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Chapter 7: Problem 197

Solve each rational equation. $$\frac{1}{a}+\frac{2}{5}=\frac{1}{2}$$

Short Answer

Expert verified
The solution is \( a = 10 \).

Step by step solution

01

Identify the Equation

Given equation is \( \frac{1}{a} + \frac{2}{5} = \frac{1}{2} \).
02

Isolate the term involving the variable

Subtract \( \frac{2}{5} \) from both sides of the equation: \( \frac{1}{a} = \frac{1}{2} - \frac{2}{5} \).
03

Find a common denominator

Rewrite \( \frac{1}{2} \) and \( \frac{2}{5} \) with a common denominator of 10: \( \frac{1}{2} = \frac{5}{10} \) and \( \frac{2}{5} = \frac{4}{10} \).
04

Perform the subtraction

Subtract \( \frac{4}{10} \) from \( \frac{5}{10} \): \( \frac{1}{a} = \frac{5}{10} - \frac{4}{10} = \frac{1}{10} \).
05

Solve for a

Take the reciprocal of both sides to solve for \( a \): \( a = 10 \).

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Key Concepts

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

Isolating the Variable
In any equation, isolating the variable is a critical step. This means you need to move all terms involving the variable to one side of the equation and everything else to the other side. In our example, we started with:
\( \frac{1}{a} + \frac{2}{5} = \frac{1}{2} \).
To isolate \( \frac{1}{a} \), we subtract \( \frac{2}{5} \) from both sides:
\( \frac{1}{a} = \frac{1}{2} - \frac{2}{5} \).
After isolating the variable, we can focus on simplifying the remaining equation.
Finding a Common Denominator
To perform operations with fractions, you need a common denominator. This makes addition and subtraction easier. For the equation \( \frac{1}{2} - \frac{2}{5} \), it's helpful to convert both fractions to have the same denominator.
For instance:
\( \frac{1}{2} = \frac{5}{10} \) and \( \frac{2}{5} = \frac{4}{10} \).
Now, both fractions have a common denominator of 10, allowing us to subtract them easily. This essential skill ensures your calculations are accurate and understandable.
Understanding Reciprocals
Reciprocals are simply flipping a fraction. For example, the reciprocal of \( \frac{1}{a} \) is \( a \). When you're solving rational equations, you often need to take reciprocals.
In our equation, the isolated form was \( \frac{1}{a} = \frac{1}{10} \).
By taking the reciprocal of both sides, we find:
\( a = 10 \).
Understanding the concept of reciprocals is crucial when dealing with rational equations.

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