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Chapter 2: Problem 68

Solve each formula for the specified variable. \(d=r t\) for \(r\)

Short Answer

Expert verified
r = \frac{d}{t}

Step by step solution

01

Identify the given formula

The given formula is d = r tWe need to solve for the variable r.
02

Isolate the variable r

To isolate r, divide both sides of the equation by t. Starting with d = r t,divide both sides by t to get:d/t = r.
03

Simplify the equation

The equation simplifies to:r = d/t.

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Key Concepts

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

Formula Manipulation
Manipulating formulas is essential in algebra. It allows us to rearrange equations so we can solve for any variable we need. For example, let's look at the formula from our exercise: \( d = rt \). Our goal is to make \( r \) the subject of the formula. This involves moving other terms to the opposite side of the equation. Understanding how to rearrange formulas helps us solve various algebraic problems more efficiently. The key steps include:
  • Identifying the variable to solve for
  • Performing algebraic operations to isolate this variable
  • Simplifying the final expression
Practicing these steps will make formula manipulation straightforward.
Isolating Variables
Isolating a variable means getting that variable by itself on one side of an equation. In our example, we want to solve for \( r \) in the equation \( d = rt \). Here's how we do it:
  • First, identify the variable you need to isolate. In this case, it's \( r \).
  • Next, perform operations to move other terms to the other side. We have \( t \) multiplying \( r \), so we divide both sides by \( t \): \[ d = rt \rightarrow \frac{d}{t} = r \]
  • This leaves you with \( r \) by itself: \( r = \frac{d}{t} \)
Always remember that whatever operation you do to one side of the equation, you must do to the other side as well. This maintains the balance of the equation. Isolating variables is a powerful skill for solving algebraic equations and is used frequently across math problems.
Algebraic Fractions
Algebraic fractions are fractions where the numerator, the denominator, or both are algebraic expressions. In our equation \( d = rt \), we end up with an algebraic fraction when isolating \( r \): \( r = \frac{d}{t} \). Here are the main points to remember:
  • Simplify fractions by canceling common factors in the numerator and denominator when possible.
  • Understand that dividing by a variable is the same as multiplying by its reciprocal.
  • When dealing with complex algebraic fractions, break them down step by step. This often involves factoring and simplifying both the numerator and the denominator.
Getting comfortable with algebraic fractions is crucial for solving various math problems. Whether you are working on simple equations or more complex ones, this skill helps in simplifying and solving equations more effectively.

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Most popular questions from this chapter

Write the answer to each problem in terms of the variable. See Example 11. A bank teller has \(t\) dollars, all in \(\$ 5\) bills. What expression represents the number of \(\$ 5\) bills the teller has?

Convert each percent to a decimal. $$ 9 \% $$

On a 75 -point algebra test, Grady scored 63 points. What percent of the total points did he score?

Write the answer to each problem in terms of the variable. See Example 11. A baseball player got 65 hits one season. He got \(h\) of the hits in one game. What expression represents the number of hits he got in the rest of the games?

Write each phrase as a mathematical expression using x as the variable. See Sections \(1.3,1.5\) \(1.6,\) and 1.8 The quotient of \(-6\) and a nonzero number
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