Problem 83 - Algebra A Combined Function

Question

Solve. See the Concept Check in this section. Solve \(D=R T\) for \(R\)

Step-by-Step Solution

Answer

\(R = \frac{D}{T}\)

Step 1: Identify the Goal

We are asked to solve the formula \(D = RT\) for \(R\). This means we need to isolate \(R\) on one side of the equation.

Step 2: Isolate Variable R

To isolate \(R\), we need to divide both sides of the equation by \(T\). This is because \(R\) is currently multiplied by \(T\). The equation becomes:\[R = \frac{D}{T}\]

Step 3: Check the Solution

Review the rearranged formula to ensure that \(R\) is indeed isolated and the formula \(D = RT\) holds true. Substituting \(R\) back should give us \(D\) when multiplied by \(T\):\[D = \left(\frac{D}{T}\right) T = D\].

Key Concepts

Solving for a Variable

When you're solving for a variable in an equation, you're essentially trying to find out what value satisfies that equation. This usually involves rearranging the equation so that the variable in question is by itself on one side. Let's take the formula \(D = RT\) as an example. Here, we are asked to solve for \(R\). To do this, we need to manipulate the equation such that \(R\) stands alone.Think of it like peeling layers to get to the core. You "move" everything unnecessary to the other side of the equation, which in mathematical terms means performing inverse operations. Since \(R\) is multiplied by \(T\) in our example, dividing both sides by \(T\) will help us isolate \(R\). By dividing both sides by \(T\), we rearrange the formula to become:

• \(R = \frac{D}{T}\)

This new arrangement clearly shows \(R\) on its own, which is exactly what we wanted. This highlights the fundamental process of solving for a variable by using algebraic operations to isolate the variable.

Isolation of Variables

The concept of variable isolation is fundamental in algebra. It means altering an equation so that the variable you are interested in is all by itself on one side of the equals sign. This makes it easier to find the solution or to understand the relationship between different variables.In the equation \(D = RT\), the goal is to isolate \(R\). This requires us to "undo" the operation that's currently being performed on \(R\), which, in this case, is multiplication by \(T\). The inverse operation of multiplication is division, so you divide both sides by \(T\) to isolate \(R\):

• \(R = \frac{D}{T}\)

Once the variable is isolated, the new equation directly represents the relationship between the variables. Each side of the equation has the same value, and the variable is in its simplest form. Variable isolation is a powerful tool for solving equations, and it builds the foundation for more complex algebraic manipulations. Whether you're dealing with linear equations, polynomials, or even systems of equations, mastering the art of isolation is key.

Proportional Relationships

Proportional relationships describe a constant rate of change between two quantities. When two quantities are directly proportional, as one increases or decreases, the other does so as well at a constant rate. In an equation like \(D = RT\), the variables \(D\), \(R\), and \(T\) are part of a proportional relationship.This equation indicates that distance \(D\) is equal to the rate \(R\) multiplied by the time \(T\). If you solve for \(R\), you get \(R = \frac{D}{T}\), which reveals how the rate \(R\) is proportional to the distance divided by time.In essence, if you double the rate \(R\), while keeping time \(T\) constant, the distance \(D\) will double. Similarly, if you double the time \(T\) at a constant rate \(R\), the distance also doubles.Understanding these relationships allows you to predict outcomes and comprehend how changing one aspect of the equation affects the others. This kind of understanding is crucial in various fields, from physics and engineering to economics and biology.