Question

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

Short answer

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

Step-by-step solution

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 \(2y\).

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: \[ 8 - 10y = 5 \]

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 \(y\): \[ y = \frac{3}{10} \]

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.

Key concepts

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}\