Problem 70 - College Algebra

Question

In Exercises \(57-76,\) solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? \(S=P+P r t\) for \(t\)

Step-by-Step Solution

Answer

The solution for \(t\) in the equation \(S=P+Prt\) is \(t = \frac{S - P}{Pr}\) .

Step 1: Subtract P from both sides

To isolate \(t\), first isolate \(Prt\) by subtracting \(P\) from both sides of the equation. That gives \(S - P = P r t\).

Step 2: Divide by Pr

Next, to completely isolate \(t\), divide both sides of the equation by \(Pr\). That gives a final equation of \(t = \frac{S - P}{Pr}\) .

Key Concepts

Solving Equations

When faced with an equation, the aim is to find the value of an unknown variable. This process is known as solving equations.
The given equation is: - \( S = P + P r t \), and the task is to solve it for \( t \).
The initial step involves making the equation simpler by performing operations that help simplify the expression.
With the provided equation, the goal is to rewrite it in such a way that the variable \( t \) stands alone on one side.
By following thoughtful actions like subtraction or division, the equation reveals the value of the hidden variable. Each step peels away a layer, bringing clarity to the problem.
Equations are like balanced scales: whatever is done to one side must be done to the other to keep the balance, ensuring you don’t change the underlying truth.
By understanding this principle, solving equations can become an enjoyable puzzle to unravel.

Variable Isolation

Variable isolation refers to getting a variable alone on one side of an equation.
This helps determine its value without interference from other numbers or variables.- For the equation \( S = P + P r t \), the goal is to have \( t \) by itself on one side.- The first step is to simplify the equation by subtracting \( P \) from both sides, giving us \( S - P = P r t \).
The subtraction removes \( P \) from the right side of the equation, focusing attention on \( P r t \).To further isolate \( t \), divide each side by \( Pr \), arriving at the result \( t = \frac{S - P}{Pr}\).
Isolation helps to clarify what \( t \) equals by removing additional influences.

Linear Equations

Linear equations are a fundamental concept in algebra. They form a straight line when graphed on a coordinate plane. - The general form of a linear equation in two variables, say \( x \) and \( y \), is \( y = mx + c \), where \( m \) is the slope, and \( c \) is the y-intercept.- In our example, we have a linear equation with variables \( S \), \( P \), \( r \), and \( t \), working with constraints to isolate one of them. - Our task becomes simpler when realizing that despite having multiple variables, the goal remains to isolate \( t \) efficiently.Because linear equations have this predictable structure, solving them often involves straightforward arithmetic manipulations.
Recognizing a problem as a linear equation helps in deploying the right strategies to efficiently find the solution.