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Chapter 2: Problem 67

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

Short Answer

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

Step by step solution

01

Write down the equation

Start by writing down the given equation: \[d = r t\]
02

Isolate the specified variable

To solve for \(t\), we need to isolate it on one side of the equation. Since \(t\) is multiplied by \(r\), we can isolate \(t\) by dividing both sides of the equation by \(r\): \[\frac{d}{r} = \frac{r t}{r}\]
03

Simplify the equation

Simplify the right side of the equation: \[\frac{d}{r} = t\]

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Key Concepts

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

algebra
Algebra is a branch of mathematics that deals with variables and the rules for manipulating these variables. It allows for the formulation of equations to describe relationships between quantities. In the given problem, we start with the equation You will encounter letters (variables) which represent unknown quantities. Algebra helps you find the values of these unknowns. The equation represents a common relationship in physics involving distance ( d ), rate ( r ), and time ( t ). To navigate algebraic concepts, always remember:
  • Equations express relationships between variables and constants.
  • Operations performed on one side of the equation must be performed on the other to maintain equality.
In our problem, we use an algebraic operation to isolate the variable t to solve for time in terms of distance and rate.
isolating variables
Isolating variables is a fundamental skill in solving equations. When you isolate a variable, you rearrange the equation so that the variable of interest appears by itself on one side of the equation. This helps in understanding how other variables influence the one you're solving for. In the given problem, your task is to isolate t in the equation Initially, t is multiplied by r . To isolate t, follow these steps:
  • Identify the operation currently involving the variable (in this case, multiplication).
  • Perform the inverse operation on both sides of the equation (here, division).
By dividing both sides by r, we get: This step removes r from the right side, leaving t alone. This method can be applied whenever you need to isolate a variable, regardless of the complexity of the equation.
basic algebraic manipulation
Basic algebraic manipulation involves performing operations to simplify or solve equations. These operations include addition, subtraction, multiplication, and division. In our example, we use division to simplify the equation Here are essential points to keep in mind when manipulating equations:
  • Always perform the same operation on both sides of the equation to keep it balanced.
  • Combine like terms where possible to simplify your equation.
  • Use inverse operations to isolate variables (e.g., use division to cancel out multiplication).
After dividing both sides by r, the equation simplifies to: This final step involves simplifying r t / r to just t by cancelling out r. These basic algebraic manipulations make it possible to solve for any variable given an equation. Practicing these steps will make isolating variables and solving equations much easier.

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