GM Lesson 047 Substitution into Linear Expressions

Learning Intentions

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

• Substitute numerical values into linear algebraic expressions.
• Evaluate linear expressions accurately.
• Interpret substituted values in practical contexts.

Prerequisites

Students should already be able to:

• Add, subtract, multiply and divide integers and decimals.
• Use the order of operations.
• Recognise pronumerals as symbols that represent numbers.
• Understand that multiplication may be written without a multiplication sign, such as .

Key Idea Summary

A linear expression contains a pronumeral with power , such as:

To substitute a value means to replace the pronumeral with a given number.

For example, if , then:

The expression can then be evaluated using correct order of operations.

Important notation:

• means .
• means .
• means .
• Substitute first, then evaluate.

Direct Instruction and Worked Examples

Lesson Timing

• Introduction and key vocabulary: 5 minutes
• Direct instruction and worked examples: 15 minutes
• Understanding checks: 5 minutes
• Exercises: 18 minutes
• Exit reflection: 2 minutes

Direct Instruction

A linear expression is an algebraic expression where the pronumeral is not squared, cubed or placed in a denominator.

Examples of linear expressions include:

To evaluate a linear expression:

1. Identify the value of the pronumeral.
2. Replace the pronumeral with that value.
3. Use order of operations to calculate the result.
4. Interpret the answer if the expression is in context.

Worked Example 1: Substituting into a Simple Linear Expression

Evaluate when .

Substitute :

Multiply first:

Add:

Therefore:

when .

Worked Example 2: Substituting into an Expression Involving Subtraction

Evaluate when .

Substitute :

Multiply first:

Subtract:

Therefore:

when .

Worked Example 3: Substituting a Negative Value

Evaluate when .

Substitute :

Multiply:

Add:

Therefore:

when .

Worked Example 4: Substitution in a Practical Context

A mobile phone plan has a fixed monthly fee of $ plus $ per gigabyte of data used.

The monthly cost can be represented by:

where is the monthly cost in dollars and is the number of gigabytes used.

Find the cost if .

Substitute :

Multiply:

Add:

The monthly cost is $ .

Worked Example 5: Interpreting a Linear Expression

A casual worker earns $ per hour and receives a fixed travel allowance of $ per shift.

The total pay for one shift is represented by:

where is the total pay in dollars and is the number of hours worked.

Find the pay for a shift of hours.

Substitute :

Multiply:

Add:

The worker earns $ for the shift.

Understanding Checks

Check 1

Evaluate when .

Check 2

Evaluate when .

Check 3

Evaluate when .

Check 4

A taxi fare is represented by:

where is the fare in dollars and is the number of kilometres travelled.

What does the represent in this context?

Check 5

Using:

find the fare for a trip of km.

Exercises

Simple Familiar Exercises

Exercise 1

Evaluate when .

Exercise 2

Evaluate when .

Exercise 3

Evaluate when .

Exercise 4

Evaluate when .

Exercise 5

Evaluate when .

Exercise 6

Evaluate when .

Exercise 7

Evaluate when .

Exercise 8

Evaluate when .

Complex Familiar Exercises

Exercise 9

Evaluate when .

Exercise 10

Evaluate when .

Exercise 11

Evaluate when .

Exercise 12

Evaluate when .

Exercise 13

Evaluate when .

Exercise 14

Evaluate when .

Exercise 15

A swimming pool charges an entry fee of $ plus $ per ride on the waterslide.

The total cost is:

where is the number of waterslide rides.

Find the total cost for rides.

Exercise 16

A delivery company charges $ plus $ per kilometre.

The delivery cost is:

where is the delivery distance in kilometres.

Find the cost of a km delivery.

Homework Problems

Homework 1

Evaluate when .

Homework 2

Evaluate when .

Homework 3

Evaluate when .

Homework 4

Evaluate when .

Homework 5

Evaluate when .

Homework 6

A taxi fare is represented by:

where is the fare in dollars and is the number of kilometres travelled.

Find the fare for a km trip.

Homework 7

A worker earns $ per hour and receives a meal allowance of $ per shift.

The total pay is:

Find the total pay for a hour shift.

Homework 8

A tank contains L of water. Water drains from the tank at L per minute.

The amount of water remaining is:

where is the number of minutes.

Find the amount of water remaining after minutes.

Homework 9

A concert ticket company charges a booking fee of $ plus $ per ticket.

The total cost is:

where is the number of tickets.

Find the cost of buying tickets.

Homework 10

Write a sentence explaining the difference between substituting into an expression and solving an equation.

Next: GM Lesson 048 Substitution into Simple Non-linear Expressions