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Chapter 1: Problem 44

Solve each equation, if possible. $$ \frac{4}{y}-5=\frac{5}{2 y} $$

Short Answer

Expert verified
The solution is \(y = \frac{3}{10}\).

Step by step solution

01

Identify a common denominator

The given equation is \(\frac{4}{y}-5=\frac{5}{2y}\). Notice that the denominators are \(y\) and \(2y\). The least common denominator (LCD) is \(2y\).
02

Eliminate the fractions

To eliminate the fractions, multiply both sides of the equation by the least common denominator, which is \(2y\). This gives: \[ 2y \times \frac{4}{y} - 2y \times 5 = 2y \times \frac{5}{2y} \] Simplify each term: \[ 8 - 10y = 5 \]
03

Solve for y

Rearrange the equation to isolate \(y\). Move the constant term to the other side by adding 10y to both sides: \[ 8 = 5 + 10y \] Next, subtract 5 from both sides: \[ 8 - 5 = 10y \] Simplify: \[ 3 = 10y \] Finally, divide both sides by 10 to find \(y\): \[ y = \frac{3}{10} \]
04

Verify the solution

Substitute \(y = \frac{3}{10}\) back into the original equation to check if it satisfies the equation. The original equation is: \[ \frac{4}{y} - 5 = \frac{5}{2y} \] Substitute \(y = \frac{3}{10}\): \[ \frac{4}{\frac{3}{10}} - 5 = \frac{5}{2 \times \frac{3}{10}} \] Simplify each fraction: \[ \frac{40}{3} - 5 = \frac{50}{6} \] Simplify \( \frac{50}{6} = \frac{25}{3} \): Compare both sides: \[ \frac{40}{3} - 5 = \frac{25}{3} \] \[ \frac{40}{3} - \frac{15}{3} = \frac{25}{3} \] This simplifies to: \[ \frac{25}{3} = \frac{25}{3} \] Since both sides are equal, 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
The first step in solving the given equation involves finding a common denominator for the fractions present. In the equation \frac{4}{y} - 5 = \frac{5}{2y} \, the fractions have denominators of \( y \) and \( 2y \). The least common denominator (LCD) is useful because it facilitates the elimination of fractions, simplifying the equation.
Here, the LCD is \( 2y \). Choosing the LCD helps ensure that, once multiplied through, the fractions disappear, allowing us to deal with simpler, whole-number terms.
Eliminating Fractions
To remove fractions from both sides of the equation, multiply every term by the least common denominator (LCD), which is \( 2y \):
\ 2y \times \frac{4}{y} - 2y \times 5 = 2y \times \frac{5}{2y}\

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Most popular questions from this chapter

Comparing Tablets The screen size of a tablet is determined by the length of the diagonal of the rectangular screen. The 12.9-inch iPad Pro \({ }^{\mathrm{TM}}\) comes in a 16: 9 format, which means that the ratio of the length to the width of the rectangular screen is \(16: 9 .\) What is the area of the iPad's screen? What is the area of a 12.3 -inch Microsoft Surface Pro \({ }^{\mathrm{TM}}\) if its screen is in a 3: 2 format? Which screen is larger? (Hint: If \(x\) is the length of a 4: 3 format screen, then \(\frac{3}{4} x\) is the width. \()\)

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