Problem 35 - Algebra Form and Function

Question

Solve for the indicated variable. Assume all constants are non-zero. $$ 2 r-t=r+2 t, \text { for } r $$

Step-by-Step Solution

Answer

Question: Given the equation 2r - t = r + 2t, solve for r, assuming all constants are non-zero. Solution: r = 3t

Step 1: Write down the given equation

We have the equation: $$ 2r - t = r + 2t $$

Step 2: Solve for r

To solve for r, we need to isolate r on one side of the equation. First, subtract r from both sides of the equation to get rid of r on the right side: $$ 2r - t - r = r + 2t - r $$ Which simplifies to: $$ r - t = 2t $$ Now, add t to both sides of the equation to isolate r: $$ r - t + t = 2t + t $$ Which simplifies to: $$ r = 3t $$

Step 3: Write the final answer

We have successfully isolated r and found the expression for r in terms of t: $$ r = 3t $$

Key Concepts

Solving Equations

Solving equations involves finding the value of unknown variables that make the equation true. For our exercise, the equation was \(2r - t = r + 2t\). There are several steps we follow:

• Identify the equation and variables involved.
• Determine what you need to solve for – in this case, the variable \(r\).
• Use algebraic operations to rearrange terms and solve for the variable.

To start solving, we first simplified the equation to bring like terms together. This step helps visualize what actions are needed to isolate the variable of interest. By carefully applying addition, subtraction, multiplication, and division operations, one can systematically approach the solution.

Variable Isolation

When isolating a variable, the goal is to have the variable alone on one side of the equation. This means moving all other terms to the opposite side. For the equation \(2r - t = r + 2t\), isolating \(r\) involves a few key steps:

• First, subtract \(r\) from both sides: \(2r - t - r = r + 2t - r\). This simplifies to \(r - t = 2t\).
• Next, add \(t\) to both sides to further isolate \(r\): \(r - t + t = 2t + t\). Simplifying gives us \(r = 3t\).

These steps demonstrate the principle of maintaining balance. Whatever operation you perform on one side of an equation, you must do to the other side as well. This ensures the equality holds true throughout the process.

Step-by-Step Solution

Step-by-step solutions are extremely helpful for students learning to solve equations. They break down each part of the process, making it easier to understand complex problems. In our original exercise, the steps included:

• Starting with the given equation: \(2r - t = r + 2t\).
• Applying operations systematically to isolate the variable \(r\).
• Writing out the simplified equation: \(r = 3t\).

By meticulously following each step, students can track what changes are made at every stage. This method builds strong problem-solving skills and enhances comprehension of algebraic principles, making it a valuable tool in learning mathematics.