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Chapter 4: Problem 48

Solve the equations by first clearing fractions. $$\frac{4}{5}=\frac{9}{15}+\frac{1}{3} n$$

Short Answer

Expert verified
n = \frac{3}{5}

Step by step solution

01

- Find the Least Common Denominator

Identify the least common denominator (LCD) for the fractions in the equation. The given fractions have denominators of 5, 15, and 3. The LCD of 5, 15, and 3 is 15.
02

- Clear the Fractions

Multiply every term in the equation by the LCD (15) to clear the fractions:\[\frac{4}{5} \times 15 = \frac{9}{15} \times 15 + \frac{1}{3} n \times 15\] which simplifies to\[12 = 9 + 5n\]
03

- Isolate the Variable

Subtract 9 from both sides to isolate the term with the variable:\[12 - 9 = 5n\] which simplifies to \[3 = 5n\]
04

- Solve for the Variable

Divide both sides by 5 to solve for n:\[n = \frac{3}{5}\]

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Key Concepts

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

Least Common Denominator
When solving linear equations that involve fractions, one helpful technique is to first identify the least common denominator (LCD). The LCD is the smallest number that each of the denominators can divide into evenly. Finding the LCD makes it easier to combine and manipulate fractions. In our example, the fractions have denominators of 5, 15, and 3. The LCD of these numbers is 15. By converting all fractions to have this common denominator, we create a foundation for clearing the fractions and simplifying the equation.
Clearing Fractions
Clearing fractions means eliminating them from the equation by using multiplication. This step is crucial because working with whole numbers often simplifies the process. After finding the LCD, multiply every term in the equation by this value. In the given example, we multiply \(\frac{4}{5}\), \(\frac{9}{15}\), and \(\frac{1}{3}n\) by 15. This operation removes the denominators, giving us:
\[12 = 9 + 5n\].
Now, the equation is free of fractions, and we can proceed with solving for the variable.
Isolating the Variable
Once the fractions are cleared, our next goal is to isolate the variable on one side of the equation. This usually involves moving other terms to the opposite side. In the example \[12 = 9 + 5n\], we subtract 9 from both sides to isolate the \(5n\) term. This operation simplifies the equation to:
\[3 = 5n\].
The variable n is now isolated with its coefficient, making it easier to solve for the variable in the next step.
Multiplying Fractions in Equations
The final step is often to solve for the variable by manipulating its coefficient. In our example, the equation \[3 = 5n\] means we need to find n. We do this by dividing both sides of the equation by the coefficient of the variable, which in this case is 5. Performing this step gives us:
\[n = \frac{3}{5}\].
This result is the solution to the original equation. Throughout these steps, understanding how to properly multiply fractions within equations helps maintain accuracy and simplifies the solution process.

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Most popular questions from this chapter

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