Problem 12 - Contemporary Precalculus

Question

Solve the equation for the indicated variable. $$\frac{1}{r}=\frac{1}{s}+\frac{1}{t} \quad \text { for } r$$

Step-by-Step Solution

Answer

Answer: \(r = \frac{st}{s+t}\).

Step 1: Find a common denominator

To combine the fractions \(\frac{1}{s}\) and \(\frac{1}{t}\), we need to find a common denominator which is the multiple of \(s\) and \(t\). In this case, the common denominator is \(s \cdot t\).

Step 2: Rewrite fractions

Rewrite the fractions with the common denominator: \(\frac{1}{r} = \frac{t}{st} + \frac{s}{st}\).

Step 3: Combine the fractions on the right side

Now that their denominators are the same, we can combine the fractions on the right side of the equation: \(\frac{1}{r} = \frac{s+t}{st}\).

Step 4: Cross-multiply

Cross-multiply the equation to get rid of the fractions. This gives us: \(r(s+t) = st\).

Step 5: Distribute and simplify

Distribute the \(r\) on the left side of the equation: \(rs+rt=st\).

Step 6: Isolate r

Group the terms that have the indicated variable, in this case \(r\): \(r(s+t)=st\). Then, isolate \(r\) by dividing both sides of the equation by the expression in parenthesis, \((s+t)\), to get the final answer: \(r=\frac{st}{s+t}\).

Key Concepts

Common Denominator

When dealing with fractions in equations, like in our example with the fractions \( \frac{1}{s} \) and \( \frac{1}{t} \), we need a uniform manner of combining them. This is where the concept of a common denominator is vital. A common denominator is a shared multiple of the denominators you're working with, allowing you to rewrite the fractions into a standard form.
Here's how it works:

• The denominators in our example are \( s \) and \( t \).
  
• A common denominator needs to be a number (or expression) that both \( s \) and \( t \) can multiply into without remainder.
  
• We find that by multiplying the two, \( s \cdot t \) becomes our common denominator.
  

This enables us to rewrite each fraction with the same denominator, which is crucial for further steps, like combining them into a single fraction. This step sets the stage for simplifying the equation efficiently and correctly.

Cross-Multiplication

After combining the fractions into a single term on one side of our equation, the next step is to tackle the fractions in the equation entirely. Cross-multiplication is an effective strategy here.
Cross-multiplying allows us to eliminate the fractions by equating the products of the terms across the equation.
The detailed breakdown is:

• Our equation after combining fractions is \( \frac{1}{r} = \frac{s+t}{st} \).
  
• By cross-multiplying, we multiply each side of the equation by the denominator of the other side.
  
• This results in \( r(s+t) = st \).
  
• Fractions disappear, revealing a clearer equation to solve for the variable.
  

Cross-multiplying provides a direct path to a simple expression, especially useful in variable isolation.

Variable Isolation

The ultimate goal in solving equations like this is to isolate the desired variable, which in initial steps may be tightly interwoven with other elements of the equation. Variable isolation involves rearranging the equation and performing operations that leave the indicated variable by itself on one side of the equation.
Let's see this process in action:

• From the cross-multiplication step, we have \( r(s+t) = st \).
  
• To isolate \( r \), we attempt to have \( r \) alone.
  
• We do this by dividing both sides of the equation by \( (s+t) \).
  
• This leads us to the solution \( r = \frac{st}{s+t} \).
  

Each step in variable isolation builds logically on the last, ensuring that calculations are precise and clear. The concept of isolating variables is fundamental in algebra and equation solving, allowing us to find solutions efficiently.