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 using 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) $A=\pi r^2$
b) $3x+5$
c) $C=2\pi r$

Worked Example 2

Use the formula $P=2l+2w$ to find the perimeter when $l=8$ and $w=3$.

Worked Example 3

Use the formula $I=Prt$ to find the simple interest when $P=500$, $r=0.04$, and $t=3$.

Worked Example 4

Use the formula $A={\displaystyle \frac{1}{2}}bh$ to find the height when $A=30$ and $b=12$.

Worked Example 5

Use the formula $F=ma$ to find the mass when $F=45$ and $a=9$.

Worked Example 6

Use the formula $C=5n+2$ to find $n$ when $C=27$.

Problems

Problem 1

State whether each of the following is an equation, and identify which are formulas or rules.
a) $V=lwh$
b) $4a-7$
c) $d=vt$

Problem 2

Use the formula $P=2l+2w$ to find the perimeter when $l=10$ and $w=4$.

Problem 3

Use the formula $I=Prt$ to find the simple interest when $P=800$, $r=0.05$, and $t=2$.

Problem 4

Use the formula $A={\displaystyle \frac{1}{2}}bh$ to find the base when $A=24$ and $h=6$.

Problem 5

Use the formula $F=ma$ to find the acceleration when $F=56$ and $m=8$.

Problem 6

Use the formula $C=4n+3$ to find $n$ when $C=31$.

Exercises

Understanding and Fluency

1. Decide whether each statement is an equation, and identify any formula or rule.
   a) $A=\pi r^2$
   b) $2x+7$
   c) $d=vt$

2. Substitute and evaluate.
   Use $P=2l+2w$.
   a) $l=5$, $w=2$
   b) $l=9$, $w=4$
   c) $l=7$, $w=6$

3. Substitute and evaluate.
   Use $C=5n+1$.
   a) $n=2$
   b) $n=6$
   c) $n=10$

4. Substitute and evaluate.
   Use $A={\displaystyle \frac{1}{2}}bh$.
   a) $b=8$, $h=5$
   b) $b=12$, $h=7$
   c) $b=9$, $h=4$

5. Substitute and solve.
   Use $C=3n+4$.
   a) $C=13$
   b) $C=25$
   c) $C=34$

6. Substitute and solve.
   Use $F=ma$.
   a) Find $m$ when $F=24$, $a=6$
   b) Find $a$ when $F=35$, $m=5$
   c) Find $m$ when $F=72$, $a=8$

7. Substitute and solve.
   Use $A={\displaystyle \frac{1}{2}}bh$.
   a) Find $h$ when $A=20$, $b=5$
   b) Find $b$ when $A=18$, $h=6$
   c) Find $h$ when $A=35$, $b=10$

8. Use the formula $d=vt$.
   a) Find $d$ when $v=60$ and $t=3$
   b) Find $v$ when $d=150$ and $t=5$
   c) Find $t$ when $d=96$ and $v=12$

Reasoning

9. Explain why a formula such as $A=\pi r^2$ is an equation.

10. A student substitutes $l=4$ and $w=3$ into $P=2l+2w$ and writes $P=2+4+2+3$. Explain the error.

11. Compare these two tasks and explain the difference.
    a) Use $C=4n+1$ to find $C$ when $n=3$
    b) Use $C=4n+1$ to find $n$ when $C=13$

12. A student says that in the formula $F=ma$, if $F=20$ and $a=4$, then $m=16$. Explain why this is incorrect.

Problem-solving

13. The cost of hiring a bike is given by the rule $C=12h+5$, where $h$ is the number of hours. Find the cost for $4$ hours.

14. The volume of a rectangular prism is given by $V=lwh$. Find the volume when $l=6$, $w=3$, and $h=5$.

15. The formula for the total cost of movie tickets is $C=15n+2$, where $n$ is the number of tickets. If the total cost is $62, how many tickets were bought?

16. A triangle has area $24{\text{ cm}}^2$ and base $8\text{ cm}$. Use $A={\displaystyle \frac{1}{2}}bh$ to find the height.

17. A car travels according to $d=vt$. If the distance is $180$ km and the speed is $90$ km/h, how long did the trip take?

18. A science experiment uses the formula $F=ma$. If the force is $54$ N and the mass is $6$ 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