Problem 57 - Elementary and Intermediate Algebra

Question

Solve for the specified variable or expression. $$ r_{1} r_{2}=r r_{2}+r r_{1} \text { for } r_{1} $$

Step-by-Step Solution

Answer

\(r_{1} = \frac{r r_{2}}{r_{2} - r}\)

Step 1: Identify the Variable to Solve For

We need to solve the given equation for the variable \(r_1\). This means isolating \(r_1\) on one side of the equation.

Step 2: Analyze the Equation

The given equation is \(r_{1} r_{2} = r r_{2} + r r_{1}\). Notice that \(r_{1}\) appears in two places in the equation: in the first term on the left-hand side and the second term on the right-hand side.

Step 3: Factor Out the Target Variable

We can rearrange the terms to group those with \(r_1\) together. Start by subtracting \(r r_{1}\) from both sides:\[ r_{1} r_{2} - r r_{1} = r r_{2} \].

Step 4: Factor \(r_1\) from the Left Side

Now, notice \(r_1\) is common in both terms on the left side, allowing you to factor it out:\[ r_{1}(r_{2} - r) = r r_{2} \].

Step 5: Solve for \(r_1\)

To isolate \(r_1\), divide both sides of the equation by \((r_{2} - r)\):\[ r_{1} = \frac{r r_{2}}{r_{2} - r} \].

Key Concepts

Variable Manipulation

Understanding variable manipulation is essential when solving equations. It involves performing operations on equations to simplify or rearrange them, ultimately making it easier to isolate the desired variable. For example, in the exercise, we started with the equation \(r_{1} r_{2} = r r_{2} + r r_{1}\). Our goal was to solve for the variable \( r_1 \).
Variable manipulation techniques such as adding, subtracting, multiplying, or dividing both sides of an equation are powerful tools. They help maintain the equation's balance while moving or simplifying components.

• Adding or subtracting terms is used to "move" terms from one side to the other. Think of it like moving furniture—with care, everything stays in balance!
• Multiplying or dividing by the same amount on both sides preserves equality while simplifying terms.

Using these operations correctly helps you reorganize equations efficiently.

Isolating Variables

Isolating a variable means getting it alone on one side of the equation. This is the core step to solving for any specific variable. During this step, you aim to have the variable of interest, like \( r_1 \), by itself.
Consider this strategy as a way of "unwrapping" the variable from the layers of numbers and operations surrounding it. In our equation, \(r_{1} r_{2} = r r_{2} + r r_{1}\), we need to shift terms around to isolate \( r_1 \).
Here's the usual plan of action:

• Identify terms that need to be "moved" to the other side of the equation.
• Move these terms using addition or subtraction.
• Reorganize or simplify remaining terms, using multiplication or division if needed.

These steps lead to having the desired variable separated, ready for the final solution.

Factoring Equations

Factoring is a handy technique in mathematics, particularly in our equation when terms stay multiplied together. Factoring involves breaking down an expression into simpler "factors" that, when multiplied, give back the original expression.
In the equation \(r_{1} r_{2} - r r_{1} = r r_{2}\), we noticed that \( r_1 \) was present in two terms. This is a perfect scenario to apply factoring! Here's how you proceed:

• Identify a common factor in multiple terms, like \( r_1 \) in this case.
• "Factor out" the common term. It’s like grouping similar items together.

For example, when we rewrite it as \( r_{1}(r_{2} - r) = r r_{2} \), we see that \( r_1 \) is factored out, revealing how it operates with other expressions. Factoring reveals simpler equations which are usually easier to solve or manipulate further.