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Chapter 1: Problem 37

Solve. \(v=\frac{d_{2}-d_{1}}{t},\) for \(d_{1}\)

Short Answer

Expert verified
\( d_{1} = d_{2} - v \times t \)

Step by step solution

01

Rearrange the equation

Start by multiplying both sides of the equation by the denominator of the right-hand side. This eliminates the denominator: \( v \times t = d_{2} - d_{1} \)
02

Isolate the term with \( d_{1} \)

Next, to isolate \( d_{1} \), move \( d_{2} \) to the other side of the equation by subtracting \( d_{2} \) from both sides: \( d_{2} - v \times t = d_{1} \)
03

Solve for \( d_{1} \)

Now, the equation is \( d_{1} = d_{2} - v \times t \) which gives the value of \( d_{1} \) in terms of \( d_{2} \), \( v \), and \( t \).

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Key Concepts

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

Solving For a Variable
When you're asked to solve for a variable, your goal is to isolate that variable on one side of the equation.
This often requires a series of algebraic manipulations.
Starting with a given equation, you will use operations like addition, subtraction, multiplication, and division.
These operations should be applied to both sides of the equation to maintain its balance.
In this exercise, we start with the equation \[v = \frac{d_{2} - d_{1}}{t}\] and we are solving for \(d_{1}\).
Rearranging Equations
Rearranging equations is a crucial step in algebraic manipulation.
The goal is to transform the equation in a way that makes it easier to isolate the desired variable.
In our example, we start by getting rid of the fraction.
We do this by multiplying both sides by the denominator \(t\):
\[ v \times t = d_{2} - d_{1}\]
This step helps to simplify the equation and makes later steps more straightforward.
You should focus on performing operations that will gradually isolate the variable you're solving for.
Isolating Terms
Isolating terms is the key to solving equations.
You achieve this by performing inverse operations to move unwanted terms to the other side of the equation.
In our example, after eliminating the fraction, we have:
\[ v \times t = d_{2} - d_{1}\]
To isolate \(d_{1}\), we subtract \(d_{2}\) from both sides:
\[ v \times t - d_{2} = -d_{1}\]
Finally, to isolate \(d_{1}\) completely, we multiply both sides by \(-1\), resulting in:
\[ d_{1} = d_{2} - v \times t\]
This step-by-step process ensures that we keep the equation balanced while isolating the desired variable.

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