Solve y-5/y-3-1=y-2/y-5

$$\frac{ y-5 }{ y-3 } -1= \frac{ y-2 }{ y-5 }$$

Solve for y

$$y=-1$$

$$y=4$$

Steps Using the Quadratic Formula

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

$$\left(y-5\right)\left(y-5\right)+\left(y-5\right)\left(y-3\right)\left(-1\right)=\left(y-3\right)\left(y-2\right)$$

Multiply y-5 and y-5 to get \left(y-5\right)^{2}.

$$\left(y-5\right)^{2}+\left(y-5\right)\left(y-3\right)\left(-1\right)=\left(y-3\right)\left(y-2\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(y-5\right)^{2}.

$$y^{2}-10y+25+\left(y-5\right)\left(y-3\right)\left(-1\right)=\left(y-3\right)\left(y-2\right)$$

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

$$y^{2}-10y+25+\left(y^{2}-8y+15\right)\left(-1\right)=\left(y-3\right)\left(y-2\right)$$

Use the distributive property to multiply y^{2}-8y+15 by -1.

$$y^{2}-10y+25-y^{2}+8y-15=\left(y-3\right)\left(y-2\right)$$

Combine y^{2} and -y^{2} to get 0.

$$-10y+25+8y-15=\left(y-3\right)\left(y-2\right)$$

Combine -10y and 8y to get -2y.

$$-2y+25-15=\left(y-3\right)\left(y-2\right)$$

Subtract 15 from 25 to get 10.

$$-2y+10=\left(y-3\right)\left(y-2\right)$$

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

$$-2y+10=y^{2}-5y+6$$

Subtract y^{2} from both sides.

$$-2y+10-y^{2}=-5y+6$$

Add 5y to both sides.

$$-2y+10-y^{2}+5y=6$$

Combine -2y and 5y to get 3y.

$$3y+10-y^{2}=6$$

Subtract 6 from both sides.

$$3y+10-y^{2}-6=0$$

Subtract 6 from 10 to get 4.

$$3y+4-y^{2}=0$$

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.

$$-y^{2}+3y+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 3 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-3±\sqrt{3^{2}-4\left(-1\right)\times 4}}{2\left(-1\right)}$$

Square 3.

$$y=\frac{-3±\sqrt{9-4\left(-1\right)\times 4}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$y=\frac{-3±\sqrt{9+4\times 4}}{2\left(-1\right)}$$

Multiply 4 times 4.

$$y=\frac{-3±\sqrt{9+16}}{2\left(-1\right)}$$

Add 9 to 16.

$$y=\frac{-3±\sqrt{25}}{2\left(-1\right)}$$

Take the square root of 25.

$$y=\frac{-3±5}{2\left(-1\right)}$$

Multiply 2 times -1.

$$y=\frac{-3±5}{-2}$$

Now solve the equation y=\frac{-3±5}{-2} when ± is plus. Add -3 to 5.

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

Divide 2 by -2.

$$y=-1$$

Now solve the equation y=\frac{-3±5}{-2} when ± is minus. Subtract 5 from -3.

$$y=-\frac{8}{-2}$$

Divide -8 by -2.

$$y=4$$

The equation is now solved.

$$y=-1 y=4$$

Steps for Completing the Square

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

$$\left(y-5\right)\left(y-5\right)+\left(y-5\right)\left(y-3\right)\left(-1\right)=\left(y-3\right)\left(y-2\right)$$

Multiply y-5 and y-5 to get \left(y-5\right)^{2}.

$$\left(y-5\right)^{2}+\left(y-5\right)\left(y-3\right)\left(-1\right)=\left(y-3\right)\left(y-2\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(y-5\right)^{2}.

$$y^{2}-10y+25+\left(y-5\right)\left(y-3\right)\left(-1\right)=\left(y-3\right)\left(y-2\right)$$

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

$$y^{2}-10y+25+\left(y^{2}-8y+15\right)\left(-1\right)=\left(y-3\right)\left(y-2\right)$$

Use the distributive property to multiply y^{2}-8y+15 by -1.

$$y^{2}-10y+25-y^{2}+8y-15=\left(y-3\right)\left(y-2\right)$$

Combine y^{2} and -y^{2} to get 0.

$$-10y+25+8y-15=\left(y-3\right)\left(y-2\right)$$

Combine -10y and 8y to get -2y.

$$-2y+25-15=\left(y-3\right)\left(y-2\right)$$

Subtract 15 from 25 to get 10.

$$-2y+10=\left(y-3\right)\left(y-2\right)$$

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

$$-2y+10=y^{2}-5y+6$$

Subtract y^{2} from both sides.

$$-2y+10-y^{2}=-5y+6$$

Add 5y to both sides.

$$-2y+10-y^{2}+5y=6$$

Combine -2y and 5y to get 3y.

$$3y+10-y^{2}=6$$

Subtract 10 from both sides.

$$3y-y^{2}=6-10$$

Subtract 10 from 6 to get -4.

$$3y-y^{2}=-4$$

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.

$$-y^{2}+3y=-4$$

Divide both sides by -1.

$$\frac{-y^{2}+3y}{-1}=-\frac{4}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$y^{2}+\frac{3}{-1}y=-\frac{4}{-1}$$

Divide 3 by -1.

$$y^{2}-3y=-\frac{4}{-1}$$

Divide -4 by -1.

$$y^{2}-3y=4$$

Divide -3, the coefficient of the x term, by 2 to get -\frac{3}{2}. Then add the square of -\frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-3y+\left(-\frac{3}{2}\right)^{2}=4+\left(-\frac{3}{2}\right)^{2}$$

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-3y+\frac{9}{4}=4+\frac{9}{4}$$

Add 4 to \frac{9}{4}.

$$y^{2}-3y+\frac{9}{4}=\frac{25}{4}$$

Factor y^{2}-3y+\frac{9}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-\frac{3}{2}\right)^{2}=\frac{25}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{3}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Simplify.

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

Add \frac{3}{2} to both sides of the equation.

$$y=4 y=-1$$