Problem 11 - 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? $$T=D+p m \text { for } D$$

Step-by-Step Solution

Answer

The solution to the exercise is \( D = T - pm \).

Step 1: Identifying the given formula

First, let's identify the given formula. In this exercise, we have the formula \( T = D + pm \). In this formula, \( T \), \( D \) and \( m \) are variables and \( p \) is a constant. We're asked to solve this formula for \( D \). This means we need to isolate \( D \) on one side of the equation.

Step 2: Isolating the variable D

To isolate \( D \), we'll need to move \( pm \) to the other side of the equation. We do this by subtracting \( pm \) from both sides of the equation. Hence, the solution will be \( D = T - pm \).

Step 3: Checking the Isolated Variable

Finally, let's double-check the result. The goal was to isolate \( D \), and now \( D \) stands by itself on one side of the equals sign. The solution \( D = T - pm \) is therefore correct.

Key Concepts

Algebraic Manipulation

Algebraic manipulation refers to the variety of methods used to transform an algebraic equation or expression into a desired form. Often, this includes operations such as adding, subtracting, multiplying, and dividing both sides of an equation by the same number or expression to maintain equality. It can also involve factoring, expanding, and using properties of exponents.

For example, in the given exercise, the manipulation began with the equation \( T = D + pm \). The challenge was to express the formula in terms of \( D \). The process required subtracting \( pm \) from both sides to yield the simplified result \( D = T - pm \). This manipulation ensured that \( D \) was isolated, while also respecting the balance of the original equation.

Isolate Variable

To isolate a variable means to rearrange an algebraic equation so that the variable of interest is alone on one side of the equals sign. This is a fundamental skill in algebra that allows for the solution of equations and formulas for a specific variable.

In our exercise, we isolated the variable \( D \) by removing all other terms from its side of the equation. This was achieved by subtracting \( pm \) from both sides, which works based on the principle that what is done to one side of an equation must be done to the other to maintain equality. The goal is to have the isolated variable described explicitly in terms of the other variables or constants present in the equation, as demonstrated with \( D = T - pm \).

Algebraic Equations

Algebraic equations are mathematical statements that assert the equality of two expressions. They consist of variables, coefficients, constants, and arithmetic operators. Equations are the cornerstone of algebra and are used to describe relationships and solve problems involving unknown quantities.

The exercise provided us with an algebraic equation \( T = D + pm \) where \( T \), \( D \), and \( m \) are variables, and \( p \) is a constant, and we needed to solve for \( D \). Recognizing the type of algebraic equation is helpful for determining the method needed to isolate the variable. For instance, this particular equation is linear, and the techniques used involved simple addition and subtraction to solve for the desired variable.