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3.05 Interpreting linear relations

Lesson

To interpret information from a linear graph or equation, we can look at pairs of coordinates. Coordinates tell us how one variable relates to the other. Each pair has an $x$x-value and a $y$y-value in the form $\left(x,y\right)$(x,y).

  • The $x$x-value tells us the value of the variable on the horizontal axis.
  • The $y$y-value tells us the value of the variable on the vertical axis.

It doesn't matter what labels we give our axes, this order is always the same.

Intercepts

Two very important points on a graph are the $x$x and $y$y-intercepts. These are the points on the graph where the line crosses the $x$x and $y$y-axes respectively. These points usually have some significance in real life contexts.

The $y$y-intercept, represented by the constant term $c$c in a linear equation of the form $y=mx+c$y=mx+c, represents things such as a fixed cost, the starting distance from a fixed point, or the amount of liquid in a vessel at time zero.

Gradient

Another key feature is the gradient, represented by $m$m in a linear equation of the form $y=mx+c$y=mx+c. This is a measure of the slope, or steepness, of a line. The gradient is most commonly associated with the concept of rates. It can represent things like the speed of a vehicle, the rate of flow of a shower, or the hourly cost of a tradesperson.

Remember!

For all linear equations of the form $y=mx+c$y=mx+c

  • The gradient is represented by $m$m
  • The $y$y-intercept is represented by $c$c

We can use our knowledge of linear relations to get a better understanding of what is actually being represented.

Let's look at some examples and see this in action.

 

Worked example

Question 1

The number of eggs farmer Joe's chickens produce each day are shown in the graph. 

What does the point $\left(6,3\right)$(6,3) represent on the graph?

Think:  The first coordinate corresponds to the values on the $x$x-axis (which in this problem would represent the number of days) and the second coordinate corresponds to values on the $y$y-axis ( which in this problem would represent the number of eggs).

Do: 

Using the given information in context, we can interpret this point to mean that in $6$6 days the chickens will produce $3$3 eggs.

Reflect: How many days does it take for the chickens to lay $1$1 egg? If it takes $6$6 days to produce three eggs, we can find the time taken to lay one egg by evaluating $6\div3=2$6÷​3=2

Notice that this is not the gradient of the line graph, as the graph represents the number off eggs laid in terms of days. The gradient in this case is $3\div6=\frac{1}{2}$3÷​6=12.

 

Practice questions

question 1

The graph shows the temperature of a room after the heater has been turned on.

Loading Graph...

  1. What is the gradient of the line?

  2. What is the $y$y-intercept?

  3. Write an equation in the form $y=mx+c$y=mx+c to represent the temperature of the room, $y$y, in terms of the time, $x$x.

  4. What does the $y$y-intercept tell you?

    The temperature of the room before the heater has been turned on.

    A

    The time at which the temperature is $0^\circ$0°.

    B

    The temperature of the room $4$4 minutes after the heater has been turned on.

    C

    The amount by which the temperature of the room increases in the first minute.

    D

    The temperature of the room before the heater has been turned on.

    A

    The time at which the temperature is $0^\circ$0°.

    B

    The temperature of the room $4$4 minutes after the heater has been turned on.

    C

    The amount by which the temperature of the room increases in the first minute.

    D
  5. Find the temperature of the room after the heater has been turned on for $44$44 minutes.

question 2

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

Number of litres ($x$x) $0$0 $10$10 $20$20 $30$30 $40$40
Cost of petrol ($y$y) $0$0 $14.70$14.70 $29.40$29.40 $44.10$44.10 $58.80$58.80
  1. Write an equation relating the number of litres of petrol pumped, $x$x, and the cost of the petrol, $y$y.

  2. How much does petrol cost per litre?

  3. How much would $14$14 litres of petrol cost at this unit price?

  4. In the equation, $y=1.47x$y=1.47x, what does $1.47$1.47 represent?

    The total cost of petrol pumped.

    A

    The number of litres of petrol pumped.

    B

    The unit rate of cost of petrol per litre.

    C

    The total cost of petrol pumped.

    A

    The number of litres of petrol pumped.

    B

    The unit rate of cost of petrol per litre.

    C

question 3

The number of fish in a river is approximated over a five year period.

The results are shown in the following table.

Time in years ($t$t) $0$0 $1$1 $2$2 $3$3 $4$4 $5$5

Number of fish ($F$F)

$6600$6600 $6300$6300 $6000$6000 $5700$5700 $5400$5400 $5100$5100
  1. Choose the graph that corresponds to this relationship.

    Loading Graph...

    A

    Loading Graph...

    B

    Loading Graph...

    C

    Loading Graph...

    D

    Loading Graph...

    A

    Loading Graph...

    B

    Loading Graph...

    C

    Loading Graph...

    D
  2. Write down the gradient of the line.

  3. What does the gradient represent in this context?

    The decrease in the fish population over the five year period.

    A

    The average number of fish in the river at a particular time.

    B

    The rate of change of the fish population over the five year period.

    C

    The rate of change of the fish population each year.

    D

    The decrease in the fish population over the five year period.

    A

    The average number of fish in the river at a particular time.

    B

    The rate of change of the fish population over the five year period.

    C

    The rate of change of the fish population each year.

    D
  4. What is the value of $F$F when the line crosses the vertical axis?

  5. Write down an equation for the line, using the given values.

  6. Hence determine the number of fish remaining in the river after $12$12 years.

  7. We want to determine the number of years until $2700$2700 fish remain in the river.

    Substitute $F=2700$F=2700 into the equation and solve for $t$t.

Outcomes

VCMNA340

Solve linear equations involving simple algebraic fractions.

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