Mr. Ward's Teaching Materials

GM Lesson 052 Transposing Linear Formulae

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GM Lesson 052 Transposing Linear Formulae

Learning Intentions

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

  • Transpose linear formulae to make a specified pronumeral the subject.
  • Apply equivalent operations to both sides of a formula.
  • Use rearranged formulae to solve practical problems.

Prerequisites

Students should already be able to:

  • Substitute values into formulae.
  • Evaluate expressions using correct order of operations.
  • Solve simple linear equations using inverse operations.
  • Check an answer by substituting it back into the original equation.
  • Recognise common syllabus formulae such as , , , and .

Key Idea Summary

A formula shows a relationship between quantities. To transpose a formula means to rearrange it so that a different pronumeral becomes the subject.

The subject of a formula is the pronumeral by itself on one side of the equation.

For example, in

is the subject.

To make the subject, divide both sides by :

The main rule is:

Whatever operation is applied to one side of a formula must also be applied to the other side.

Common inverse operations:

Operation in formulaInverse operation
AddSubtract
SubtractAdd
MultiplyDivide
DivideMultiply

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.
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1. Transposing by Undoing Multiplication

Consider the volume formula for a prism:

where is volume, is base area and is perpendicular height.

To make the subject:

Since is multiplied by , divide both sides by :

So:

Worked Example 1

A prism has volume and base area . Find the height.

Use:

Transpose first:

Substitute:

Therefore, the height is .

2. Transposing when there is Addition or Subtraction

The perimeter of a rectangle can be written as:

where is perimeter, is length and is width.

To make the subject:

Subtract from both sides:

Divide both sides by :

So:

Worked Example 2

A rectangle has perimeter and length . Find the width.

Use:

Transpose:

Substitute:

Therefore, the width is .

3. Transposing the Straight-line Formula

The slope-intercept form of a straight line is:

where is the gradient and is the -intercept.

To make the subject:

Subtract from both sides:

Divide both sides by :

So:

Worked Example 3

For the line

find when , and .

Use the transposed formula:

Substitute:

Therefore, .

4. Transposing a Financial Formula

The simple interest formula is:

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

To make the subject:

Since is multiplied by and , divide both sides by :

So:

Worked Example 4

An investment earns simple interest of $ over years at an annual interest rate of . Find the principal.

Use:

Transpose:

Convert the percentage to a decimal:

Substitute:

Therefore, the principal was $ .

Understanding Checks

Check 1

In the formula

which pronumeral is the subject?

Check 2

Transpose

to make the subject.

Check 3

Transpose

to make the subject.

Check 4

Transpose

to make the subject.

Check 5

A cylinder has volume

If is known and is unknown, what operation would isolate ?

Check 6

Explain why the following rearrangement is incorrect:

Check 7

A student writes:

Check the rearrangement by explaining which operation was used.

Exercises

Simple Familiar Exercises

Exercise 1

Transpose the formula

to make the subject.

Exercise 2

Transpose the formula

to make the subject.

Exercise 3

Transpose the formula

to make the subject.

Exercise 4

Transpose the formula

to make the subject.

Exercise 5

Transpose the formula

to make the subject.

Exercise 6

Transpose the formula

to make the subject.

Exercise 7

Transpose the formula

to make the subject.

Exercise 8

Transpose the formula

to make the subject.

Complex Familiar Exercises

Exercise 9

The perimeter of a rectangle is given by:

Transpose the formula to make the subject.

Exercise 10

The perimeter of a rectangle is given by:

Transpose the formula to make the subject.

Exercise 11

The volume of a prism is given by:

Find when and .

Exercise 12

The circumference of a circle is given by:

Find when . Use .

Exercise 13

The simple interest formula is:

Find when , and .

Exercise 14

The straight-line formula is:

Find when , and .

Exercise 15

The straight-line formula is:

Find when , and .

Exercise 16

The area of a parallelogram is given by:

Find when and .

Homework Problems

Problem 1

Transpose

to make the subject.

Problem 2

Transpose

to make the subject.

Problem 3

Transpose

to make the subject.

Problem 4

Transpose

to make the subject.

Problem 5

Transpose

to make the subject.

Problem 6

Transpose

to make the subject.

Problem 7

A prism has volume and base area .

Use

to find the height.

Problem 8

A circle has circumference .

Use

to find the radius. Use .

Problem 9

An investment earns simple interest of $ over years at an annual interest rate of .

Use

to find the principal.

Problem 10

The cost of a taxi trip is modelled by:

where is the cost in dollars and is the distance in kilometres.

Transpose the formula to make the subject.

Then find the distance travelled when the fare is $ .

Problem 11

The temperature conversion formula is:

Transpose the formula to make the subject.

Then find when .

Problem 12

A line is written as:

A point on the line is and the gradient is .

Transpose the formula to make the subject, then find the -intercept.

Next: GM Lesson 053 Finding a Pronumeral in Simple Non-linear Equations

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