Topics
3. Linear equations and inequalities
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Australia
Year 10

3.02 Linear models

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Introduction

We can now use our knowledge of linear relationships to model and solve problems in real world situations.

Before we begin, there are a few differences to be aware of between the linear relationships we have seen so far on the coordinate plane, and the ones we use to model real world situations.

Linear models

The first thing to notice is that the independent and dependent variables, x and y, will often represent physical quantities such as time, distance, cost, mass or temperature. Instead of being labelled x and y, they may be labelled with letters or names that better represent those quantities.

For example, if we were modelling the rate of fuel being used in a car, the independent variable may represent the distance travelled, in kilometres, and the dependent variable may represent the volume of fuel, in litres, in the car's fuel tank. Instead of x and y, we might use d and V as our variables. Each of these variables has units of measurement associated with them.

Fuel consumption of a mid-sized Sports Utility Vehicle
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d \text{ (km)}
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V \text{ (L)}

As we know already, the two key features of a linear function, or straight-line graph, are the gradient and the y-intercept. In linear modelling situations, the y-intercept is often referred to as the vertical intercept, because the vertical axis may be labelled with a variable other than y.

The vertical intercept represents an initial value. In our example above, the vertical intercept is 63 litres. It represents the volume of fuel in a full tank, before the car began its journey.

In a real-world context the gradient represents a rate of change. Using our example above, the gradient would represent the volume of fuel used per distance travelled. In other words, the gradient is a measure of the car's fuel consumption.

The graph above is decreasing as the fuel is being used, so it has a negative gradient. If we divide the 'rise' of -63 by the 'run' of 900 we get a gradient of -0.07. This means fuel is being consumed at a rate of 0.07 litres per kilometre (or 7 litres per 100 kilometres).

Because most physical quantities like distance, volume or time do not contain negative values, the graphs of most linear models tend to exist only in the first quadrant of the coordinate plane (like the example above). This is not always the case though. Temperature is a physical quantity that can have negative values.

Examples

Example 1

A bucket that is full of water has hole made in its side.

The graph below shows the amount of water remaining in a bucket (in litres) over time (in minutes).

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\text{time (mins)}
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\text{quantity (L)}
a

What is the gradient of the function?

Worked Solution
Create a strategy

Use the formula: \text{Gradient}= \dfrac{y_2 - y_1}{x_2 - x_1}

Apply the idea

By choosing two points that lie on the line we can let (x_1,y_1)=(0,32) and (x_2,y_2)=(8,28).

\displaystyle m\displaystyle =\displaystyle \frac{28 - 32}{8 - 0}Substitute the coordinates
\displaystyle =\displaystyle -\dfrac{4}{8}Evaluate the difference
\displaystyle =\displaystyle -\dfrac{1}{2}Simplify
Reflect and check

In context, this means that for every 1 minute that passes the bucket loses \dfrac{1}{2} a litre of water.

b

What is the y-value of the y-intercept?

Worked Solution
Create a strategy

The y-intercept is the value of at which the line crosses the y-axis or when x=0.

Apply the idea

Looking at the graph, the y-value at which the line crosses the y-axis is 32.

Reflect and check

In context, this means that initially (when x=0) there was 32 litres of water in the bucket.

c

Write an equation to represent the amount of water remaining in the bucket, y, as a function of time, x.

Worked Solution
Create a strategy

Use the form y = mx + c

Apply the idea

From parts (a) and (b), we found that m = -\dfrac{1}{2} and c = 32.

\displaystyle y\displaystyle =\displaystyle -\dfrac{1}{2} \times x + 32Substitute m and c
\displaystyle y\displaystyle =\displaystyle -\dfrac{1}{2}x + 32Simplify
d

What does the slope tell you?

A
The amount of water remaining in the bucket after 2 minutes.
B
The time it takes the amount of water remaining in the bucket to drop by one litre.
C
The time it takes for the bucket to be completely empty.
D
The amount of water that is flowing out of the hole every minute.
Worked Solution
Create a strategy

The slope represents the rate of change of the amount of water in the bucket for a change in time.

Apply the idea

From the graph, we can see that the amount of water is decreasing as the time increases. So the slope represents the amount of water that is flowing out of the hole every minute, option D.

e

What does the y-intercept tell you?

A
The capacity of the bucket.
B
The amount of water remaining in the bucket after 32 minutes.
C
The size of the hole.
D
The amount of water remaining in the bucket when it is empty.
Worked Solution
Create a strategy

The y-intercept represents the amount of water in the bucket before any water has been flowed out through the hole.

Apply the idea

Since the value of y-intercept is the initial amount of water before the flow out, it can be assumed that it is the capacity of the bucket, option A.

f

Find the amount of water remaining in the bucket after 56 minutes.

Worked Solution
Create a strategy

Substitute the given time in the equation found in part c.

Apply the idea
\displaystyle y\displaystyle =\displaystyle -\dfrac{1}{2} \times 56 + 32Substitute x=56
\displaystyle =\displaystyle -\dfrac{56}{2} + 32Evaluate the multiplication
\displaystyle =\displaystyle -28 + 32Simplify the fraction
\displaystyle =\displaystyle 4 \text{ Litres}Evaluate

Example 2

Use the graph to convert -20\degree \text{F} to Celsius.

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\degree\text{ F}
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\degree\text{ C}
A
-40\degree\text{C}
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-34\degree\text{C}
C
-29\degree\text{C}
D
-23\degree\text{C}
Worked Solution
Create a strategy

Using the graph, move directly to the right from that point of intersection and look for the value that reached the vertical axis is the temperature in degrees Celsius that -20\degree \text{F} is equivalent to.

Apply the idea
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-40
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\degree\text{ F}
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\degree\text{ C}

The point of intersection is -29 which means -20\degree\text{F} = -29\degree\text{C} .

So the correct answer is option C.

Idea summary

In linear modelling situations:

  • The vertical intercept represents the initial value.

  • The gradient represents the rate of change.

  • The independent variable goes on the horizontal axis.
  • The dependent variable goes on the vertical axis.

Linear relationship from a table of values

Real world situations often involve the collection of data, which is commonly displayed in a table of values. The independent variable is always displayed in the first row of the table, followed by the dependent variable, in the second row.

We may first want to decide whether the values in the table represent a linear relationship. To do this, we could plot the values on a coordinate plane. If we can draw a straight line through all of the points, then we have a linear relationship.

Once we know that the relationship between the variables is linear, we can work out the gradient and vertical intercept, and express the relationship as a linear equation (or function).

Examples

Example 3

Petrol costs a certain amount per litre. The table shows the cost of various amounts of petrol in dollars:

\text{Number of litres }(x)010203040
\text{Cost of petrol }(y)016.4032.8049.2065.60
a

Write an equation linking the number of litres of petrol pumped (x) and the cost of the petrol (y).

Worked Solution
Create a strategy

The cost of petrol is equal to the price per litre times the number of litres pumped.

Apply the idea

Find the price per litre:

\displaystyle y\displaystyle =\displaystyle \dfrac{16.40}{10}Divide 16.40 by 10
\displaystyle =\displaystyle 1.64Evaluate

Since we get the price per litre, we can write the equation as y=1.64x.

b

How much does petrol cost per litre?

Worked Solution
Create a strategy

Divide the cost of the petrol (y) by the number of litres (x).

Apply the idea
\displaystyle \text{Cost per litre}\displaystyle =\displaystyle \dfrac{16.40}{10}Divide 16.40 by 10
\displaystyle =\displaystyle \$1.64Evaluate
c

How much would 47 litres of petrol cost at this unit price?

Worked Solution
Create a strategy

Using the equation from part (a), subtitute x=47.

Apply the idea
\displaystyle y\displaystyle =\displaystyle 1.64xWrite the equation
\displaystyle =\displaystyle 1.64 \times 47Substitute x=47
\displaystyle =\displaystyle \$77.08Evaluate
d

In the equation, y=1.64x, what does 1.64 represent?

A
The unit rate of cost of petrol per litre.
B
The number of litres of petrol pumped.
C
The total cost of petrol pumped.
Worked Solution
Create a strategy

By looking on part (a), 1.64 represents the rate at which y increases.

Apply the idea

The correct answer is option A: The unit rate of cost of petrol per litre.

Example 4

Kerry currently pays \$50 a month for her internet service. She is planning to switch to a fibre optic cable service.

a

Write an equation for the total cost T of Kerry's current internet service over a period of n months.

Worked Solution
Create a strategy

To find the total cost, we need to multiply the cost of internet a month by the number of months.

Apply the idea

T=\text{Total cost},\, n=\text{ Number of months},\, \$50=\text{Cost of internet per month}

T= 50n

b

For the fibre optic cable service, Kerry pays a one-off amount of \$1200 for the installation costs and then a monthly fee of \$25. Write an equation of the total cost T of Kerry's new internet service over n months.

Worked Solution
Create a strategy

Multiply the monthly fee by the number of months, and then add it to the installation fee.

Apply the idea

\text{Total cost}=T ,\,\text{Monthly fee} = \$25 ,\, \text{Number of months}=n,\,\text{Installation fee}=\$1200

T = 1200 + 25 n

c

Fill in the table of values for the total cost of the current internet service, given by T=50n.

n16121824
T\text{ (dollars)}
Worked Solution
Create a strategy

Substitute the value of n in T=50n.

Apply the idea
\displaystyle T\displaystyle =\displaystyle 50\times 1Substitute n=1
\displaystyle =\displaystyle 50Evaluate
\displaystyle T\displaystyle =\displaystyle 50\times 6Substitute n=6
\displaystyle =\displaystyle 300Evaluate
\displaystyle T\displaystyle =\displaystyle 50\times 12Substitute n=12
\displaystyle =\displaystyle 600Evaluate
\displaystyle T\displaystyle =\displaystyle 50\times 18Substitute n=18
\displaystyle =\displaystyle 900Evaluate
\displaystyle T\displaystyle =\displaystyle 50\times 24Substitute n=24
\displaystyle =\displaystyle 1200Evaluate

The complete table of values for the total cost of the current internet service:

n16121824
T\text{ (dollars)}503006009001200
d

Fill in the table of values for the total cost of the fibre optic cable service, given by T=25n+1200.

n16121824
T\text{ (dollars)}
Worked Solution
Create a strategy

Substitute the value of n in T=25n + 1200.

Apply the idea
\displaystyle T\displaystyle =\displaystyle 25 \times 1 + 1200Substitute n=1
\displaystyle =\displaystyle 1225Evaluate
\displaystyle T\displaystyle =\displaystyle 25 \times 6 + 1200Substitute n=6
\displaystyle =\displaystyle 1350Evaluate
\displaystyle T\displaystyle =\displaystyle 25 \times 12 + 1200Substitute n=12
\displaystyle =\displaystyle 1500Evaluate
\displaystyle T\displaystyle =\displaystyle 25 \times 18 + 1200Substitute n=18
\displaystyle =\displaystyle 1650Evaluate
\displaystyle T\displaystyle =\displaystyle 25 \times 24 + 1200Substitute n=24
\displaystyle =\displaystyle 1800Evaluate

The complete table of values for the total cost of the fibre optic internet service:

n16121824
T\text{ (dollars)}12251350150016501800
e

Using the same set of axes, sketch a graph that corresponds to the total cost of Kerry's current internet service and the total cost of her new internet service.

Worked Solution
Create a strategy

Sketch the equation given in part (c) and part (d).

Apply the idea
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T
f

Using the graph from the previous question, determine how many months it will take for Kerry to break even on her new internet service.

Worked Solution
Create a strategy

Use the graph from part (e) and look for the point of intersection of the two lines.

Apply the idea

The point of intersection of the two lines is 48which means that it will take 48 months for Kerry to break even on her new internet service.

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T
Idea summary

The independent variable is always displayed in the first row of the table, followed by the dependent variable, in the second row.

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