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9.05 Applications of simultaneous equations

Lesson

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=7x   equation (3)

 

We can now substitute this into equation (2):

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

 

$3x+4\left(7-x\right)$3x+4(7x) $=$= $26$26

substituting (3) into (2)

$3x+28-4x$3x+284x $=$= $26$26

 

$28-x$28x $=$= $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$7x

 

  $=$= $7-2$72

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

Consider the following phone plans:

GO SMALL plan: This plan has a $\$20$$20 monthly base charge and charges $90$90 cents per minute for all calls.

GO MEDIUM plan: This plan has a $\$26$$26 monthly base charge and then charges $70$70 cents per minute for all calls.

  1. Complete the following table of values for various total monthly call times for the two plans:

    Call time (in minutes) Total cost for GO SMALL plan Total cost for GO MEDIUM plan
    $20$20 $\editable{}$ $\editable{}$
    $30$30 $\editable{}$ $\editable{}$
    $40$40 $\editable{}$ $\editable{}$
    $50$50 $\editable{}$ $\editable{}$
  2. Sketch the graph of the two plans.

    Loading Graph...

  3. Using the graphs, determine how many minutes of calls would need to be made so that the monthly bill costs the same on both plans.

Question 2

The length of a rectangle measures $12$12 units more than the width, and the perimeter of the rectangle is $56$56 units.

Let $y$y be the width and $x$x be the length of the rectangle.

  1. Use the fact that the length of the rectangle is $12$12 units more than the width to set up an equation relating $x$x and $y$y (we'll call this equation (1)).

  2. Use the fact that the perimeter of the rectangle is equal to $56$56 to set up another equation relating $x$x and $y$y (we'll call this equation (2)).

  3. First solve for $y$y to find the width.

    $x=y+12$x=y+12 equation (1)
    $x+y=28$x+y=28 equation (2)
  4. Now solve for $x$x to find the length.

    $x=y+12$x=y+12 equation (1)
    $x+y=28$x+y=28 equation (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

David decides to start his own yoga class. The cost and revenue of running the class have been graphed.

Loading Graph...

  1. How much revenue does David make for each student?

  2. How many students must attend his class so that David can cover the costs of running the class?

  3. How much profit does David make if there are $8$8 students in his class?

Question 4

The cost for a furniture manufacturer to make an armchair is $\$500$$500 per armchair plus a fixed setup cost of $\$3500$$3500. The armchairs will sell for $\$650$$650 each.

  1. Write an expression to represent the cost of manufacturing $x$x armchairs.

  2. Write an expression to represent the revenue generated from the sale of $x$x armchairs.

  3. Find the break-even point.

Outcomes

1.3.3.2

solve practical problems that involve finding the point of intersection of two straight-line graphs, such as determining the break-even point where cost and revenue are represented by linear equations

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