Changing the subject of a formula involves rearranging it to solve for a different variable. This skill is crucial in algebra and enables students to manipulate equations to focus on the desired variable, maintaining the equation's integrity.

Introduction to Formula Rearrangement

The process of changing the subject in an equation is essential for solving problems where the variable of interest is not isolated. This involves using algebraic operations to rearrange the equation.

Techniques for Rearrangement

Key to changing the subject of a formula is understanding how to apply inverse operations to both sides of the equation to isolate the desired variable.

Example 1: Rearranging Area of a Triangle

Given the formula for the area of a triangle $A = \frac{1}{2}bh$, to make $h$ the subject:

1. Start with $A = \frac{1}{2}bh.$

2. Rearrange to isolate $h$: Multiply both sides by 2 and then divide by $b$.

Solution:

Example 2: Rearranging Speed Formula

Given the formula for speed $v = \frac{d}{t}$, to make $t$ the subject:

1. Start with $v = \frac{d}{t}$.

2. Rearrange to isolate $t$: Multiply both sides by $t$ and then divide by $v$.

Solution:

Example 3:  Converting Fahrenheit to Celsius

Given the formula relating the temperature Fahrenheit (°F): $°F = (\frac{9}{5})°C + 32$, change the subject to solve for the temperature in Celsius (°C).

Solution:

1. Isolate °C.

2. °C is multiplied by $\frac{9}{5}$ and then added to 32.

3. To isolate °C, perform the following steps:

• Subtract 32 from both sides to remove the constant term.
• Divide both sides by $\frac{9}{5}$ to isolate the term with °C.

$°F = (\frac{9}{5})°C + 32$

$°F - 32 = (\frac{9}{5})°C + 32 - 32$

$°F - 32 = (\frac{9}{5})°C$

$(°F - 32) × (\frac{5}{9}) = (\frac{9}{5})°C × (\frac{5}{9})$

$°C = (\frac{5}{9}) (°F - 32)$

Therefore, the new formula with °C as the subject is: $°C = (\frac{5}{9}) (°F - 32).$

Real-World Application

Physics Example: Finding Mass

Given $F = ma$, to find $m$, the mass:

1. Start with $F = ma$.

2. Rearrange to isolate $m$: Divide both sides by $a$.

Solution:

Chemistry Example: Finding Volume

Given concentration formula $C = \frac{m}{V}$, to find $V$, the volume:

1. Start with $C = \frac{m}{V}$.

2. Rearrange to isolate $V$: Multiply both sides by $V$ and then divide by $C$.

Solution: