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Chapter 1: Problem 44

Solve the equation for the indicated variable. $$ V=C\left(1-\frac{n}{N}\right) ; n $$

Short Answer

Expert verified
The solution to the equation for the variable \(n\) is given by: $$ n = N - \frac{NV}{C} $$

Step by step solution

01

Expand the equation

First, expand the equation by distributing the constant \(C\) in the parenthesis: $$ V = C - \frac{Cn}{N} $$
02

Rearrange to isolate terms containing \(n\)

Next, rearrange the equation to isolate terms containing \(n\) on one side: $$ \frac{Cn}{N} = C - V $$
03

Remove the fraction

To remove the fraction, multiply both sides of the equation by \(N\): $$ CN = N(C - V) $$
04

Distribute the \(N\)

Distribute the \(N\) on the right side of the equation: $$ CN = NC - NV $$
05

Simplify the equation and solve for \(n\)

Notice that the C terms on both sides can be combined to give us a simplified equation. Divide both sides by \(C\): $$ N = N - \frac{NV}{C} $$ Next, solve for \(n\) by isolating it on one side of the equation: $$ n = N - \frac{NV}{C} $$ Now you have found the expression for \(n\) in terms of the other variables.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Variable Isolation
Variable isolation is a key step in solving algebraic equations. It involves moving all terms with the variable you are solving for onto one side of the equation. This simplifies the equation, making it easier to solve for the desired variable.
For example, consider the exercise equation \( V=C\left(1-\frac{n}{N}\right) \). We want to solve for \( n \). This means isolating the \( n \) term. Let's see how this is achieved:
  • First, expand the equation to \( V = C - \frac{Cn}{N} \), which separates the fraction from the constant \( C \).
  • Next, rearrange it to keep terms with \( n \) on one side: \( \frac{Cn}{N} = C - V \).
  • This rearrangement makes it clear which operations are needed to further isolate \( n \).
Understanding variable isolation helps in visualizing which operations need to be performed to simplify and ultimately solve the equation.
Fraction Elimination
Dealing with fractions in equations can be tricky, but eliminating them makes the equation easier to work with. When you have a fraction, a common strategy is to multiply both sides by the denominator. This cancels out the fraction.
In the example \( \frac{Cn}{N} = C - V \), let's remove the fraction by multiplying both sides by \( N \):
  • This gives \( CN = N(C - V) \), effectively removing the denominator.
  • Once fractions are eliminated, simpler arithmetic can be used to further solve the equation.
Multiplying by the denominator is a reliable method to simplify equations, especially when trying to isolate variables.
Equation Rearrangement
Rearranging an equation is crucial in the problem-solving process. This involves shifting terms around to group similar elements together, which helps in simplifying the equation significantly.
For this step, using the example equation after fraction elimination, \( CN = NC - NV \):
  • Simplify the expression \( CN = NC - NV \) by distributing and gathering like terms on one side.
  • In our case, isolating terms related to \( n \) helped streamline the solution process.
  • Solving for \( n \) involves simplifying further, leading to the formula \( n = N - \frac{NV}{C} \).
Rearranging equations brings clarity and sets a clear path for isolating and solving for variables.

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Most popular questions from this chapter

Solve the equation. $$ \frac{m}{m-2}-\frac{27}{7}=\frac{2}{m^{2}-m-2} $$

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