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Chapter 7: Problem 66

Solve. \(a=\frac{b}{c},\) for \(c\)

Short Answer

Expert verified
c = \frac{b}{a}

Step by step solution

01

Understand the equation

The given equation is \(a = \frac{b}{c}\). We need to solve this equation for \(c\).
02

Isolate the variable \(c\)

To isolate \(c\), multiply both sides of the equation by \(c\) to get rid of the fraction: \(a \cdot c = b\).
03

Solve for \(c\)

Now, divide both sides by \(a\) to solve for \(c\): \(c = \frac{b}{a}\).

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Key Concepts

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

isolating variables
When you solve equations, you often need to get a specific variable by itself. This process is called 'isolating the variable'. This means you'll perform operations to move other numbers or variables to the opposite side of the equation. First, identify the variable you need to isolate. For instance, in the equation \(a = \frac{b}{c}\), we aim to isolate \(c\). Next, use operations like addition, subtraction, multiplication, or division to rearrange the equation. Here, we multiply both sides by \(c\). So, we get \(a \cdot c = b\). Then, divide both sides by \(a\) to isolate \(c\), which gives us \(c = \frac{b}{a}\).
algebraic manipulation
Algebraic manipulation involves using basic arithmetic operations to rearrange equations and expressions. These operations include addition, subtraction, multiplication, and division. When manipulating equations, remember the key rule: whatever you do to one side of the equation, you must do to the other side. For example, in \(a = \frac{b}{c}\), if you multiply both sides by \(c\) to remove the fraction, you get \(a \cdot c = b\). This keeps the equation balanced. By understanding and applying these operations correctly, you can simplify and solve even complex equations.
rational equations
Rational equations are equations that involve fractions. These fractions, or ratios, have variables in the numerator, denominator, or both. An example of a rational equation is \(a = \frac{b}{c}\). Solving rational equations often requires finding a common denominator or eliminating the fraction altogether. In our example, we multiplied both sides of the equation by \(c\) to get rid of the denominator, giving us \(a \cdot c = b\). After this step, solving for the variable becomes easier. It's crucial to carefully perform each step to ensure accuracy when working with rational equations.

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