Problem 22 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(1-26,\) solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? $$S=P+P r t \text { for } t$$

Step-by-Step Solution

Answer

\(t = \frac{S - P}{Pr}\)

Step 1: Identify the variable you are solving for

We want to solve for \(t\), so this is the variable we are focussing on.

Step 2: Rearrange the equation to isolate \(t\)

First, move the \(P\) term to the other side of the equation. Subtract \(P\) from both sides to get \(S-P = Prt\). Next, isolate \(t\) on one side of the equation by dividing all terms by \(Pr\). So, \(t = \frac{S - P}{Pr}\). This way \(t\) is presented as a function of the other variables.

Key Concepts

Solving for a Variable

In algebra, one of the key skills is solving for a particular variable within an equation. This involves finding the value of the variable by performing operations that transform the equation. In the given exercise, the goal is to solve for the variable \(t\) in the formula \(S = P + Prt\). The aim is to express \(t\) explicitly using other given variables such as \(S\), \(P\), and \(r\). It's important to follow certain steps with care:

• Identify the variable to be solved. Here, it is \(t\).
• Perform operations that isolate \(t\) on one side of the equation.

This requires some rearranging of terms and perhaps dividing, so that \(t\) is the subject of the formula. Mastering this skill is fundamental in solving a wide range of algebra problems.

Linear Equations

Linear equations are foundational in mathematics and are one of the simplest forms of algebraic equations. They are equations of the first degree, which means the highest power of the variable is one. In this exercise, the equation \(S = P + Prt\) is a linear equation in terms of \(t\).
Linear equations generally represent a straight line when plotted on a graph. They have the standard form of \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants, while \(x\) can be any variable.
The process of solving means finding the particular value or values that make the equation true. Linear equations are straightforward in that the operations required to solve them are direct and involve basic arithmetic steps. Understanding linear equations allows one to move into more complex algebra efficiently.

Rearranging Equations

Rearranging equations is a necessary skill in algebra that involves changing the form of an equation without changing its equality. This process helps isolate a specific variable. In the formula \(S = P + Prt\), rearranging is used to solve for \(t\). Here’s how it works:

• First, the equation needs to be manipulated to get \(t\) by itself on one side.
• Start by subtracting \(P\) from both sides of the equation to help remove terms involving \(t\) from the original pile.
• Look at \(S - P = Prt\). The next step is to divide both sides by \(Pr\) to finally isolate \(t\).
• The result is \(t = \frac{S - P}{Pr}\).

Rearranging equations is crucial because it shows how different variables relate within the formula. This skill is used across many different types of math problems, helping to simplify and solve them effectively.