Solve 8y-5/2y+5=53y+7/-3y+2

$$\frac{ 8y-5 }{ 2y+5 } = 5 \frac{ 3y+7 }{ -3y+2 }$$

Solve for y

$$y=\frac{-\sqrt{749}i-19}{18}\approx -1.055555556-1.520436909i$$

$$y=\frac{-19+\sqrt{749}i}{18}\approx -1.055555556+1.520436909i$$

Steps Using the Quadratic Formula

Variable y cannot be equal to any of the values -\frac{5}{2},\frac{2}{3} since division by zero is not defined. Multiply both sides of the equation by \left(3y-2\right)\left(2y+5\right), the least common multiple of 2y+5,-3y+2.

$$\left(3y-2\right)\left(8y-5\right)=5\left(-5-2y\right)\left(3y+7\right)$$

Use the distributive property to multiply 3y-2 by 8y-5 and combine like terms.

$$24y^{2}-31y+10=5\left(-5-2y\right)\left(3y+7\right)$$

Use the distributive property to multiply 5 by -5-2y.

$$24y^{2}-31y+10=\left(-25-10y\right)\left(3y+7\right)$$

Use the distributive property to multiply -25-10y by 3y+7 and combine like terms.

$$24y^{2}-31y+10=-145y-175-30y^{2}$$

Add 145y to both sides.

$$24y^{2}-31y+10+145y=-175-30y^{2}$$

Combine -31y and 145y to get 114y.

$$24y^{2}+114y+10=-175-30y^{2}$$

Subtract -175 from both sides.

$$24y^{2}+114y+10-\left(-175\right)=-30y^{2}$$

The opposite of -175 is 175.

$$24y^{2}+114y+10+175=-30y^{2}$$

Add 30y^{2} to both sides.

$$24y^{2}+114y+10+175+30y^{2}=0$$

Add 10 and 175 to get 185.

$$24y^{2}+114y+185+30y^{2}=0$$

Combine 24y^{2} and 30y^{2} to get 54y^{2}.

$$54y^{2}+114y+185=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 54 for a, 114 for b, and 185 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-114±\sqrt{114^{2}-4\times 54\times 185}}{2\times 54}$$

Square 114.

$$y=\frac{-114±\sqrt{12996-4\times 54\times 185}}{2\times 54}$$

Multiply -4 times 54.

$$y=\frac{-114±\sqrt{12996-216\times 185}}{2\times 54}$$

Multiply -216 times 185.

$$y=\frac{-114±\sqrt{12996-39960}}{2\times 54}$$

Add 12996 to -39960.

$$y=\frac{-114±\sqrt{-26964}}{2\times 54}$$

Take the square root of -26964.

$$y=\frac{-114±6\sqrt{749}i}{2\times 54}$$

Multiply 2 times 54.

$$y=\frac{-114±6\sqrt{749}i}{108}$$

Now solve the equation y=\frac{-114±6\sqrt{749}i}{108} when ± is plus. Add -114 to 6i\sqrt{749}.

$$y=\frac{-114+6\sqrt{749}i}{108}$$

Divide -114+6i\sqrt{749} by 108.

$$y=\frac{-19+\sqrt{749}i}{18}$$

Now solve the equation y=\frac{-114±6\sqrt{749}i}{108} when ± is minus. Subtract 6i\sqrt{749} from -114.

$$y=\frac{-6\sqrt{749}i-114}{108}$$

Divide -114-6i\sqrt{749} by 108.

$$y=\frac{-\sqrt{749}i-19}{18}$$

The equation is now solved.

$$y=\frac{-19+\sqrt{749}i}{18} y=\frac{-\sqrt{749}i-19}{18}$$

Steps for Completing the Square

Variable y cannot be equal to any of the values -\frac{5}{2},\frac{2}{3} since division by zero is not defined. Multiply both sides of the equation by \left(3y-2\right)\left(2y+5\right), the least common multiple of 2y+5,-3y+2.

$$\left(3y-2\right)\left(8y-5\right)=5\left(-5-2y\right)\left(3y+7\right)$$

Use the distributive property to multiply 3y-2 by 8y-5 and combine like terms.

$$24y^{2}-31y+10=5\left(-5-2y\right)\left(3y+7\right)$$

Use the distributive property to multiply 5 by -5-2y.

$$24y^{2}-31y+10=\left(-25-10y\right)\left(3y+7\right)$$

Use the distributive property to multiply -25-10y by 3y+7 and combine like terms.

$$24y^{2}-31y+10=-145y-175-30y^{2}$$

Add 145y to both sides.

$$24y^{2}-31y+10+145y=-175-30y^{2}$$

Combine -31y and 145y to get 114y.

$$24y^{2}+114y+10=-175-30y^{2}$$

Add 30y^{2} to both sides.

$$24y^{2}+114y+10+30y^{2}=-175$$

Combine 24y^{2} and 30y^{2} to get 54y^{2}.

$$54y^{2}+114y+10=-175$$

Subtract 10 from both sides.

$$54y^{2}+114y=-175-10$$

Subtract 10 from -175 to get -185.

$$54y^{2}+114y=-185$$

Divide both sides by 54.

$$\frac{54y^{2}+114y}{54}=-\frac{185}{54}$$

Dividing by 54 undoes the multiplication by 54.

$$y^{2}+\frac{114}{54}y=-\frac{185}{54}$$

Reduce the fraction \frac{114}{54} to lowest terms by extracting and canceling out 6.

$$y^{2}+\frac{19}{9}y=-\frac{185}{54}$$

Divide \frac{19}{9}, the coefficient of the x term, by 2 to get \frac{19}{18}. Then add the square of \frac{19}{18} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+\frac{19}{9}y+\left(\frac{19}{18}\right)^{2}=-\frac{185}{54}+\left(\frac{19}{18}\right)^{2}$$

Square \frac{19}{18} by squaring both the numerator and the denominator of the fraction.

$$y^{2}+\frac{19}{9}y+\frac{361}{324}=-\frac{185}{54}+\frac{361}{324}$$

Add -\frac{185}{54} to \frac{361}{324} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}+\frac{19}{9}y+\frac{361}{324}=-\frac{749}{324}$$

Factor y^{2}+\frac{19}{9}y+\frac{361}{324}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y+\frac{19}{18}\right)^{2}=-\frac{749}{324}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+\frac{19}{18}\right)^{2}}=\sqrt{-\frac{749}{324}}$$

Simplify.

$$y+\frac{19}{18}=\frac{\sqrt{749}i}{18} y+\frac{19}{18}=-\frac{\sqrt{749}i}{18}$$

Subtract \frac{19}{18} from both sides of the equation.

$$y=\frac{-19+\sqrt{749}i}{18} y=\frac{-\sqrt{749}i-19}{18}$$