Problem 65 - 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? \(T=D+p m\) for \(p\)

Step-by-Step Solution

Answer

The formula solved for \(p\) is \(p = (T-D) / m\).

Step 1: Rearrange the formula

To solve the equation \(T=D+p m\) for \(p\), it is necessary to first subtract \(D\) from both sides of the equation such that \(p m\) is on one side of the equation. This results in \(T-D = p m\).

Step 2: Solve for p

Next, divide both sides of the equation by \(m\) to isolate \(p\). This gives \(p = (T-D) / m\). So, \(p\) is equal to \(T-D\) divided by \(m\).

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra. It involves finding the value of unknown variables that satisfy the equation. The original exercise asked us to solve the formula \( T = D + pm \) for \( p \). Let's look at the step-by-step process again to make sure every part of it is crystal clear.

• We need to isolate \( p \) on one side to find its value.
• This begins by removing any terms that are not directly connected to \( p \).
• In our formula, \( D \) is the term we need to shift to the other side of the equation.
• To do this, subtract \( D \) from both sides, which undoes the addition and helps in isolating \( p \).

After subtracting \( D \), the equation becomes \( T - D = pm \). Now the left side is ready for the last step to finally solve for \( p \). You then need to perform one final operation: division.
Divide both sides by \( m \) to solve for \( p \), resulting in \( p = \frac{T - D}{m} \). This methodical process of isolating variables by performing inverse operations is key in solving equations.

Formulas

Formulas are expressions that define relationships between different quantities. Solving a formula often involves rearranging it to express one variable in terms of the others. This is what we focused on in the exercise with the formula \( T = D + pm \). Recognizing whether the formula has a practical application is useful as well.

• A formula like \( T = D + pm \) might describe a real-world situation.
• Try considering what each term might represent, such as \( T \) being a total amount, \( D \) a constant, and \( pm \) an additional amount based on some rate or multiplier.

Rearranging formulas is common when solving for a particular variable. By isolating \( p \) on one side, you can better understand how changes in \( T \), \( D \), or \( m \) will affect \( p \). This approach helps to systematically delve into complex scenarios by understanding the roles of each component in the formula.

Variables

Variables are symbols used in algebra to represent numbers. They allow you to formulate and solve problems even when some values are unknown. In our exercise, \( T \), \( D \), \( p \), and \( m \) are all variables, each representing different quantities. Grasping how to manipulate these variables is key to mastering algebra.

• Think of variables as placeholders or symbols that can take on different values.
• This flexibility allows formulas to be widely applicable in various contexts.

Understanding the role of a specific variable in an equation or formula is crucial. Each variable has a unique role; for instance, in \( T = D + pm \), \( p \) might be a price per item, \( m \) the number of items, enhancing the formula's practical context.
Hence, comprehending how changing one variable affects another helps to predict outcomes and make informed decisions based on algebraic models. With practice, you'll find manipulating and understanding variables in equations becomes intuitive.