9.05 Applications of simultaneous equations

The magic of simultaneous equations comes to life when we see how useful they are in real life applications. Simultaneous equations are often used when we have at least two unknown quantities and at least two pieces of information involving these quantities. Our first step is to define variables to represent these quantities and then translate the pieces of information into equations. Lastly, by using either the substitution method, elimination method or a graphical method, we solve the equations simultaneously.

 

Worked example

Rachel is at a cafe buying drinks for her friends. Some of them want coffee, while others want tea. At this particular cafe, a regular cup of coffee costs $\$4$$4 and a regular cup of black tea costs $\$3$$3. If she buys $7$7 drinks for a total cost of $\$26$$26, how many of each type of drink did she purchase?

Think:

We want to determine the number of cups of coffee and the number of cups of tea that Rachel bought. These are two unknowns, so we can choose variables to represent them. We then also know the total number of drinks that Rachel bought, and the total cost of the drinks - so we have two pieces of information, and can use this to construct two simultaneous equations.

Do:

Let $x$x be the number cups of tea, and let $y$y be the number of cups of coffee.

Using the first piece of information, we know that the total number of drinks purchased is $7$7 - that is, the sum of $x$x and $y$y is $7$7. So we have our first equation:

$x+y=7$x+y=7

equation (1)

 

Using the second piece of information, we know that the total cost of the drinks is $\$26$$26. Now the price of a cup of tea is $\$3$$3, so the total cost of the cups of tea will be $3x$3x dollars. Similarly, the total cost of the cups of coffee will be $4y$4y. Putting this together, we have the second equation:

$3x+4y=26$3x+4y=26

equation (2)

 

Now that we have our two equations, we can either use substitution or elimination to solve the system. Let's use the method of substitution - to do so, however, we will first rearrange equation (1) to make $y$y the subject:

$x+y=7$x+y=7		equation (1)
$y=7-x$y=7−x		equation (3)

 

We can now substitute this into equation (2):

$3x+4y$3x+4y	$=$=	$26$26
$3x+4\left(7-x\right)$3x+4(7−x)	$=$=	$26$26	substituting (3) into (2)
$3x+28-4x$3x+28−4x	$=$=	$26$26
$28-x$28−x	$=$=	$26$26
$-x$−x	$=$=	$-2$−2
$x$x	$=$=	$2$2

 

So we have that Rachel purchased $2$2 cups of tea (remember, this is what we chose $x$x to represent). We can now use this to find $y$y (the number of cups of coffee) by substituting back into any of our original equations. In this case, it will be simplest to use equation (3):

$y$y	$=$=	$7-x$7−x
	$=$=	$7-2$7−2	substituting $x=2$x=2
	$=$=	$5$5

 

So we have found that Rachel bought $2$2 cups of tea and $5$5 cups of coffee.

 

Practice questions

Question 1

Question 2

 

Break-even point

It's really important for businesses to make a profit! Otherwise, they won't be around for very long. So businesses should know their break-even point, the amount of money they need to take in to cover all their expenses. At this point there is no profit or loss, and their income is equal to their expenses.

To find this point, we use break-even analysis. Break-even analysis looks at the graphs of cost and revenue together to determine where they cross. This point of intersection is the break-even point where income equals expenses.

Here is an example of a break-even analysis for a single day of operation for the company Lovely Lemonade.

The horizontal axis is the number of drinks, or units, they sell. The expenses line (red) starts off higher than the income line (green). So for a low number of units sold, Lovely Lemonade will lose money. For example, if they only sell $1000$1000 units, the green line tells us they earn $\$200$$200, the red line tells us they spend $\$600$$600, and so overall they make $\$200-\$600=-\$400$$200−$600=−$400. The negative sign means overall they lose $\$400$$400.

But eventually, the income line becomes higher than the expenses line, so for a high number of units sold, Lovely Lemonade will make money. For example, if they sell $5000$5000 units, the graph tells us that they will earn $\$1100$$1100 but only spend $\$1000$$1000. Overall they make $\$1100-\$1000=\$100$$1100−$1000=$100, and the positive sign means a profit of $\$100$$100.

The important point is when the two lines meet - this is the break-even point. In this example we can see that selling $4000$4000 units means they don't make any profit, but they don't lose any money either - both their income and their expenses are $\$900$$900. Reaching this amount should be an important first goal for Lovely Lemonade every single day!

We can also use break-even analysis to compare two different pricing plans for the same product or service to see which one offers the best deal for a particular situation.

Here's an example comparison between two energy plans provided by rival companies Thorgate and Callisto:

In this scenario the break-even point at $\left(40,10\right)$(40,10) tells us that both plans charge $\$10$$10 for $40$40 kWh. For energy amounts less than the break-even point we can see that Thorgate is cheaper, and for energy amounts more than the break-even point Callisto is cheaper. Knowing how much energy someone plans on using can then determine the best plan for them.

 

Practice questions

Question 3

Question 4