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Chapter 2: Problem 46

Solve \(d=r t\) for \(r\).

Short Answer

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

Step by step solution

01

- Understand the given equation

The given equation is the formula for distance: d = r twhere d represents distance, r represents rate or speed, and t represents time.
02

- Isolate the variable r

To solve for r, we need to get r by itself on one side of the equation. Currently, r is being multiplied by t. To isolate r, we need to divide both sides of the equation by t.
03

- Divide both sides by t

Divide both sides of the equation by t: \( \frac{d}{t} = \frac{rt}{t} \)Since \( \frac{t}{t} \) equals 1, it simplifies to: \( \frac{d}{t} = r \)
04

- Write the final solution

The solution to the equation d = r t solved for r is: \( r = \frac{d}{t} \)

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Key Concepts

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

Isolating Variables
Isolating variables is an essential skill in algebra. It involves manipulating an equation to get the variable you're solving for by itself on one side of the equation. This makes it easier to understand what the variable represents and how it relates to other quantities in the equation.

To isolate a variable, you often need to use opposite operations. For example, if the variable is being added to another number, you would subtract that number from both sides of the equation. If it's being multiplied, you divide both sides by the same number. By methodically applying these opposite operations, you can simplify complex equations and solve for the unknown variable.
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying equations using various algebraic properties. This could include adding, subtracting, multiplying, or dividing both sides of an equation by the same number.

Let's go through a basic example: If you have the equation \(3x + 5 = 20\), and you want to solve for \(x\), you would:
  • Subtract 5 from both sides: \(3x = 15\)
  • Divide both sides by 3: \(x = 5\)
In this way, algebraic manipulation helps make equations more manageable to solve.
Remember to always perform the same operation on both sides of the equation to maintain the balance.
Distance-Rate-Time Formula
The distance-rate-time formula is a key concept in algebra and physics. It is given by the equation \(d = rt\), where:
  • \(d\) is the distance traveled
  • \(r\) is the rate or speed
  • \(t\) is the time taken
This formula can be rearranged to solve for different variables depending on the information available.

For example, to solve for rate \(r\): Divide both sides of the equation by \(t\): \( \frac{d}{t} = r \)

This rearrangement of formulas allows us to find missing values and can be applied to various real-world problems, such as calculating travel time or speed.

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Most popular questions from this chapter

For exercises 29-36, solve the equation for \(x\). Solve \(L=C-H-\frac{1}{5} T\) for \(C\).

For problems \(89-92\), do the arithmetic with a calculator. The area \(A\) of a rectangular playground is \(28,800 \mathrm{ft}^{2}\). The length \(L\) is \(180 \mathrm{ft}\). Solve the formula \(A=L W\) for \(L\), and use it to find the width \(W\) of the playground.

In the first five meets of the season, a collegiate gymnast had vault scores of \(9.625,9.750,9.825,9.725\) and \(9.550\). Find the score she needs in the next meet to have an average score of at least 9.700.

For exercises 53-62, (a) clear the fractions or decimals and solve. (b) check the direction of the inequality sign. $$ -0.98 p+4 \geq-0.18 p+8 $$

Find the enrollment at Oregon State University before the increase. Round to the nearest whole number. Oregon State University saw the biggest growth, a \(5.3\) percent jump of 1,302 students. Portland State University grew by only \(1.5\) percent, but remains the state's largest university with 28,958 students. (Source: www.oregonlive .com, Nov. 10. 2011)
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