Solve 3y-8/5y-2=y-2/y+5

$$\frac { 3 y - 8 } { 5 y - 2 } = \frac { y - 2 } { y + 5 }$$

Solve for y

$$y=4$$

$$y = \frac{11}{2} = 5\frac{1}{2} = 5.5$$

Steps Using the Quadratic Formula

Variable y cannot be equal to any of the values -5,\frac{2}{5} since division by zero is not defined. Multiply both sides of the equation by \left(5y-2\right)\left(y+5\right), the least common multiple of 5y-2,y+5.

$$\left(y+5\right)\left(3y-8\right)=\left(5y-2\right)\left(y-2\right)$$

Use the distributive property to multiply y+5 by 3y-8 and combine like terms.

$$3y^{2}+7y-40=\left(5y-2\right)\left(y-2\right)$$

Use the distributive property to multiply 5y-2 by y-2 and combine like terms.

$$3y^{2}+7y-40=5y^{2}-12y+4$$

Subtract 5y^{2} from both sides.

$$3y^{2}+7y-40-5y^{2}=-12y+4$$

Combine 3y^{2} and -5y^{2} to get -2y^{2}.

$$-2y^{2}+7y-40=-12y+4$$

Add 12y to both sides.

$$-2y^{2}+7y-40+12y=4$$

Combine 7y and 12y to get 19y.

$$-2y^{2}+19y-40=4$$

Subtract 4 from both sides.

$$-2y^{2}+19y-40-4=0$$

Subtract 4 from -40 to get -44.

$$-2y^{2}+19y-44=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 19 for b, and -44 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-19±\sqrt{19^{2}-4\left(-2\right)\left(-44\right)}}{2\left(-2\right)}$$

Square 19.

$$y=\frac{-19±\sqrt{361-4\left(-2\right)\left(-44\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$y=\frac{-19±\sqrt{361+8\left(-44\right)}}{2\left(-2\right)}$$

Multiply 8 times -44.

$$y=\frac{-19±\sqrt{361-352}}{2\left(-2\right)}$$

Add 361 to -352.

$$y=\frac{-19±\sqrt{9}}{2\left(-2\right)}$$

Take the square root of 9.

$$y=\frac{-19±3}{2\left(-2\right)}$$

Multiply 2 times -2.

$$y=\frac{-19±3}{-4}$$

Now solve the equation y=\frac{-19±3}{-4} when ± is plus. Add -19 to 3.

$$y=-\frac{16}{-4}$$

Divide -16 by -4.

$$y=4$$

Now solve the equation y=\frac{-19±3}{-4} when ± is minus. Subtract 3 from -19.

$$y=-\frac{22}{-4}$$

Reduce the fraction \frac{-22}{-4} to lowest terms by extracting and canceling out 2.

$$y=\frac{11}{2}$$

The equation is now solved.

$$y=4 y=\frac{11}{2}$$

Steps for Completing the Square

Variable y cannot be equal to any of the values -5,\frac{2}{5} since division by zero is not defined. Multiply both sides of the equation by \left(5y-2\right)\left(y+5\right), the least common multiple of 5y-2,y+5.

$$\left(y+5\right)\left(3y-8\right)=\left(5y-2\right)\left(y-2\right)$$

Use the distributive property to multiply y+5 by 3y-8 and combine like terms.

$$3y^{2}+7y-40=\left(5y-2\right)\left(y-2\right)$$

Use the distributive property to multiply 5y-2 by y-2 and combine like terms.

$$3y^{2}+7y-40=5y^{2}-12y+4$$

Subtract 5y^{2} from both sides.

$$3y^{2}+7y-40-5y^{2}=-12y+4$$

Combine 3y^{2} and -5y^{2} to get -2y^{2}.

$$-2y^{2}+7y-40=-12y+4$$

Add 12y to both sides.

$$-2y^{2}+7y-40+12y=4$$

Combine 7y and 12y to get 19y.

$$-2y^{2}+19y-40=4$$

Add 40 to both sides.

$$-2y^{2}+19y=4+40$$

Add 4 and 40 to get 44.

$$-2y^{2}+19y=44$$

Divide both sides by -2.

$$\frac{-2y^{2}+19y}{-2}=\frac{44}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$y^{2}+\frac{19}{-2}y=\frac{44}{-2}$$

Divide 19 by -2.

$$y^{2}-\frac{19}{2}y=\frac{44}{-2}$$

Divide 44 by -2.

$$y^{2}-\frac{19}{2}y=-22$$

Divide -\frac{19}{2}, the coefficient of the x term, by 2 to get -\frac{19}{4}. Then add the square of -\frac{19}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-\frac{19}{2}y+\left(-\frac{19}{4}\right)^{2}=-22+\left(-\frac{19}{4}\right)^{2}$$

Square -\frac{19}{4} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-\frac{19}{2}y+\frac{361}{16}=-22+\frac{361}{16}$$

Add -22 to \frac{361}{16}.

$$y^{2}-\frac{19}{2}y+\frac{361}{16}=\frac{9}{16}$$

Factor y^{2}-\frac{19}{2}y+\frac{361}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-\frac{19}{4}\right)^{2}=\frac{9}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{19}{4}\right)^{2}}=\sqrt{\frac{9}{16}}$$

Simplify.

$$y-\frac{19}{4}=\frac{3}{4} y-\frac{19}{4}=-\frac{3}{4}$$

Add \frac{19}{4} to both sides of the equation.

$$y=\frac{11}{2} y=4$$