Solve {[(1/2x+1)(1/2y-1/2)-1/2(xy-1)+frac{1}{4}xy](-frac{1}{2}y^2)}:(-2y)

$$\{ [ ( \frac { 1 } { 2 } x + 1 ) ( \frac { 1 } { 2 } y - \frac { 1 } { 2 } ) - \frac { 1 } { 2 } ( x y - 1 ) + \frac { 1 } { 4 } x y ] ( - \frac { 1 } { 2 } y ^ { 2 } ) \} : ( - 2 y )$$

Evaluate

$$\frac{y\left(2y-x\right)}{16}$$

Solution Steps

Cancel out y in both numerator and denominator.

$$\frac{-\frac{1}{2}y\left(\left(\frac{1}{2}y-\frac{1}{2}\right)\left(\frac{1}{2}x+1\right)-\frac{1}{2}\left(xy-1\right)+\frac{1}{4}xy\right)}{-2}$$

Divide -\frac{1}{2}y\left(\left(\frac{1}{2}y-\frac{1}{2}\right)\left(\frac{1}{2}x+1\right)-\frac{1}{2}\left(xy-1\right)+\frac{1}{4}xy\right) by -2 to get \frac{1}{4}y\left(\left(\frac{1}{2}y-\frac{1}{2}\right)\left(\frac{1}{2}x+1\right)-\frac{1}{2}\left(xy-1\right)+\frac{1}{4}xy\right).

$$\frac{1}{4}y\left(\left(\frac{1}{2}y-\frac{1}{2}\right)\left(\frac{1}{2}x+1\right)-\frac{1}{2}\left(xy-1\right)+\frac{1}{4}xy\right)$$

Use the distributive property to multiply \frac{1}{4}y by \left(\frac{1}{2}y-\frac{1}{2}\right)\left(\frac{1}{2}x+1\right)-\frac{1}{2}\left(xy-1\right)+\frac{1}{4}xy.

$$\frac{1}{4}y\left(\frac{1}{2}y-\frac{1}{2}\right)\left(\frac{1}{2}x+1\right)-\frac{1}{8}y\left(xy-1\right)+\frac{1}{16}y^{2}x$$

Use the distributive property to multiply \frac{1}{4}y by \frac{1}{2}y-\frac{1}{2}.

$$\left(\frac{1}{8}y^{2}-\frac{1}{8}y\right)\left(\frac{1}{2}x+1\right)-\frac{1}{8}y\left(xy-1\right)+\frac{1}{16}y^{2}x$$

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

$$\frac{1}{16}y^{2}x+\frac{1}{8}y^{2}-\frac{1}{16}yx-\frac{1}{8}y-\frac{1}{8}y\left(xy-1\right)+\frac{1}{16}y^{2}x$$

Use the distributive property to multiply -\frac{1}{8}y by xy-1.

$$\frac{1}{16}y^{2}x+\frac{1}{8}y^{2}-\frac{1}{16}yx-\frac{1}{8}y-\frac{1}{8}xy^{2}+\frac{1}{8}y+\frac{1}{16}y^{2}x$$

Combine \frac{1}{16}y^{2}x and -\frac{1}{8}xy^{2} to get -\frac{1}{16}y^{2}x.

$$-\frac{1}{16}y^{2}x+\frac{1}{8}y^{2}-\frac{1}{16}yx-\frac{1}{8}y+\frac{1}{8}y+\frac{1}{16}y^{2}x$$

Combine -\frac{1}{8}y and \frac{1}{8}y to get 0.

$$-\frac{1}{16}y^{2}x+\frac{1}{8}y^{2}-\frac{1}{16}yx+\frac{1}{16}y^{2}x$$

Combine -\frac{1}{16}y^{2}x and \frac{1}{16}y^{2}x to get 0.

$$\frac{1}{8}y^{2}-\frac{1}{16}yx$$

Expand

$$-\frac{xy}{16}+\frac{y^{2}}{8}$$

Solution Steps

Cancel out y in both numerator and denominator.

$$\frac{-\frac{1}{2}y\left(\left(\frac{1}{2}y-\frac{1}{2}\right)\left(\frac{1}{2}x+1\right)-\frac{1}{2}\left(xy-1\right)+\frac{1}{4}xy\right)}{-2}$$

Divide -\frac{1}{2}y\left(\left(\frac{1}{2}y-\frac{1}{2}\right)\left(\frac{1}{2}x+1\right)-\frac{1}{2}\left(xy-1\right)+\frac{1}{4}xy\right) by -2 to get \frac{1}{4}y\left(\left(\frac{1}{2}y-\frac{1}{2}\right)\left(\frac{1}{2}x+1\right)-\frac{1}{2}\left(xy-1\right)+\frac{1}{4}xy\right).

$$\frac{1}{4}y\left(\left(\frac{1}{2}y-\frac{1}{2}\right)\left(\frac{1}{2}x+1\right)-\frac{1}{2}\left(xy-1\right)+\frac{1}{4}xy\right)$$

Use the distributive property to multiply \frac{1}{4}y by \left(\frac{1}{2}y-\frac{1}{2}\right)\left(\frac{1}{2}x+1\right)-\frac{1}{2}\left(xy-1\right)+\frac{1}{4}xy.

$$\frac{1}{4}y\left(\frac{1}{2}y-\frac{1}{2}\right)\left(\frac{1}{2}x+1\right)-\frac{1}{8}y\left(xy-1\right)+\frac{1}{16}y^{2}x$$

Use the distributive property to multiply \frac{1}{4}y by \frac{1}{2}y-\frac{1}{2}.

$$\left(\frac{1}{8}y^{2}-\frac{1}{8}y\right)\left(\frac{1}{2}x+1\right)-\frac{1}{8}y\left(xy-1\right)+\frac{1}{16}y^{2}x$$

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

$$\frac{1}{16}y^{2}x+\frac{1}{8}y^{2}-\frac{1}{16}yx-\frac{1}{8}y-\frac{1}{8}y\left(xy-1\right)+\frac{1}{16}y^{2}x$$

Use the distributive property to multiply -\frac{1}{8}y by xy-1.

$$\frac{1}{16}y^{2}x+\frac{1}{8}y^{2}-\frac{1}{16}yx-\frac{1}{8}y-\frac{1}{8}xy^{2}+\frac{1}{8}y+\frac{1}{16}y^{2}x$$

Combine \frac{1}{16}y^{2}x and -\frac{1}{8}xy^{2} to get -\frac{1}{16}y^{2}x.

$$-\frac{1}{16}y^{2}x+\frac{1}{8}y^{2}-\frac{1}{16}yx-\frac{1}{8}y+\frac{1}{8}y+\frac{1}{16}y^{2}x$$

Combine -\frac{1}{8}y and \frac{1}{8}y to get 0.

$$-\frac{1}{16}y^{2}x+\frac{1}{8}y^{2}-\frac{1}{16}yx+\frac{1}{16}y^{2}x$$

Combine -\frac{1}{16}y^{2}x and \frac{1}{16}y^{2}x to get 0.

$$\frac{1}{8}y^{2}-\frac{1}{16}yx$$