Problem 24 - Algebra for College Students

Question

The formula \(F=\frac{9}{5} C+32\) can be solved for \(C\) to obtain \(C=\frac{5}{9}(F-32)\). Which one of the following is not equivalent to this? A. \(C=\frac{5}{9} F-\frac{160}{9}\) B. \(C=\frac{5 F}{9}-\frac{160}{9}\) C. \(C=\frac{5 F-160}{9}\) D. \(C=\frac{5}{9} F-32\)

Step-by-Step Solution

Answer

Option D is not equivalent.

Step 1 - Understand the given expression for Fahrenheit to Celsius conversion

The provided formula to convert Fahrenheit (\(F\)) to Celsius (\(C\)) is: \[ F = \frac{9}{5}C + 32 \] We need to determine which of the provided expressions is not equivalent to the rearranged formula for \(C\).

Step 2 - Solve the formula for C

Start with the given formula and solve for \(C\): \[ F = \frac{9}{5}C + 32 \] Subtract 32 from both sides: \[ F - 32 = \frac{9}{5}C \] Multiply both sides by \(\frac{5}{9}\): \[ C = \frac{5}{9}(F - 32) \]

Step 3 - Expand the expression

Expand the right-hand side of the equation: \[ C = \frac{5}{9}F - \frac{5}{9} \times 32 \] Calculate \(\frac{5}{9} \times 32\): \[ \frac{5}{9} \times 32 = \frac{160}{9} \] Therefore, \[ C = \frac{5}{9}F - \frac{160}{9} \]

Step 4 - Compare with the given options

Compare the expanded formula \( C = \frac{5}{9}F - \frac{160}{9} \) with the options given: A. \( C = \frac{5}{9}F - \frac{160}{9} \) which matches our expanded formula. B. \( C = \frac{5}{9}F - \frac{160}{9} \) is equivalent to option A. C. \( C = \frac{5F - 160}{9} \) simplifies to \( C = \frac{5}{9}F - \frac{160}{9} \). D. \( C = \frac{5}{9}F - 32 \) which does not match our expanded equation and is thus the incorrect choice.

Key Concepts

Fahrenheit to Celsius conversion

Before diving into solving equations, it’s important to understand the basics of temperature conversion between Fahrenheit (°F) and Celsius (°C). The conversion formulas are crucial when dealing with these two temperature scales. The general formula to convert Fahrenheit to Celsius is: \[F = \frac{9}{5}C + 32\] This formula means that if you know the temperature in Celsius, you can find the temperature in Fahrenheit by multiplying by 9/5 and then adding 32. This relationship is based on how the scales are defined. Understanding this formula helps you see the linear relationship between Fahrenheit and Celsius.

Algebraic manipulation

To solve the temperature conversion formula for Celsius, we need to rearrange the equation using algebraic manipulation. Start with the given formula: \[F = \frac{9}{5}C + 32\] The goal is to isolate \(C\) on one side of the equation. Follow these steps to manipulate the equation:

• Subtract 32 from both sides: \(F - 32 = \frac{9}{5}C\)
• Multiply both sides by \(\frac{5}{9}\) to isolate \(C\): \(C = \frac{5}{9}(F - 32)\)

By performing these algebraic steps, you’ll have rearranged the original formula to solve for Celsius, which is handy when you only have the Fahrenheit temperature and need the Celsius value.

Equivalent expressions

Equivalent expressions are different algebraic expressions that represent the same value. In our problem, we derived \[C = \frac{5}{9}(F - 32)\] When expanded, this becomes: \[C = \frac{5}{9}F - \frac{160}{9}\] Equivalently, it can be rewritten in different forms by distributing and simplifying terms. For example:

• \( C = \frac{5F - 160}{9} \) simplifies to \( C = \frac{5}{9}F - \frac{160}{9} \)
• \( C = \frac{5F}{9} - \frac{160}{9} \) is another equivalent form

In algebra, recognizing these equivalent forms is essential when comparing and checking your solutions. It's also important to spot the non-equivalent expression, which in this case is option D: \( C = \frac{5}{9}F - 32 \).

Temperature conversion formulas

Different temperature scales use distinct formulas for conversion. The Fahrenheit to Celsius and Celsius to Fahrenheit conversions are the most common. Here's a quick recap:

• **Fahrenheit to Celsius**: \(F = \frac{9}{5}C + 32\) and solving for \(C\) gives \(C = \frac{5}{9}(F - 32)\)
• **Celsius to Fahrenheit**: Use the rearranged form \(F = \frac{9}{5}C + 32\)

Another handy conversion is finding Kelvin from Celsius, which requires adding 273.15: \[K = C + 273.15\] Knowing and understanding these temperature conversion formulas is key to solving a variety of problems in science and everyday contexts. Mastering them makes it easier to switch between temperature scales effortlessly.