GM Lesson 051 Finding a Pronumeral in Linear Equations

Learning Intentions

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

• Find the value of a pronumeral in a linear equation.
• Use inverse operations to isolate the unknown.
• Check solutions by substitution.

Prerequisites

Students should already be able to:

• Substitute numerical values into algebraic expressions and formulae.
• Evaluate formulae using correct order of operations.
• Identify known and unknown quantities in a formula.
• Perform inverse operations, including addition/subtraction and multiplication/division.
• Work with (substitute into) common syllabus formulae such as:

Key Idea Summary

A linear equation contains an unknown pronumeral that is not raised to a power other than .

To find the value of the pronumeral, we use inverse operations to undo the equation step by step.

The goal is to isolate the unknown.

For example:

Since has been added, subtract from both sides:

Since has been multiplied by , divide both sides by :

A solution can be checked by substituting it back into the original equation.

Therefore, is correct.

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

Part A: One-step Linear Equations

A one-step equation needs only one inverse operation.

Worked Example 1

Find :

Subtract from both sides:

Check:

So, .

Worked Example 2

Find :

Divide both sides by :

Check:

So, .

Part B: Two-step Linear Equations

A two-step equation requires two inverse operations.

Undo addition or subtraction first, then undo multiplication or division.

Worked Example 3

Find :

Subtract from both sides:

Divide both sides by :

Check:

So, .

Worked Example 4

Find :

Add to both sides:

Multiply both sides by :

Check:

So, .

Part C: Finding a Pronumeral in a Syllabus Formula

In formula problems, some values are given and one pronumeral is unknown.

Substitute the known values first, then solve the resulting equation.

Worked Example 5: Simple Interest

The simple interest formula is:

where is simple interest, is principal, is the interest rate per year and is the number of years.

A loan earns simple interest of $ at an interest rate of per year for years.

Find the principal .

First write .

Substitute into the formula:

Simplify the right side:

Divide both sides by :

So, the principal was $ .

Check:

The answer is correct.

Worked Example 6: Circumference of a Circle

The circumference formula is:

A circular garden bed has circumference .

Find the radius , correct to decimal places.

Substitute into the formula:

Divide both sides by :

So, the radius is approximately .

Worked Example 7: Linear Function

The slope-intercept form of a linear function is:

Given , and , find .

Substitute the known values:

Subtract from both sides:

Divide both sides by :

Check:

So, .

Understanding Checks

Check 1

Find :

Check 2

Find :

Check 3

Find :

Check 4

Find :

given:

Check 5

A linear model is given by:

Find when .

Check 6

Explain why substitution can be used to check whether a solution is correct.

Exercises

Simple Familiar Exercises

Exercise 1

Find :

Exercise 2

Find :

Exercise 3

Find :

Exercise 4

Find :

Exercise 5

Find :

Exercise 6

Find :

Exercise 7

Find :

Exercise 8

Find :

Complex Familiar Exercises

Exercise 9

Find using:

where:

Exercise 10

Find using:

where:

Exercise 11

Find using:

where:

Give your answer correct to decimal places.

Exercise 12

Find using:

where:

Exercise 13

Find using:

where:

Exercise 14

Find using:

where:

Exercise 15

Find using:

where:

Exercise 16

Find using:

where:

Homework Problems

Homework 1

Find :

Homework 2

Find :

Homework 3

Find using:

where:

Homework 4

Find using:

where:

Homework 5

Find using:

where:

Give your answer correct to decimal places.

Homework 6

A linear cost model is:

Find when the total cost is $ .

Homework 7

A prism has volume:

The volume is and the base area is .

Find the height .

Homework 8

Solve and check by substitution:

Next: GM Lesson 052 Transposing Linear Formulae