Problem 86 - Intermediate Algebra

Question

Solve for the specified variable. $$ I_{Q}=\frac{100 M}{C} \text { for } C $$

Step-by-Step Solution

Answer

\( C = \frac{100M}{I_Q} \)

Step 1: Understand the Equation

We have the equation \( I_Q = \frac{100M}{C} \). Our task is to solve for \( C \). This means we need \( C \) as the subject of the equation.

Step 2: Clear the Denominator

To isolate \( C \), first remove the fraction by multiplying both sides of the equation by \( C \). This gives \( C \cdot I_Q = 100M \).

Step 3: Solve for C

Now, we divide both sides by \( I_Q \) to solve for \( C \). This results in \( C = \frac{100M}{I_Q} \).

Key Concepts

Isolating Variables

One of the foundational skills in algebra is isolating a variable. This is crucial when you want to solve an equation for a specific variable, making it the "subject" of the equation. The term "isolating" essentially means getting the variable you're solving for on one side of the equation by itself. Here are some steps to effectively isolate a variable:

• Identify the variable you need to isolate. In our exercise, it was \( C \).
• Perform operations to "undo" what’s being done to the variable. Pay attention to operations like addition, subtraction, multiplication, and division.
• Remember to do the same operation on both sides of the equation. This maintains the equation's balance and ensures correctness.

In our example equation \( I_Q = \frac{100M}{C} \), we multiplied both sides by \( C \) to first remove \( C \) from the denominator, allowing us to then isolate it.

Manipulating Equations

Manipulating equations involves performing algebraic operations to both simplify and solve them. It's the art of rearranging terms to make solving easier. Key operations include adding, subtracting, multiplying, or dividing both sides of the equation by the same number or expression.

• Rearrange terms to get closer to the form you need. Start by dealing with terms directly affecting your target variable.
• Simplify as necessary. Look for opportunities to combine like terms or reduce fractions.
• Keep equations balanced. Every operation affects both sides, so ensure each side stays equivalent.

For example, when manipulating \( I_Q = \frac{100M}{C} \) to solve for \( C \), we multiplied both sides by \( C \) for simplification. Then, we divided by \( I_Q \) to isolate \( C \), resulting in \( C = \frac{100M}{I_Q} \). This manipulation leads directly to a clearer and more useful equation for \( C \).

Algebraic Fractions

Algebraic fractions are fractions that involve variables in their numerators, denominators, or both. Handling these fractions is quite similar to handling numerical fractions, but with added consideration due to the presence of variables.

• First, understand the components of the fraction, distinguishing between constant numbers and variables.
• Identify the operation you need to perform to modify the fraction. This might include clearing the fraction by multiplying through by the denominator.
• Be cautious of zero. Variables in denominators can't be zero, as division by zero is undefined.

In our example \( I_Q = \frac{100M}{C} \), we approached the problem of an algebraic fraction by multiplying both sides by \( C \) to eliminate the fraction. This straightforward step was essential in simplifying the equation and in making further manipulations to reach the solution. Understanding how to manage fractions with variables is vital for tackling more complex algebraic equations effectively.