Problem 84 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Formula

We are given the formula to solve: \( D = R T \). This is the equation that needs to be rearranged to solve for \( T \).

Step 2: Isolate T

To solve for \( T \), we need to rearrange the equation so that \( T \) is by itself on one side of the equation. Since \( T \) is multiplied by \( R \), we can divide both sides of the equation by \( R \). This gives us: \( T = \frac{D}{R} \).

Step 3: Conclusion

Now the formula is rearranged to solve for \( T \), so \( T = \frac{D}{R} \) is the solution. This means if you know the values of \( D \) and \( R \), you can substitute them into the formula to find \( T \).

Key Concepts

Rearranging Formulas

Rearranging formulas is all about changing the form of an equation to isolate and solve for a specific variable. This skill is vital in math and science as it enables you to approach and solve problems more flexibly. For instance, given a formula like \( D = RT \), our goal is to rewrite it so we can easily find the value of \( T \) when \( D \) and \( R \) are known.

• If you want to solve for a particular variable, identify that variable in the equation.
• Think of the equation as a balance. Whatever you do to one side, you must do to the other to keep it balanced.
• Apply basic arithmetic operations like addition, subtraction, multiplication, or division to rearrange the formula.

By rearranging formulas, you develop a deeper understanding of the relationships within the equations themselves.

Isolating Variables

Isolating variables involves manipulating an equation to get one variable alone on one side. To better illustrate, take the equation \( D = RT \), where we want to isolate \( T \). Here, \( T \) is currently multiplied by \( R \). To isolate it, divide both sides of the equation by \( R \), effectively separating \( T \) from \( R \).

• Understand what operations are being performed on the variable you want to isolate.
• Use inverse operations to reverse these actions until the variable stands alone.
• The equation should retain its meaning, so make sure each step maintains equality.

This practice strengthens your algebraic thought process, making it easier to solve varied mathematical problems.

Algebraic Manipulation

Algebraic manipulation refers to using algebraic operations to simplify or rewrite equations and expressions. This concept is often used when rearranging formulas or isolating variables. When transforming \( D = RT \) into \( T = \frac{D}{R} \), we're using algebraic manipulation to divide both sides by \( R \), maintaining the equation's equality.

• Ensure each manipulation respects the equality of the equation.
• Break down complex expressions into simpler parts to manage them more easily.
• Regularly check your work to avoid errors during manipulation.

Mastering algebraic manipulation is crucial for solving not just equations but also for tackling real-world problems using algebra.