Problem 25 - Algebra for College Students

Question

$$ F=\frac{9}{5} C+32 \text { for } C $$

Step-by-Step Solution

Answer

The formula for \( C \) is \( C = \frac{5}{9}(F - 32) \).

Step 1: Identify the formula

The given formula is for converting Celsius temperatures to Fahrenheit: \[ F = \frac{9}{5}C + 32 \] We need to rearrange it to solve for \( C \).

Step 2: Isolate the Celsius term

Subtract 32 from both sides of the equation to focus on the term involving \( C \): \[ F - 32 = \frac{9}{5}C \]

Step 3: Solve for Celsius

To solve for \( C \), multiply both sides by the reciprocal of \( \frac{9}{5} \), which is \( \frac{5}{9} \):\[ C = \frac{5}{9}(F - 32) \] This rearranges the formula to solve for \( C \).

Key Concepts

Celsius to Fahrenheit

The process of converting Celsius to Fahrenheit is quite straightforward but integral to understanding degrees of heat measurement across different systems. The key formula used to convert a Celsius temperature to Fahrenheit is:\[ F = \frac{9}{5}C + 32 \]This formula helps to translate the Celsius degree into the Fahrenheit system, which is commonly used in the United States. Here’s how it works:

• Take the temperature in Celsius and multiply it by \( \frac{9}{5} \). This step converts the Celsius measurement into the basic scale used for Fahrenheit.
• Add 32 to the product of the above calculation. The "+32" accounts for the offset between the two temperature scales.

Now you have your Fahrenheit temperature. Whether you're following a science experiment or cooking your favorite recipe, understanding this conversion can be incredibly helpful!

Formula Manipulation

Manipulating mathematical formulas is a fundamental skill in problem-solving, especially for converting one unit to another. In this context, formula manipulation involves rearranging an equation to isolate and solve for a specific variable.In our exercise, the main equation is:\[ F = \frac{9}{5}C + 32 \]To rearrange this for the variable \( C \), we need to perform a series of algebraic operations. Here's how you can manipulate this equation:1. Subtract 32 from both sides to begin isolating \( C \): \[ F - 32 = \frac{9}{5}C \]2. Notice that the term \( \frac{9}{5}C \) means "\( C \) multiplied by \( \frac{9}{5} \)."3. To bring \( C \) alone, multiply both sides by the reciprocal of \( \frac{9}{5} \), which is \( \frac{5}{9} \). This cancels out \( \frac{9}{5} \) on the side of \( C \).Suddenly, you have:\[ C = \frac{5}{9}(F - 32) \]Through formula manipulation, you've reversed the equation’s role, and now you can readily convert Fahrenheit back to Celsius!

Solving Equations

Solving equations is the final step in both verification and application of formula manipulations. This concept involves finding the value of an unknown variable that turns the equation into a true statement. Once you've manipulated a formula, as shown above, solving it is a breeze!Consider the rearranged equation from our exercise:\[ C = \frac{5}{9}(F - 32) \]Steps to solve a practical example:1. **Plug in the value:** If you know the Fahrenheit temperature, say 77°F, substitute it into the equation: \[ C = \frac{5}{9}(77 - 32) \]2. **Perform arithmetic inside the parentheses first:** \[ 77 - 32 = 45 \]3. **Calculate the final value by multiplying the fraction:** \[ C = \frac{5}{9} \times 45 = 25 \]And there you have it! The degree Celsius is found.When solving equations, always remember to carry out operations in the correct order and double-check your work. This careful approach ensures that your solution is both accurate and reliable.