Problem 45 - Algebra and Trigonometry

Question

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 isolated variable \(p\) in the given formula is \(p=(T-D)/m\).

Step 1: Identify the term with the variable to be isolated

In the formula \(T=D+p m\), \(p m\) is the term that contains the variable we want to isolate, which is \(p\).

Step 2: Remove the rest of the terms from the side of the equation

Subtract \(D\) from both sides of the equation to isolate the term \(p m\) on one side. This results in the equation: \(T-D=p m\).

Step 3: Solve for the specified variable

To isolate \(p\), divide both sides of the equation by \(m\). As a result, the equation becomes \(p=(T-D)/m\).

Key Concepts

Solving Equations

Solving equations is a method used in mathematics to find the value of unknown variables that make the equation true. In an equation, both sides are considered equal, and solving it involves reforming or simplifying the equation until the unknown number, or 'variable,' stands alone on one side of the equation.
For instance, in the equation \(T = D + pm\), we're tasked to find the value of \(p\) that holds the equation true for given values of \(T\), \(D\), and \(m\). Solving such equations typically involves:

• Moving unwanted terms from one side to the other
• Reversing operations like addition or multiplication

This process continues until all that remains is the variable itself, now solved or 'isolated' on one side, revealing its value.

Variable Isolation

In mathematics, isolating a variable is important to solve equations. It involves arranging the equation so that the variable you're interested in is by itself on one side of the equation.
Here's how you can do it:

• Identify the variable that needs to be isolated: In our equation \(T = D + pm\), the goal is to isolate \(p\).
• Remove other terms from the variable’s side: Subtract \(D\) from both sides to clear it from the equation \(T = D + pm\), resulting in \(T - D = pm\).
• Use inverse operations to separate the variable: Divide both sides by \(m\) to complete the isolation: \(p = \frac{T-D}{m}\).

The objective of these steps is to leave the target variable alone on one side of the equation, effectively solving for that variable.

Linear Equations

Linear equations are equations of the first degree, meaning they involve variables raised only to the power of one. They take the form \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable.
The example equation \(T = D + pm\) is linear in terms of \(p\) because it can be rearranged to show \(p\) isolated on one side, demonstrating its linear nature.
Properties of linear equations include:

• Consistent straight-line graphs
• Constant rate of change, slope for a line
• One intersection with axes when plotted

Understanding and solving linear equations is fundamental, as they form the basis for more complex algebraic expressions and models.