Problem 11 - Algebra 1: Concepts and Skills

Question

In Exercises 11 and \(12,\) use the temperature conversion formula \(F=\frac{9}{5} c+32\). Solve the formula for degrees Celsius \(C .\) Show all your steps.

Step-by-Step Solution

Answer

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

Step 1: Write down the provided formula

The provided temperature conversion formula is \(F=\frac{9}{5}C+32\).

Step 2: Isolate the term containing 'C'

To isolate 'C', first subtract 32 from each side of the equation. The result is \(F - 32 = \frac{9}{5}C\).

Step 3: Solve for 'C'

To solve for 'C', multiply each side of the equation by the reciprocal of \(\frac{9}{5}\), which is \( \frac{5}{9}\). The result is \(C = \frac{5}{9}(F - 32)\).

Key Concepts

Algebraic Manipulation

Algebraic manipulation involves rearranging equations and expressions to make them easier to work with. In the context of temperature conversion, we begin with the formula that converts Celsius to Fahrenheit: \( F = \frac{9}{5}C + 32 \). This equation tells us how to find the Fahrenheit temperature if we know the Celsius temperature. However, if we want to convert Fahrenheit back to Celsius, we need to rearrange this formula to solve for "C".

To do this, we perform algebraic operations that change the structure of the equation without changing its truth. In essence, we use basic arithmetic operations such as addition, subtraction, multiplication, and division to isolate the variable of interest. During this process, maintaining the balance of the equation is crucial; whatever action you apply to one side of the equation, you must also apply to the other side.

Solving Equations

Solving equations is a key skill in algebra that involves finding the value of the variable that makes the equation true. In the temperature conversion formula, our goal is to solve for "C". This task starts with the equation \( F = \frac{9}{5}C + 32 \).

We first need to remove the constant term on the right-hand side of the equation, which is "32". By subtracting 32 from both sides of the equation, it becomes \( F - 32 = \frac{9}{5}C \). This step is crucial as it helps in aligning terms with the variable we are solving for on one side of the equation. Through consistent strategies like these, we find values that satisfy the equation given specific conditions.

• Subtract equal amounts from both sides to maintain balance.
• Be methodical with each arithmetic operation to avoid mistakes.

Isolating Variables

Isolating variables is an essential technique when dealing with equations. It refers to the process of manipulating an equation so that the variable we are interested in is alone on one side of the equation. After rearranging our temperature equation to \( F - 32 = \frac{9}{5}C \), the next step is to isolate "C".

The presence of the coefficient \(\frac{9}{5}\) attached to "C" must be addressed. To isolate "C", one needs to multiply each side of the equation by the reciprocal of the coefficient, which in this case is \(\frac{5}{9}\). By doing this, those coefficients cancel out on the right-hand side, leaving "C" alone. The equation now becomes \( C = \frac{5}{9}(F - 32) \).

• Use reciprocal multiplication to cancel out fractions effectively.
• Double-check the equation to ensure the correct variable isolation.

Formula Derivation

Formula derivation refers to the process of creating a new formula by manipulating an existing one. In this case, starting with the Fahrenheit-to-Celsius conversion formula \( F = \frac{9}{5}C + 32 \), we derive a new form of the equation that allows us to convert temperatures from Fahrenheit back to Celsius.

The derived formula, \( C = \frac{5}{9}(F - 32) \), is essential for practical purposes such as understanding weather data from different regions or scientific studies that require temperature comparisons. The derivation process involves using consistent algebraic techniques such as isolating terms and solving for specific variables, ensuring the initial relationship between Celsius and Fahrenheit is preserved in the new format.

• Derivation helps create useful equations tailored to specific needs.
• This approach consolidates understanding of temperature relationships.