Problem 61 - Intermediate Algebra

Question

Solve each formula for the specified variable. See Example 5. $$ v=\frac{1}{t}\left(d_{1}-d_{2}\right) \quad \text { for } t $$

Step-by-Step Solution

Answer

\( t = \frac{d_1 - d_2}{v} \)

Step 1: Understand the Formula

The given formula is \( v = \frac{1}{t}(d_1 - d_2) \) and we need to solve for \( t \). This means we want to rearrange the formula so that \( t \) is isolated on one side of the equation.

Step 2: Eliminate the Fraction

Multiply both sides of the equation by \( t \) to remove the fraction on the right side: \( v \cdot t = d_1 - d_2 \).

Step 3: Isolate \( t \)

Now solve for \( t \) by dividing both sides by \( v \): \( t = \frac{d_1 - d_2}{v} \). This step isolates \( t \) on the left side of the equation.

Key Concepts

Formula Rearrangement

Formula rearrangement is the process of making a different variable the subject of an equation. In this exercise, we started with the formula:

• \( v = \frac{1}{t}(d_{1} - d_{2}) \)
• and rearranged it to make \( t \) the subject.

To rearrange a formula, you must perform operations to both sides of the equation that will help you isolate the desired variable. In this case, our goal was to make \( t \) the subject, so it required reversing the operations around \( t \) one step at a time.

Start by identifying the operations applied to \( t \) in the original equation. Here, it's initially in the denominator, inside a multiplication by its reciprocal. By multiplying both sides of the equation by \( t \), we eliminated the fraction, making it easier to solve.
Sometimes it helps to note each operation's inverse effectively. If a variable is being divided, you multiply. If multiplied, you divide, and so on. Always apply the same operations to both sides of the equation to maintain equality. This systematic approach helps simplify complex equations and brings order to the algebraic manipulation you're performing.

Having a clear strategy for rearranging can greatly reduce the potential for errors and increase confidence in handling such equations.

Solving Equations

Solving equations involves finding the value of one variable in terms of the others present in the equation. The goal is to achieve an equation where the unknown variable is alone on one side. In our exercise, we started with:

• \( v \cdot t = d_{1} - d_{2} \)
• and aimed to solve for \( t \).

Each operation you perform is aimed at isolating the unknown variable. When solving for \( t \), we divided both sides by \( v \):

• \( t = \frac{d_{1} - d_{2}}{v} \).

This division works because \( v \) is being multiplied by \( t \), and dividing removes that relationship, isolating \( t \) on one side.
Double-checking your work helps assure you've performed the correct operations. With practice, solving equations becomes more intuitive, and understanding this process is key to tackling more advanced problems.
Once you grasp the steps, you can handle equations of any complexity.

Isolation of Variables

Isolation of a variable is about re-arranging an equation so that a single variable stands alone on one side. In our scenario, we aimed to isolate \( t \) when starting with:

• \( v \cdot t = d_{1} - d_{2} \)

The equation was rearranged to make \( t \) alone:

• \( t = \frac{d_{1} - d_{2}}{v} \).

Isolating \( t \) involved eliminating \( v \) from the left side by division.

The concept requires careful manipulation, like reversing any arithmetic operations affecting the variable of interest. To do this successfully, understand the inverse of arithmetic operations:

• Subtraction is the inverse of addition.
• Division is the inverse of multiplication.

By mastering isolation, you'll strengthen your ability to manipulate equations and solve various algebraic tasks efficiently.

This skill is fundamental to advanced math, as it lies at the core of constructing and solving equations accurately. Approach each problem methodically, ensuring you respect operations' rules while focusing on isolating your variable efficiently.