Solve 1/y-2-1/y-1=1/6

$$\frac { 1 } { y - 2 } - \frac { 1 } { y - 1 } = \frac { 1 } { 6 }$$

Solve for y

$$y=-1$$

$$y=4$$

Steps Using the Quadratic Formula

Variable y cannot be equal to any of the values 1,2 since division by zero is not defined. Multiply both sides of the equation by 6\left(y-2\right)\left(y-1\right), the least common multiple of y-2,y-1,6.

$$6y-6-\left(6y-12\right)=\left(y-2\right)\left(y-1\right)$$

To find the opposite of 6y-12, find the opposite of each term.

$$6y-6-6y+12=\left(y-2\right)\left(y-1\right)$$

Combine 6y and -6y to get 0.

$$-6+12=\left(y-2\right)\left(y-1\right)$$

Add -6 and 12 to get 6.

$$6=\left(y-2\right)\left(y-1\right)$$

Use the distributive property to multiply y-2 by y-1 and combine like terms.

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

Swap sides so that all variable terms are on the left hand side.

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

Subtract 6 from both sides.

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

Subtract 6 from 2 to get -4.

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

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}.

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

Square -3.

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

Multiply -4 times -4.

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

Add 9 to 16.

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

Take the square root of 25.

$$y=\frac{-\left(-3\right)±5}{2}$$

The opposite of -3 is 3.

$$y=\frac{3±5}{2}$$

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

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

Divide 8 by 2.

$$y=4$$

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

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

Divide -2 by 2.

$$y=-1$$

The equation is now solved.

$$y=4 y=-1$$

Steps for Completing the Square

Variable y cannot be equal to any of the values 1,2 since division by zero is not defined. Multiply both sides of the equation by 6\left(y-2\right)\left(y-1\right), the least common multiple of y-2,y-1,6.

$$6y-6-\left(6y-12\right)=\left(y-2\right)\left(y-1\right)$$

To find the opposite of 6y-12, find the opposite of each term.

$$6y-6-6y+12=\left(y-2\right)\left(y-1\right)$$

Combine 6y and -6y to get 0.

$$-6+12=\left(y-2\right)\left(y-1\right)$$

Add -6 and 12 to get 6.

$$6=\left(y-2\right)\left(y-1\right)$$

Use the distributive property to multiply y-2 by y-1 and combine like terms.

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

Swap sides so that all variable terms are on the left hand side.

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

Subtract 2 from both sides.

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

Subtract 2 from 6 to get 4.

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

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.

$$y=4 y=-1$$