equation to get 12B + 3S = 144, then subtract this from the first equation, you get 9S = 36, or S = 4. So if the sports package costs £4 per month, Company B's basic package price of £8 per month is cheaper.

## The fluctuating cost of wine

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Simultaneous equations are often used to analyse problems in the field of economics, particularly when describing the relationship between quantity, price and other variables such as consumer satisfaction or demand. The Metal Project has an interesting video clip explaining the relationship between supply, demand and the price of French wine using simultaneous equations (see Resources).

## Hire a car....

Company A charges £50 per day to hire a car, and 15p per mile travelled. Company B charges £35 per day and 20p a mile. How do you know which is the better deal?

This one is similar to the mobile phone problem, in that what you are looking for is the break-even point, that is, the point at which both deals cost are the same. If M = total miles to be driven and C = the total cost for each company, the equations we have are C = 50 + 0.15m and C = 35 + 0.2m. It follows that 50 + 0.15 m = 35 + 0.20m, that 0.05 m = 15, and that m = 300.

In other words, for a journey of 300 miles the cost of each company would be the same. For a shorter journey, company A is cheaper.

## ....Or buy a rail card?

Of course, you could always travel by train. Young people between the ages of 16 and 25 in the UK can buy a rail card for £30 (rate current July 2013). This gets you one third off the price of each journey you make. A simultaneous equation can be used to calculate how many journeys you would have to make before it makes sense to invest in a rail card. The Metal Project has a worked example (see Resources).

According to the website Further Mathematics Support, Google's search engine technology uses a complex and inventive system of simultaneous equations to rank websites in order of importance - which might explain their famously tough interview process.

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