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Chapter 8: Problem 70

Solve for the indicated variable. $$W=\frac{C A}{m} \text { for } m$$

Short Answer

Expert verified
In short, given the equation \(W = \frac{CA}{m}\), we can solve for the variable \(m\) by performing algebraic operations and obtain the result: \(m = \frac{CA}{W}\).

Step by step solution

01

Identify the equation

We are given the equation: \(W = \frac{CA}{m}\)
02

Multiply both sides by 'm'

To eliminate the fraction, we multiply both sides of the equation by the denominator 'm': \[mW = CA\]
03

Divide both sides by 'W'

To isolate 'm', we divide both sides of the equation by 'W': \[m = \frac{CA}{W}\] Now, the variable 'm' is isolated on one side of the equation and it is expressed in terms of the other variables: \[m = \frac{CA}{W}\]

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Key Concepts

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

Algebraic Manipulation
Algebraic manipulation is a fundamental skill in solving equations. It involves the use of various algebraic operations to simplify or rewrite equations so that they are easier to solve. In the given problem, we start with an equation where a variable, in this case, \(m\), is part of a fraction.
  • To tackle fractions, we often multiply by the denominator to eliminate it. This converts complex fractions into simpler equations.
  • Algebraic manipulation is about using properties of equality effectively, such as the fact that what you do to one side of an equation, you must do to the other.
By applying these operations, the original fraction becomes a simpler equation where \(m\) is more manageable. This skill is essential for rearranging equations into useful forms and for understanding the relationships between different variables.
Isolating Variables
Isolating a variable means getting it alone on one side of the equation. This process is crucial in solving literal equations, where we solve for one variable in terms of others. In our exercise:
  • We isolate \(m\) by performing operations that reverse the operations applied to it in the equation.
  • Initially, \(m\) is in the denominator of a fraction, so we multiply both sides by \(m\) to remove the fraction: \(mW = CA\).
  • Next, we need \(m\) alone, so we divide by \(W\): \(m = \frac{CA}{W}\).
These steps ensure that \(m\) is expressed in terms of \(C\), \(A\), and \(W\). Isolating the variable is all about reversing and undoing operations until you achieve that goal.
Literal Equations
Literal equations feature multiple variables and often require solving for one specific variable. These are common in formulas across different subjects.
  • Unlike typical numerical equations, solving literal equations involves algebraic manipulations to express one variable in terms of the others.
  • For the given exercise, \(W = \frac{CA}{m}\), our goal was to solve for \(m\).
  • In essence, we rearrange the equation to make one variable, \(m\), subject to the equation. This often involves several steps of algebraic manipulation.
Practicing with literal equations enhances understanding of how variables are related in an equation and the process of isolating a specific variable amidst others.

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Most popular questions from this chapter

Solve each equation. $$\frac{1}{6} j-j=-10$$

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Add or subtract, as indicated. $$\frac{a}{4 a^{2}-9}-\frac{4}{3-2 a}$$

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