 Hong Kong
Stage 1 - Stage 3

# Finding Linear Equations in Context

Lesson

Now that we know how

• to graph linear relationships
• to find the equations of linear functions
• to use algebra and graphs to extract information
• to find intercepts and constant values, and
• that the gradient of a linear function represents constant change.

we can put this to use to solve a range of real life applications.

It's all the same mathematics, but this time you will have a context to apply it to.

Some examples will be the best way to show you how the mathematics we have learnt can be applied to everyday situations.

#### Examples

##### Question 1

A carpenter charges a callout fee of $\$150$$150 plus \45$$45 per hour.

1. Write an equation to represent the total amount charged, $y$y, by the carpenter as a function of the number of hours worked, $x$x.

2. What is the gradient of the function?

3. What does this gradient represent?

The total amount charged increases by $\$45$$45 for each additional hour of work. A The minimum amount charged by the carpenter. B The total amount charged increases by \1$$1 for each additional $45$45 hours of work.

C

The total amount charged for $0$0 hours of work.

D
4. What is the value of the $y$y-intercept?

5. What does this $y$y-intercept represent?

Select all that apply.

The total amount charged increases by $\$150150 for each additional hour of work.

A

The maximum amount charged by the carpenter.

B

The callout fee.

C

The minimum amount charged by the carpenter.

D
6. Find the total amount charged by the carpenter for $6$6 hours of work.

##### Question 2

The table shows the linear relationship between the length of a mobile phone call and the cost of the call.

 Length of call (minutes) Cost (dollars) $1$1 $2$2 $3$3 $7.6$7.6 $14.4$14.4 $21.2$21.2
1. Write an equation to represent the cost of a call, $y$y, as a function of the length of the call, $x$x.

2. What is the slope of the function?

3. What does the slope tell you?

The connection fee

A

The cost of each additional minute

B

The cost of the phone

C

The cost of a $1$1-minute call

D
4. What is the $y$y-intercept?

5. What does this $y$y-intercept tell you?

The cost of each additional minute

A

The cost of the phone

B

The cost of a $1$1-minute call

C

The connection fee

D
6. Find the cost of a $6$6-minute call.

##### Question 3

The graph shows the amount of water remaining in a bucket that was initially full before a hole was made in its side.

1. What is the gradient of the function?

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

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

4. What does the slope tell you?

The amount of water remaining in the bucket after $2$2 minutes.

A

The amount of water that is flowing out of the hole every minute.

B

The time it takes for the bucket to be completely empty.

C

The time it takes the amount of water remaining in the bucket to drop by one litre.

D
5. What does the $y$y-intercept tell you?

The capacity of the bucket.

A

The amount of water remaining in the bucket after $30$30 minutes.

B

The amount of water remaining in the bucket when it is empty.

C

The size of the hole.

D
6. Find the amount of water remaining in the bucket after $54$54 minutes.