Problem 7 - Elementary and Intermediate Algebra

Question

a. Solve \(d=r t\) for \(t\) b. Solve \(I=P r t\) for \(P\)

Step-by-Step Solution

Answer

a. \( t = \frac{d}{r} \). b. \( P = \frac{I}{r t} \).

Step 1: Identify the Equation

The problem gives the equation \( d = r t \) and asks us to solve for \( t \).

Step 2: Isolate the Variable

Since \( t \) is multiplied by \( r \), divide both sides by \( r \) to isolate \( t \). This gives \( t = \frac{d}{r} \).

Step 3: Identify the Second Equation

The problem gives another equation \( I = P r t \) and asks us to solve for \( P \).

Step 4: Isolate the Variable

To solve for \( P \), divide both sides of the equation by \( r t \). This yields \( P = \frac{I}{r t} \).

Key Concepts

Solving Equations

Understanding how to solve equations is an essential skill in algebra and mathematics in general. Equations provide a way to express relationships between different variables. To solve an equation means to find the value of the unknown variable that makes the equation true. This process usually involves manipulating the equation using mathematical operations.There are some key steps you typically follow:

• Identify the equation and determine which variable to solve for.
• Use inverse operations to move terms around, simplifying the equation.
• Check the solution by substituting it back into the original equation.

Let's take an example: consider the equation \(d = rt\). If you need to find \(t\), you would rearrange the equation to isolate \(t\) on one side. This systematic approach is applicable to many types of algebraic equations, making it a fundamental concept to master.

Algebra Concepts

Algebra opens up the door to a deeper understanding of mathematical relationships. At its core, it involves using symbols, typically letters, to represent numbers and values in equations and expressions. This allows us to generalize arithmetic operations and find solutions to various problems.A few important algebra concepts include:

• Variables: These are symbols, like \(x\), \(y\), or \(t\), that stand in for unknown values.
• Constants: Fixed values that don't change, like \(d\), \(r\) in our examples.
• Coefficients: Numbers that multiply a variable, such as \(r\) in \(d=rt\).

Algebraic equations utilize these concepts to model real-world scenarios, helping us solve problems methodically. Recognizing these concepts and understanding their functions within equations is key to mastering algebra.

Variable Isolation

Variable isolation involves rearranging an equation so that the variable of interest is on one side by itself. This allows us to easily identify its value. In any algebraic equation, isolating a variable involves using inverse operations to "undo" operations applied to the variable.Here's how you can isolate a variable:

• Look at the operations performed on the variable and use the opposite operation to cancel them out. For instance, if the variable is multiplied by a number, divide by that number.
• Apply the same operation to both sides of the equation to maintain equality.
• Continue this process until the variable is by itself on one side of the equation.

In our example equation \(d=rt\), to isolate \(t\), you divide both sides by \(r\), resulting in \(t=\frac{d}{r}\). Similarly, in \(I=Prt\), solving for \(P\) means dividing by \(rt\), giving \(P=\frac{I}{rt}\). This step-by-step manipulation is a vital part of problem-solving in algebra.