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3. Linear equations and inequalities
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Australia
Year 10

3.08 Applications of simultaneous equations

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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.

Examples

Example 1

Consider the following phone plans:

  • GO SMALL plan: This plan has a \$20 monthly base charge and charges 90\text{c} per minute for all calls.

  • GO MEDIUM plan: This plan has a \$26 monthly base charge and then charges 70 \text{c} per minute for all calls.

a

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

\text{Call time} \\ \text{(in minutes)}\text{Total cost for} \\ \text{GO SMALL plan } (S)\text{Total cost for} \\ \text{GO MEDIUM plan } (M)
20
30
40
50
Worked Solution
Create a strategy

Add the monthly base charge to the number of minutes times the rate per minute.

Apply the idea

For the call time of 20 minutes we can find the cost for the GO SMALL plan by:

\displaystyle \text{GO SMALL cost}\displaystyle =\displaystyle 20+0.9\times 20Add the base charge to the rate times the minutes
\displaystyle =\displaystyle \$38Evaluate

We can find the cost for the GO MEDIUM plan by:

\displaystyle \text{GO MEDIUM cost}\displaystyle =\displaystyle 26+0.7\times 20Add the base charge to the rate times the minutes
\displaystyle =\displaystyle \$40Evaluate

Continuing in this way for the rest of the call times gives us the following table:

\text{Call time} \\ \text{(in minutes)}\text{Total cost for} \\ \text{GO SMALL plan } (S)\text{Total cost for} \\ \text{GO MEDIUM plan } (M)
2038.0040.00
3047.0047.00
4056.0054.00
5065.0061.00
b

Sketch a graph for the two plans.

Worked Solution
Create a strategy

Use the points from the table to graph both plans.

Apply the idea
5
10
15
20
25
30
35
40
45
\text{Call time}
5
10
15
20
25
30
35
40
45
50
55
60
\text{Cost}

To include the points from the table the vertical or cost axes needs to go up to 65, and the horizontal or call time axes needs to go up to 50.

Label your lines with S and M.

To graph the lines we can use two points for each plan from the table. For the GO SMALL plan we can use (20,38) and (30,47). For the GO MEDIUM plan we can use (20,40) and (30,47).

c

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.

Worked Solution
Create a strategy

Use the point of intersection.

Apply the idea

The point of intersection is at (30,47). This means that for a call time of 30 minutes both plans cost \$47.

So 30 minutes of calls would need to be made.

Example 2

The length of a rectangle is 12 units more than the width, and the perimeter of the rectangle is 56 units. Let y be the width and x be the length of the rectangle.

a

Use the fact that the length of the rectangle is 12 units more than the width to set up an equation for x and y.

Worked Solution
Create a strategy

Turn the following into a mathematical statement: The length is 12 more than the width.

Apply the idea
\displaystyle \text{length}\displaystyle =\displaystyle \text{width} + 12Write the equation in words
\displaystyle x\displaystyle =\displaystyle y+12Replace width with y and length with x
b

Use the fact that the perimeter of the rectangle is 56 units to set up an another equation for x and y.

Worked Solution
Create a strategy

Use the perimeter of a rectangle formula P=2x+2y.

Apply the idea
\displaystyle 2x+2y\displaystyle =\displaystyle PWrite the formula
\displaystyle 2x+2y\displaystyle =\displaystyle 56Substitute the perimeter
\displaystyle x+y\displaystyle =\displaystyle 28Divide both sides by 2
c

First solve for y to find the width.

Worked Solution
Create a strategy

Use the substitution method.

Apply the idea
\displaystyle x\displaystyle =\displaystyle y+12(1)
\displaystyle x+y\displaystyle =\displaystyle 28(2)
\displaystyle y+12+y\displaystyle =\displaystyle 28Substitute (1) into (2)
\displaystyle 2y+12\displaystyle =\displaystyle 28Add like terms
\displaystyle 2y\displaystyle =\displaystyle 16Subtract 12 from both sides
\displaystyle y\displaystyle =\displaystyle 8Divide both sides by 2
d

Now solve for x to find the length

Worked Solution
Create a strategy

Substitute the value of y found into equation 1.

Apply the idea
\displaystyle x\displaystyle =\displaystyle y+12(1)
\displaystyle x\displaystyle =\displaystyle 8+12Substitute y=8 into (1)
\displaystyle x\displaystyle =\displaystyle 20Evaluate
Idea summary

To solve a problem with two unknown quantities:

  • Define variables to represent the quantities.

  • Translate the information into two equations.

  • Use the substitution method, elimination method or graphical method to solve the equations simultaneously.

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.

A graph showing a Lovely Lemonde's break-even analysis. Ask your teacher for more information.

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 units, the green line tells us they earn \$200, the red line tells us they spend \$600, and so overall they make \$200 - \$600 = -\$400. The negative sign means overall they lose \$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 units, the graph tells us that they will earn \$1100 but only spend \$1000. Overall they make \$1100 - \$1000 = \$100, and the positive sign means a profit of \$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 units means they don't make any profit, but they don't lose any money either - both their income and their expenses are \$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:

A graph showing Thorgate ad Callisto pricing comparison. Ask your teacher for more information.

In this scenario the break-even point at (40, 10) tells us that both plans charge \$10 for 40kWh. 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.

Examples

Example 3

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

1
2
3
4
5
6
7
8
9
10
11
\text{Students}
6
12
18
24
30
36
42
48
54
\text{C/R}
a

Find the revenue that David makes for each student.

Worked Solution
Create a strategy

Use the revenue line on the graph.

Apply the idea
1
2
3
4
5
6
7
8
9
10
11
\text{Students}
6
12
18
24
30
36
42
48
54
\text{C/R}

For 1 student, x=1. We move from x=1 on the x-axis up to the revenue line, then across to the y-axis to get y=6.

So the revenue per student is \$6.

b

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

Worked Solution
Create a strategy

The number of students is given by the intersection point of the revenue and cost curves.

Apply the idea

The point of intersection is (6,36). So if David get 6 students, he can cover the cost of running the class.

c

How much profit does David make if there are 9 students in his class?

Worked Solution
Create a strategy

Find the difference between the revenue and the cost.

Apply the idea
1
2
3
4
5
6
7
8
9
10
11
\text{Students}
6
12
18
24
30
36
42
48
54
\text{C/R}

To find the revenue and cost for 9 students, we find x=9 on the x-axis and move up to the two lines and then across to the y-axis.

We can see that cost is \$45 and the revenue is \$54.

\displaystyle \text{Profit}\displaystyle =\displaystyle 54-45Subtract the cost from the revenue
\displaystyle =\displaystyle \$9Evaluate

Example 4

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

a

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

Worked Solution
Create a strategy

Add the fixed cost and the variable cost.

Apply the idea

The fixed cost is \$3500.

The variable cost is \$500 per armchair, which is \$500\times x.

\displaystyle \text{Cost}\displaystyle =\displaystyle 3500+500xAdd the fixed and variable costs
b

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

Worked Solution
Create a strategy

The amount of revenue generated is equal to the number of pieces sold multiplied by the price at which they sell.

Apply the idea
\displaystyle \text{Revenue}\displaystyle =\displaystyle 650xMultiply the selling price and the number of armchairs sold
c

Find the break-even point.

Worked Solution
Create a strategy

Equate the expressions for the cost and the revenue and solve for x.

Apply the idea

At the break-even point, the cost is equal to the revenue, so we equate their expressions.

\displaystyle 650x\displaystyle =\displaystyle 3500+500xEquate the revenue and cost
\displaystyle 650x-500x\displaystyle =\displaystyle 3500Subtract 500x from both sides
\displaystyle 150x\displaystyle =\displaystyle 3500Simplify
\displaystyle x\displaystyle =\displaystyle \dfrac{3500}{150}Divide both sides by 150
\displaystyle \approx\displaystyle 23.3333333 \ldotsEvaluate

So the break even point is when x=24 since we can only sell whole numbers of armchairs.

Idea summary

The break-even point is where the cost and revenue are equal.

To find the break-even point, we can either:

  • Look at the graphs of cost and revenue together to determine where they intersect.

  • Equate the expressions for revenue and cost and solve for the independent variable.

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