Problem 76 - Elementary Algebra

Question

Solve for the indicated variable. $$ \text { Solve for } C: \quad F=95 C+32 $$

Step-by-Step Solution

Answer

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

Step 1: Rearrange the equation

We start with the equation given in the problem: \[ F = 9/5 \cdot C + 32 \] To isolate \( C \), we need to move the constant term \( 32 \) to the other side. This is achieved by subtracting \( 32 \) from both sides, resulting in: \[ F - 32 = 9/5 \cdot C \]

Step 2: Solve for C

Now, we are left with \[ F - 32 = 9/5 \cdot C \]. To solve for \( C \), we need \( C \) by itself on one side of the equation. We do this by multiplying both sides of the equation by the reciprocal of \( 9/5 \), which is \( 5/9 \). This isolates \( C \) on the right side of the equation: \[ C = \frac{5}{9}(F - 32) \].

Key Concepts

Solving Linear Equations

Solving linear equations is one of the fundamental skills in algebra. It comes in handy across different mathematical problems, including those involving temperature conversion. When you're solving a linear equation, you're basically finding out what the variable equals, often denoted as 'x', 'C', or another letter. You will usually start with an equation that has the variable mixed with other numbers and terms. The goal is to get the variable all by itself on one side of the equation. Here's a quick walkthrough of the basic steps:

• Identify the variable you need to solve for in the equation.
• Use basic arithmetic to move any numbers from one side of the equation to the other. This usually involves adding, subtracting, multiplying, or dividing.
• Look for any fractions and deal with them by multiplying through by the denominator or using the reciprocal.

Keep practicing, and soon enough, solving linear equations will become second nature!

Variable Isolation

The term "variable isolation" refers to the process of manipulating an equation so that a specific variable stands alone on one side of the equation. This is a crucial step not only in algebra, but also in many scientific formulas. To isolate a variable, you'll often need to perform operations on both sides of the equation simultaneously. This ensures that the equation remains balanced.

In our example, isolating the variable involved moving the constant (32) and dealing with the fraction (9/5).

• First, identify what operations are applied to the variable and reverse them.
• Use addition or subtraction to get rid of constants on one side.
• If there's a coefficient (like 9/5), multiply by its reciprocal to isolate the variable.

Remember: whatever you do to one side of the equation, you must do to the other to keep it balanced. Practicing this technique can help you tackle more complex problems with confidence.

Temperature Conversion

Temperature conversion is a practical application of algebra that many students will encounter.

In particular, it's important to understand how to switch between Celsius and Fahrenheit, which are two very common temperature scales.The formula used in the example is: \[ F = \frac{9}{5}C + 32 \]Here, \( F \) stands for Fahrenheit and \( C \) stands for Celsius.To convert Fahrenheit to Celsius:

• Subtract 32 from the Fahrenheit temperature.
• Multiply the result by \( \frac{5}{9} \).

This conversion formula is invaluable in various scientific, weather-related, and everyday settings.

Understanding this algebraic manipulation helps make temperature conversion quick and easy in many situations.