Problem 17 - Algebra for College Students

Question

Solve each formula for the specified variable. $$\begin{aligned}&F=\frac{9}{5} C+32 \text { for } C\\\&\text { (Celsius to Fahrenheit) }\end{aligned}$$

Step-by-Step Solution

Answer

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

Step 1: Understand the formula

The given formula is used to convert Celsius (C) to Fahrenheit (F). It is given by: \[ F = \frac{9}{5}C + 32 \]

Step 2: Isolate the term involving C

To solve for C, first isolate the term involving C by subtracting 32 from both sides of the equation. \[ F - 32 = \frac{9}{5}C \]

Step 3: Solve for C

To isolate C, multiply both sides of the equation by \( \frac{5}{9} \): \[ C = \frac{5}{9}(F - 32) \]

Key Concepts

Solving Equations

When solving equations, we aim to find the value of the unknown variable that makes the equation true. This involves transforming the equation step by step while keeping it balanced by performing the same operation on both sides. Here's what you need to know to solve equations effectively:

• Understand the equation: First, identify the variable you need to solve for and understand the relationship between the variables given.
• Isolate the variable: Use fundamental algebraic operations like addition, subtraction, multiplication, and division to isolate the variable on one side of the equation.
• Simplify the equation: This might involve combining like terms, distributing coefficients, or reducing fractions.
• Check your work: Substituting your solution back into the original equation helps ensure accuracy.

Let's see this in action with our exercise on converting Celsius to Fahrenheit. We start with the formula given: \[ F = \frac{9}{5}C + 32 \] The goal is to isolate the variable C. By systematically using algebraic operations, we can find the formula to convert Fahrenheit back to Celsius.

Linear Equations

Linear equations are algebraic equations where the highest power of the variable is one. They can be written in the form: \[ ax + b = c \] where \(a\), \(b\), and \(c\) are constants. Linear equations can represent many practical scenarios, such as temperature conversion:

• Standard form: Convert the given formula into standard form if necessary. Our formula \(F = \frac{9}{5}C + 32\) is already in this format.
• Isolate the variable: Steps to isolate the variable involve reversing the operations done to the variable. We first subtract 32 from both sides: \[ F - 32 = \frac{9}{5}C \]
• Balance the equation: To solve for \(C\), we multiply both sides by the reciprocal of the coefficient of \(C\): \[ C = \frac{5}{9}(F - 32) \]

Linear equations are straightforward to solve since they involve simple arithmetic operations. Understanding these basics is key to solving any linear equation you might encounter.

Temperature Conversion

Temperature conversion is a common topic in both everyday life and various scientific fields. Different scales, such as Celsius, Fahrenheit, and Kelvin, are used to measure temperature, and converting between them requires specific formulas:

• Celsius to Fahrenheit: \[ F = \frac{9}{5}C + 32 \] This formula states that to convert from Celsius to Fahrenheit, you multiply by \(\frac{9}{5}\) and then add 32.
• Fahrenheit to Celsius: To find the inverse relationship, we rearrange our original formula and solve for \(C\): \[ C = \frac{5}{9}(F - 32) \]

Practical uses of these conversions are abundant, from cooking to weather forecasting, and scientific experiments. Remember to keep the steps clear in your mind:
1. Recognize the given temperature and target scale.
2. Use the appropriate formula and solve step by step.
With practice, converting temperatures between different scales becomes a quick and easy process!