Problem 60 - Algebra and Trigonometry with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

C = \frac{N}{D+1-P}, where D+1-P \neq 0.

Step 1: Identify the Terms Involving the Variable

We begin by identifying all the terms in the equation that involve the variable we want to solve for, which is \(C\). In the equation \(CD + C = PC + N\), the terms that contain \(C\) are \(CD\), \(C\), and \(PC\).

Step 2: Move All Terms with C to One Side

Rearrange the equation to have all terms containing \(C\) on one side. Subtract \(PC\) from both sides to obtain \(CD + C - PC = N\).

Step 3: Factor Out C

Notice that \(C\) is common in the left side of the equation. Factor \(C\) out to get \(C(D + 1 - P) = N\).

Step 4: Isolate C by Dividing

To solve for \(C\), divide both sides of the equation by \(D+1-P\). This yields \(C = \frac{N}{D+1-P}\).

Step 5: Consider the Domain

Ensure that \(D+1-P eq 0\) since division by zero is undefined.

Key Concepts

Variables

Variables are the letters we use in equations to represent numbers. In the equation \(CD + C = PC + N\), the letter \(C\) is our variable. It's a placeholder that can stand for various numbers, and we are trying to find its value.
Variables are important in algebra because they allow us to express relationships between numbers in a general way. This flexibility helps us solve problems in many contexts by simply plugging in different numbers for the variables.

Factoring

Factoring is a technique used to simplify algebraic expressions. It involves writing an expression as a product of its factors. In our equation \(CD + C - PC = N\), factoring helps to isolate our variable \(C\).
When you factor out \(C\), you are essentially reversing the distribution process. Initially, \(C\) is distributed over \(D\), \(1\), and \(-P\). By factoring it out, we express it in the form \(C(D+1-P)\), which simplifies the equation and makes it easier to solve for \(C\).

• Factoring can greatly reduce the complexity of equations.
• Always check if a common factor exists among the terms.
• Once factored, you can perform operations to isolate the variable.

Equation Manipulation

Equation manipulation involves rearranging and simplifying expressions to solve for a desired variable. This skill is fundamental in solving algebraic equations.
In the given problem, we manipulated the original equation \(CD + C = PC + N\) by subtracting \(PC\) from both sides. This moved all terms with \(C\) over to one side, setting the stage for factoring.
Effective equation manipulation requires a firm understanding of mathematical operations and their properties.

• Keep an equation balanced by performing the same operation on both sides.
• Use inverse operations like subtraction, addition, multiplication, and division wisely.
• Know when to simplify expressions to make equations easier to handle.

Domain Considerations

Domain considerations are crucial when dealing with equations, especially those involving division or square roots. The domain of an expression refers to the set of values that the variable can take, leading to real or permissible outcomes. In our solution, \(C = \frac{N}{D+1-P}\), we must ensure that division by zero does not occur.

This condition tells us that \(D+1-P eq 0\) because if it were zero, the fraction would become undefined. It's essential to analyze and state such restrictions upfront to avoid any mathematical errors.

• Always check for potential division by zero.
• Ensure all variables are defined within the context of the problem.
• Identify any restrictions or special conditions early in the solving process.

Understanding domain considerations ensures that our solutions are valid and applicable in real-life situations.