Solve 81/41-y+15=71/y

$$\frac { 81 } { 41 - y } + 15 = \frac { 71 } { y }$$

Solve for y

$$y = \frac{\sqrt{413629} + 767}{30} \approx 47.004665122$$

$$y = \frac{767 - \sqrt{413629}}{30} \approx 4.128668211$$

Steps Using the Quadratic Formula

Variable y cannot be equal to any of the values 0,41 since division by zero is not defined. Multiply both sides of the equation by y\left(y-41\right), the least common multiple of 41-y,y.

$$-y\times 81+y\left(y-41\right)\times 15=\left(y-41\right)\times 71$$

Multiply -1 and 81 to get -81.

$$-81y+y\left(y-41\right)\times 15=\left(y-41\right)\times 71$$

Use the distributive property to multiply y by y-41.

$$-81y+\left(y^{2}-41y\right)\times 15=\left(y-41\right)\times 71$$

Use the distributive property to multiply y^{2}-41y by 15.

$$-81y+15y^{2}-615y=\left(y-41\right)\times 71$$

Combine -81y and -615y to get -696y.

$$-696y+15y^{2}=\left(y-41\right)\times 71$$

Use the distributive property to multiply y-41 by 71.

$$-696y+15y^{2}=71y-2911$$

Subtract 71y from both sides.

$$-696y+15y^{2}-71y=-2911$$

Combine -696y and -71y to get -767y.

$$-767y+15y^{2}=-2911$$

Add 2911 to both sides.

$$-767y+15y^{2}+2911=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$15y^{2}-767y+2911=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 15 for a, -767 for b, and 2911 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-767\right)±\sqrt{\left(-767\right)^{2}-4\times 15\times 2911}}{2\times 15}$$

Square -767.

$$y=\frac{-\left(-767\right)±\sqrt{588289-4\times 15\times 2911}}{2\times 15}$$

Multiply -4 times 15.

$$y=\frac{-\left(-767\right)±\sqrt{588289-60\times 2911}}{2\times 15}$$

Multiply -60 times 2911.

$$y=\frac{-\left(-767\right)±\sqrt{588289-174660}}{2\times 15}$$

Add 588289 to -174660.

$$y=\frac{-\left(-767\right)±\sqrt{413629}}{2\times 15}$$

The opposite of -767 is 767.

$$y=\frac{767±\sqrt{413629}}{2\times 15}$$

Multiply 2 times 15.

$$y=\frac{767±\sqrt{413629}}{30}$$

Now solve the equation y=\frac{767±\sqrt{413629}}{30} when ± is plus. Add 767 to \sqrt{413629}.

$$y=\frac{\sqrt{413629}+767}{30}$$

Now solve the equation y=\frac{767±\sqrt{413629}}{30} when ± is minus. Subtract \sqrt{413629} from 767.

$$y=\frac{767-\sqrt{413629}}{30}$$

The equation is now solved.

$$y=\frac{\sqrt{413629}+767}{30} y=\frac{767-\sqrt{413629}}{30}$$

Steps for Completing the Square

Variable y cannot be equal to any of the values 0,41 since division by zero is not defined. Multiply both sides of the equation by y\left(y-41\right), the least common multiple of 41-y,y.

$$-y\times 81+y\left(y-41\right)\times 15=\left(y-41\right)\times 71$$

Multiply -1 and 81 to get -81.

$$-81y+y\left(y-41\right)\times 15=\left(y-41\right)\times 71$$

Use the distributive property to multiply y by y-41.

$$-81y+\left(y^{2}-41y\right)\times 15=\left(y-41\right)\times 71$$

Use the distributive property to multiply y^{2}-41y by 15.

$$-81y+15y^{2}-615y=\left(y-41\right)\times 71$$

Combine -81y and -615y to get -696y.

$$-696y+15y^{2}=\left(y-41\right)\times 71$$

Use the distributive property to multiply y-41 by 71.

$$-696y+15y^{2}=71y-2911$$

Subtract 71y from both sides.

$$-696y+15y^{2}-71y=-2911$$

Combine -696y and -71y to get -767y.

$$-767y+15y^{2}=-2911$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$15y^{2}-767y=-2911$$

Divide both sides by 15.

$$\frac{15y^{2}-767y}{15}=-\frac{2911}{15}$$

Dividing by 15 undoes the multiplication by 15.

$$y^{2}-\frac{767}{15}y=-\frac{2911}{15}$$

Divide -\frac{767}{15}, the coefficient of the x term, by 2 to get -\frac{767}{30}. Then add the square of -\frac{767}{30} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-\frac{767}{15}y+\left(-\frac{767}{30}\right)^{2}=-\frac{2911}{15}+\left(-\frac{767}{30}\right)^{2}$$

Square -\frac{767}{30} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-\frac{767}{15}y+\frac{588289}{900}=-\frac{2911}{15}+\frac{588289}{900}$$

Add -\frac{2911}{15} to \frac{588289}{900} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}-\frac{767}{15}y+\frac{588289}{900}=\frac{413629}{900}$$

Factor y^{2}-\frac{767}{15}y+\frac{588289}{900}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-\frac{767}{30}\right)^{2}=\frac{413629}{900}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{767}{30}\right)^{2}}=\sqrt{\frac{413629}{900}}$$

Simplify.

$$y-\frac{767}{30}=\frac{\sqrt{413629}}{30} y-\frac{767}{30}=-\frac{\sqrt{413629}}{30}$$

Add \frac{767}{30} to both sides of the equation.

$$y=\frac{\sqrt{413629}+767}{30} y=\frac{767-\sqrt{413629}}{30}$$