GM Lesson 063 Choosing and Checking Linear Equation Models

Learning Intentions

By the end of this lesson, students will be able to:

• Decide whether a practical situation can be modelled by a linear equation.
• Solve the resulting linear equation accurately.
• Check whether the solution is reasonable for the original context.

Prerequisites

Students should already be able to:

• Solve one-step and two-step linear equations.
• Solve equations with variables on both sides.
• Translate simple worded descriptions into equations.
• Interpret solutions using units and context.

Key Idea Summary

A practical situation can often be modelled by a linear equation when there is:

• a fixed starting amount
• a constant rate of change
• an unknown quantity that is only multiplied by a constant and added or subtracted

A common linear model has the structure:

or

where:

• is the total amount
• is the fixed amount
• is the constant rate
• is the number of units

A solution should always be checked against the original context. A mathematically correct answer may still be unreasonable if it gives a negative time, a fraction of a person, too many items for the situation, or a value outside the possible range.

Direct Instruction and Worked Examples

Time Allocation

Time Allocation

• Introduction, warmup and vocabulary: 5 minutes
• Direct instruction: 15 minutes
• Understanding checks: 5 minutes
• Exercises: 20 minutes
• Homework: 20 to 30 minutes outside the lesson it was taught in.

Link to original

Modelling Decision Process

When solving a practical problem, use the following process:

1. Identify the unknown.
2. Decide whether the relationship is linear.
3. Define a variable.
4. Write a linear equation.
5. Solve the equation.
6. Check the solution in context.

A useful test for linearity is:

If a quantity repeatedly increases or decreases by the same amount, the model may be linear.

If a quantity repeatedly increases or decreases by the same percentage or factor, the model is usually not linear.

Worked Example 1: Choosing a Linear Model

A gym charges a joining fee of $ and then $ per week. Mia has paid a total of $ . How many weeks has she been a member?

Solution

The situation has a fixed fee and a constant weekly charge, so a linear model is appropriate.

Let be the number of weeks.

Solve:

Mia has been a member for weeks.

Check:

The answer is reasonable because weeks is a whole number of weeks and gives the correct total cost.

Worked Example 2: Checking Whether the Model is Reasonable

A plumber charges a call-out fee of $ plus $ per hour. A customer is charged $ . How long did the plumber work?

Solution

The situation has a fixed charge and a constant hourly rate, so a linear equation is appropriate.

Let be the number of hours worked.

Solve:

The plumber worked for hours.

Check:

The solution is reasonable because hours means hours and minutes, which is possible in this context.

Worked Example 3: Rejecting a Linear Model

A phone battery loses of its remaining charge each hour. The phone starts at charge. Can this situation be modelled by a linear equation?

Solution

If the phone loses of the remaining charge each hour, then the amount lost changes each hour.

After hour:

After hours:

The battery does not decrease by the same number of percentage points each hour.

The changes are:

and

Since the change is not constant, this situation is not linear. A linear equation is not an appropriate model.

Worked Example 4: Interpreting a Solution in Context

A school is hiring buses for an excursion. Each bus holds students. There are students attending. The school wants to know the minimum number of buses required.

A student writes:

where is the number of buses.

Solve and interpret the answer.

Solution

Mathematically, the solution is approximately buses.

However, the school cannot hire of a bus.

Since buses would only hold:

students, this is not enough.

The school must hire buses.

The checked contextual answer is:

The equation gives the calculation, but the context determines how the answer must be interpreted.

Understanding Checks

Check 1

A taxi fare has a flag fall of $ and costs $ per kilometre.

Is this situation linear? Explain why.

Expected response: Yes. There is a fixed starting cost and a constant cost per kilometre.

Check 2

A population doubles every day.

Is this situation linear? Explain why.

Expected response: No. The population changes by a constant multiplier, not a constant amount.

Check 3

Write an equation for this situation:

A streaming service charges $ per month. A customer has paid $ in total.

Expected response:

where is the number of months.

Check 4

Solve the equation from Check 3.

Expected response:

The customer has paid for months.

Check 5

A solution to a ticket problem gives , where is the number of people. What should you consider before giving the final answer?

Expected response: The number of people must be a whole number, so the decimal answer must be interpreted in context.

Exercises

Simple Familiar Exercises

Exercise 1

A car park charges $ entry plus $ per hour.

a. Explain why this situation is linear.
b. Write an equation for a total cost of $ .
c. Solve the equation.
d. Check whether your answer is reasonable.

Exercise 2

A mobile phone plan costs $ per month plus $ per extra text message. A customer is charged $ .

a. Define a variable.
b. Write a linear equation.
c. Solve the equation.
d. Interpret the answer in context.

Exercise 3

A plumber charges $ call-out fee plus $ per hour. The total bill is $ .

a. Write an equation.
b. Solve the equation.
c. Check your solution by substitution.

Exercise 4

A student saves $ each week. After some number of weeks, the student has saved $ .

a. Write a linear equation.
b. Solve the equation.
c. State the answer using appropriate units.

Exercise 5

For each situation, decide whether a linear equation is appropriate.

a. A worker earns $ per hour.
b. A car loses of its value each year.
c. A tank is filled at litres per minute.
d. A bacteria population triples every hour.
e. A venue charges $ hire fee plus $ per guest.

Complex Familiar Exercises

Exercise 6

A school formal venue charges a fixed booking fee of $ plus $ per student. The total cost is $ .

a. Explain why a linear model is appropriate.
b. Let be the number of students. Write an equation.
c. Solve the equation.
d. Check that the answer is reasonable for the context.

Exercise 7

A water tank initially contains litres of water. Water is drained at a constant rate of litres per minute.

a. Explain why this situation can be modelled linearly.
b. Write an equation for when the tank contains litres.
c. Solve the equation.
d. Interpret the answer.

Exercise 8

A concert ticket company charges $ booking fee plus $ per ticket. A group has a maximum budget of $ .

a. Write an equation for the exact number of tickets that would use the full budget.
b. Solve the equation.
c. Explain why the exact mathematical answer may not be the final practical answer.
d. Determine the maximum number of tickets the group can buy.

Exercise 9

A fitness challenge starts with a km walk in Week . Each week after that, the distance increases by km.

a. Is this situation linear? Explain.
b. Write a model for the distance in Week .
c. Find the week when the walking distance first reaches km.

Exercise 10

A delivery company charges by weight. The cost is $ for handling plus $ per kilogram.

A customer is charged $ .

a. Define the unknown.
b. Write and solve a linear equation.
c. Check the solution.
d. Explain whether the answer is reasonable if packages can be weighed to the nearest kg.

Homework Problems

Problem 1

A tutor charges $ for travel plus $ per hour. A family pays $ .

a. Write a linear equation.
b. Solve the equation.
c. Interpret the answer in context.

Problem 2

A savings account starts with $ . The owner deposits $ each week.

a. Explain why this situation is linear.
b. Write an equation for when the balance reaches $ .
c. Solve the equation.

Problem 3

For each situation, state whether a linear model is appropriate. Give a reason.

a. A cyclist travels at a constant speed of km/h.
b. A house increases in value by each year.
c. A candle burns down by cm per hour.
d. A prize pool is shared equally between different numbers of winners.

Problem 4

A bus company needs to transport students. Each bus holds students.

a. Write an equation to find the exact number of buses needed.
b. Solve the equation.
c. Explain why the exact answer must be adjusted.
d. State the minimum number of buses required.

Problem 5

A function room charges $ hire fee plus $ per guest. The total cost must not exceed $ .

a. Write an equation for the number of guests that would make the cost exactly $ .
b. Solve the equation.
c. Determine the greatest whole number of guests possible.
d. Check your answer in context.

Next: GM Lesson 064 Slope-Intercept Form