Problem 11 - Introductory Algebra for College Students

Question

Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? \(T=D+p m\) for \(D\)

Step-by-Step Solution

Answer

The given equation \(T=D+pm\) solved for \(D\), results in \(D=T-pm\).

Step 1: Identify the Equation

The given equation is \(T=D+pm\). We are asked to solve this equation for \(D\).

Step 2: Subtract pm from both sides

The goal is to isolate \(D\) on one side of the equation. To achieve this, subtract \(pm\) from both sides of the equation. This gives us a new equation, \(T-pm = D\).

Step 3: Final step

The equation \(T-pm = D\) is the equation solved for \(D\).

Key Concepts

Algebraic Equations

Algebraic equations are at the heart of solving mathematical problems, especially in algebra. These equations express a relationship between variables and constants. They consist of two expressions set equal to each other, with one or more unknowns to be solved for. In our specific problem, we have the equation

\( T = D + pm \).

It represents a linear relationship between the variables. The variable we wish to solve for is denoted by 'D,' while 'T' and 'pm' play the roles of the other elements in the relationship. By manipulating this equation, we can uncover the value of 'D' in terms of the other variables and constants. It is essential to understand the structure of algebraic equations because it allows us to handle a vast array of problem-solving situations. Whether it be in finance, physics, or even daily life concerns, knowing how to work with these equations empowers us to find solutions effectively.

Isolate Variables

To isolate a variable means to rearrange an algebraic equation so that the variable we are interested in stands alone on one side of the equals sign, with a coefficient of 1. The process of isolating variables is important because it allows one to solve for that specific variable, providing insights or answers to the problem at hand.

In our given exercise, we aimed to isolate 'D.' Beginning with \( T = D + pm \) and using algebraic manipulation, we subtracted 'pm' from both sides, leading us to the equation \( T - pm = D \), thus successfully isolating 'D.' This step—subtracting 'pm'—is a strategic move that balances the equation and retains its equality, allowing us to see that the value of 'D' is simply 'T' minus 'pm.' Knowing how to isolate variables opens the door to solving virtually any simple or complex algebra problem.

Algebraic Manipulation

The term algebraic manipulation refers to the process of rearranging and simplifying an equation using a variety of algebraic techniques such as adding, subtracting, multiplying, dividing, factoring, and expanding. These techniques are used to solve equations or to present them in a more useable form.

In our exercise, we engaged in algebraic manipulation when we subtracted 'pm' from both sides of our initial equation \( T = D + pm \), which yielded our solution \( T - pm = D \). The technique of 'doing to one side what we do to the other' ensures that the equation remains in balance. A solid grasp of algebraic manipulation is crucial in solving algebraic equations because it provides a methodical approach to isolating variables and simplifying expressions. The ability to manipulate equations with confidence can greatly ease the complexity of algebraic challenges.