082. Using Formulas by Substitution and Solving

Learning Intentions

• To understand that formulas and rules are types of equation
• Apply a formula by substituting and evaluating
• apply a formula by substituting and solving an equation

Pre-requisite Summary

• Understand that an equation states that two expressions are equal
• Know that pronumerals can stand for different quantities in a rule or formula
• Know that a formula is a general rule written Use pronumerals
• Be able to Substitute a number for a pronumeral correctly using brackets when needed
• Be able to follow order of operations when evaluating expressions
• Be able to Solve one-step and two-step equations using inverse operations
• Understand that when solving a formula for an unknown quantity, the equation must remain balanced

Worked Examples

Worked Example 1

State whether each of the following is an equation, and Identify which are formulas or rules.

a)

b)

c)

Worked Example 2

Use the formula to Solve the perimeter when and .

Worked Example 3

Use the formula to find the simple interest when , , and .

Worked Example 4

Use the formula to find the height when and .

Worked Example 5

Use the formula to find the mass when and .

Worked Example 6

Use the formula to find when .

Problems

Problem 1

State whether each of the following is an equation, and identify which are formulas or rules.

a)

b)

c)

Problem 2

Use the formula to find the perimeter when and .

Problem 3

Use the formula to find the simple interest when , , and .

Problem 4

Use the formula to find the base when and .

Problem 5

Use the formula to find the acceleration when and .

Problem 6

Use the formula to find when .

Exercises

Understanding and Fluency

Exercise 1.

Decide whether each statement is an equation, and identify any formula or rule.

a)

b)

c)

Exercise 2.

Substitute and Evaluate.

Use .

a) ,

b) ,

c) ,

Exercise 3.

Substitute and evaluate.

Use .

a)

b)

c)

Exercise 4.

Substitute and evaluate.

Use .

a) ,

b) ,

c) ,

Exercise 5.

Substitute and solve.

Use .

a)

b)

c)

Exercise 6.

Substitute and solve.

Use .

a) Find when ,

b) Find when ,

c) Find when ,

Exercise 7.

Substitute and solve.

Use .

a) Find when ,

b) Find when ,

c) Find when ,

Exercise 8.

Use the formula .

a) Find when and

b) Find when and

c) Find when and

Reasoning

Exercise 9.

Explain why a formula such as is an equation.

Exercise 10.

A student substitutes and into and writes . Explain the error.

Exercise 11.

Compare these two tasks and explain the difference.

a) Use to find when

b) Use to find when

Exercise 12.

A student says that in the formula , if and , then . Explain why this is incorrect.

Problem-solving

Exercise 13.

The cost of hiring a bike is given by the rule , where is the number of hours. Find the cost for hours.

Exercise 14.

The volume of a rectangular prism is given by . Find the volume when , , and .

Exercise 15.

The formula for the total cost of movie tickets is , where is the number of tickets. If the total cost is $62, how many tickets were bought?

Exercise 16.

A triangle has area and base . Use to find the height.

Exercise 17.

A car travels according to . If the distance is km and the speed is km/h, how long did the trip take?

Exercise 18.

A science experiment uses the formula . If the force is N and the mass is kg, find the acceleration.

Potential Misunderstandings

• A student may think a formula is different from an equation because it uses several pronumerals
• A student may not recognise that formulas and rules show equality between two expressions
• A student may substitute values into the wrong pronumerals
• A student may forget to use brackets when substituting negative numbers
• A student may ignore order of operations when evaluating a formula
• A student may think substituting and evaluating is the same as solving for an unknown pronumeral
• A student may solve a substituted formula incorrectly by applying the wrong inverse operation
• A student may divide or subtract incorrectly when rearranging after substitution
• A student may think the answer to a solved formula must always be the pronumeral on the left side

Next: 083. Solving Real-World Problems with Equations