Problem 50 - Precalculus: Functions and Graphs

Question

Solve the formula for the specifled variable. \(C D+C=P C+N\) for \(C\)

Step-by-Step Solution

Answer

\( C = \frac{N}{D + 1 - P} \)

Step 1: Rearrange the Equation

We start with the equation \( C D + C = P C + N \). Our goal is to solve for \( C \). First, get all terms involving \( C \) on one side of the equation. We can subtract \( PC \) from both sides: \( C D + C - PC = N \).

Step 2: Factor Out \( C \)

On the left-hand side of the equation, we have terms that include \( C \): \( C(D + 1) - PC = N \). Now, we can factor \( C \) out: \( C(D + 1 - P) = N \).

Step 3: Solve for \( C \)

Now, isolate \( C \) by dividing both sides of the equation by \( D + 1 - P \). This gives us \( C = \frac{N}{D + 1 - P} \).

Key Concepts

Variable Isolation

When solving equations, an important concept to understand is variable isolation. This means rearranging the equation so that the variable you are solving for stands alone on one side. For example, consider the equation: \( C D + C = P C + N \). Our goal here is to solve for \( C \).

The first step in variable isolation is to group all terms involving \( C \) on one side of the equation. This is often done by adding or subtracting terms from both sides. In our current example, we subtract \( P C \) from both sides to achieve this, which gives us \( C D + C - P C = N \).

By doing so, we have successfully isolated terms with the variable \( C \) for the next steps of factoring and solving.

Factoring Expressions

Once we have gathered all terms with the variable on one side, the next step is often to factor these expressions. Factoring helps to simplify the equation and prepare it for solving the isolated variable.

In our example, after rewriting \( C D + C - PC = N \), we noticed that both \( C D + C \) and \( -PC \) contain \( C \) as a common factor.

This allows us to factor \( C \) out of the expression, leaving us with \( C(D + 1 - P) = N \).

Factoring helps transform the equation into a simpler form, where you can then focus on isolating the variable \( C \) by performing additional operations.

Linear Equations

Understanding linear equations is fundamental when solving for a variable. A linear equation is an algebraic expression where the variable is only raised to the power of one.

In the equation we are dealing with: \( C(D + 1 - P) = N \), once we have isolated \( C \) as a factor, the equation remains linear because \( C \) is not raised to any power greater than one.

The primary goal with linear equations is to isolate the variable, which we can do by performing arithmetic operations like division.

For example, to solve for \( C \) in our equation, we divide both sides by \( D + 1 - P \), giving us the solution: \( C = \frac{N}{D + 1 - P} \).

Linear equations are simpler to solve compared to quadratic or higher-degree equations, making mastery of basic isolation and manipulation techniques highly beneficial.