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Solve for \( y \). \[ \frac{y+5}{10 y}+\frac{3}{4}=\frac{1}{y} \] If there is more than one solution, separate them with commas. If there is no solution, click on "No solution". \( y=\square \)

Ask by Long Ruiz. in the United States
Jan 05,2025

Real Tutor Solution

Tutor-Verified Answer

Answer

\( y = \frac{10}{17} \)

Solution

Solve the equation \( \frac{y+5}{10y}+\frac{3}{4}=\frac{1}{y} \). Solve the equation by following steps: - step0: Solve for \(y\): \(\frac{y+5}{10y}+\frac{3}{4}=\frac{1}{y}\) - step1: Find the domain: \(\frac{y+5}{10y}+\frac{3}{4}=\frac{1}{y},y\neq 0\) - step2: Multiply both sides of the equation by LCD: \(\left(\frac{y+5}{10y}+\frac{3}{4}\right)\times 20y=\frac{1}{y}\times 20y\) - step3: Simplify the equation: \(17y+10=20\) - step4: Move the constant to the right side: \(17y=20-10\) - step5: Subtract the numbers: \(17y=10\) - step6: Divide both sides: \(\frac{17y}{17}=\frac{10}{17}\) - step7: Divide the numbers: \(y=\frac{10}{17}\) - step8: Check if the solution is in the defined range: \(y=\frac{10}{17},y\neq 0\) - step9: Find the intersection: \(y=\frac{10}{17}\) The solution to the equation is \( y = \frac{10}{17} \).

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The Deep Dive

To solve the equation \[ \frac{y+5}{10y} + \frac{3}{4} = \frac{1}{y}, \] we start by eliminating the fractions. To do this, we can multiply both sides by \( 40y \) (the least common multiple of the denominators): \[ 40y \left( \frac{y+5}{10y} \right) + 40y \left( \frac{3}{4} \right) = 40y \left( \frac{1}{y} \right). \] This simplifies to: \[ 4(y + 5) + 30y = 40. \] Expanding this gives: \[ 4y + 20 + 30y = 40. \] Combining like terms results in: \[ 34y + 20 = 40. \] Next, we isolate \( y \): \[ 34y = 40 - 20, \] \[ 34y = 20, \] \[ y = \frac{20}{34} = \frac{10}{17}. \] Thus, the solution is: \[ y = \frac{10}{17}. \]

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