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Chapter 6: Problem 23

Solve. If no solution exists, state this. $$ \frac{1}{3 t}+\frac{1}{t}=\frac{1}{2} $$

Short Answer

Expert verified
\( t = \frac{8}{3} \)

Step by step solution

01

- Identify the common denominator

The common denominator for the fractions \( \frac{1}{3t} \) and \( \frac{1}{t} \) is \( 3t \).
02

- Rewrite fractions with the common denominator

Rewrite \( \frac{1}{3t} \) and \( \frac{1}{t} \) with the common denominator: \[ \frac{1}{3t} + \frac{3}{3t} = \frac{1 + 3}{3t} = \frac{4}{3t}. \]
03

- Set up the equation

Set the sum of the fractions equal to the right-hand side of the equation: \[ \frac{4}{3t} = \frac{1}{2}. \]
04

- Cross-multiply to solve for t

Cross-multiply to eliminate the fractions: \[ 4 \times 2 = 3t, \] resulting in \[ 8 = 3t. \]
05

- Solve for t

Divide both sides by 3 to isolate \( t \): \[ t = \frac{8}{3}. \]
06

- Check your solution

Substitute \( t = \frac{8}{3} \) back into the original equation to ensure it holds true: \[ \frac{1}{3 \cdot \frac{8}{3}} + \frac{1}{\frac{8}{3}} = \frac{1}{2}. \] This simplifies to \[ \frac{1}{8} + \frac{3}{8} = \frac{4}{8} = \frac{1}{2}, \] so the solution is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Denominator
When dealing with fractions, having a common denominator makes the equation easier to work with. In this problem, the fractions are \( \frac{1}{3t} \) and \( \frac{1}{t} \). To combine them, we use the common denominator, which is \( 3t \). Essentially, we look for a denominator that both denominators divide into.
It鈥檚 similar to finding the least common multiple (LCM). Here, the LCM of \( 3t \) and \( t \) is easily \( 3t \). Now, we rewrite each fraction with \( 3t \) as the denominator: \( \frac{1}{3t} \) stays the same and \( \frac{1}{t} \) becomes \( \frac{3}{3t} \). This step prepares us for adding the fractions together.
Fraction Addition
Once fractions have a common denominator, adding them becomes straightforward. From our common denominator step, we have:
\( \frac{1}{3t} + \frac{3}{3t} \)
We can directly add the numerators while keeping the denominator constant:
  • Numerator: \( 1 + 3 = 4 \)
  • Denominator: \( 3t \)
So, \( \frac{1}{3t} + \frac{3}{3t} = \frac{4}{3t} \). This simplified form lets us set it equal to the given fraction on the right side of the equation, \( \frac{1}{2} \).
Cross-Multiplication
Next, we solve the equation \( \frac{4}{3t} = \frac{1}{2} \) using cross-multiplication. This method helps eliminate the fractions by transforming the equation into a simpler form. Here's how it works:
We multiply the numerator of one fraction by the denominator of the other fraction:
  • \( 4 \times 2 = 8 \)
  • \( 3t \times 1 = 3t \)
This results in the equation \( 8 = 3t \), which is a linear equation and much easier to solve.
Isolating Variables
Our final goal is to solve for \( t \). From the cross-multiplication step, we have the equation \( 8 = 3t \). To isolate the variable \( t \), we need to get it alone on one side of the equation. We do this by dividing both sides by 3:
  • \( 8 eq 3 = 3t \)
  • \( \text{then} \frac{8}{3} = t \)
Thus, \( t = \frac{8}{3} \). Always check your solution by plugging it back into the original equation to ensure the left and right sides are equal. Substituting \( t = \frac{8}{3} \) back in verifies our solution as correct.

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