Problem 61 - Precalculus: Functions and Graphs

Question

Solve for the specified variable. $$P+N=\frac{C+2}{C} \text { for } C$$

Step-by-Step Solution

Answer

C = \frac{2}{(P+N) - 1}

Step 1: Reorganize the Equation

Start with the original equation:\[P+N=\frac{C+2}{C}\]Reorganize to identify the term involving \(C\):\[(P+N) = \frac{C+2}{C}\].

Step 2: Clear the Fraction

Multiply both sides by \(C\) to eliminate the fraction:\[C(P+N) = C + 2\].

Step 3: Isolate Terms with C

Bring all terms involving \(C\) to one side of the equation:\[C(P+N) - C = 2\].

Step 4: Factor Out C

Factor \(C\) out of the left side of the equation:\[C((P+N) - 1) = 2\].

Step 5: Solve for C

Divide both sides by \((P+N) - 1\) to solve for \(C\):\[C = \frac{2}{(P+N) - 1}\].

Key Concepts

Variable Isolation

One of the core concepts in solving equations is isolating the variable. This is the process of transforming an equation so that a specific variable appears on one side of the equation by itself. In our example, we aim to solve for the variable \( C \).
When isolating the variable, follow these guidelines:

• Identify the variable you want to solve for – in this case, it's \( C \).
• Rearrange the equation so that all terms involving the variable are on one side of the equation and all other terms are on the opposite side.
• Once isolated, you can further simplify or rearrange the equation, if necessary.

By focusing on isolating \( C \), you make it easier to understand how changes to the other terms affect the solution. This simplification is critical when dealing with complex equations or multiple variables.

Equation Rearrangement

Rearranging an equation is a strategy used to simplify it and prepare it for further solving steps, like isolating the desired variable.
For the equation given by \( P + N = \frac{C+2}{C} \), rearranging entails:

• Moving terms around in a logical sequence to help with further simplification.
• Ensuring that like terms are grouped together, which can make the operations easier to handle.
• Making sure not to change the equality of the equation while shifting terms.

By moving terms with \( C \) to one side, you prepare it for operations like factoring, as seen in the solution, which eases the path to isolating the variable. Rearranging is crucial because it sets the stage for clear and straightforward calculation steps.

Fraction Elimination

Eliminating fractions is an important technique when dealing with algebraic equations. Fractions can make equations look more complicated, so removing them simplifies the solving process.
In our example, we have the equation \( P + N = \frac{C+2}{C} \), which includes a fraction involving \( C \).

• To eliminate this fraction, multiply both sides of the equation by the denominator (\( C \) in this case), which gives you a simpler, fraction-free equation to work with.
• This step helps convert the equation into a more accessible form, making it easier to proceed with isolating your variable.
• Be mindful of distributing multiplication correctly on both sides of the equation.

By removing the fraction, you can continue solving the equation without the extra complexity fractions introduce, leading to easier manipulation and solution of the equation.