Solve (x+y)(2x-2y)+(3y-2x)*(3/2x-1/3y)-(x+2y)(1/2y-x)

$$( x + y ) ( 2 x - 2 y ) + ( 3 y - 2 x ) \times ( \frac { 3 } { 2 } x - \frac { 1 } { 3 } y ) - ( x + 2 y ) ( \frac { 1 } { 2 } y - x )$$

Evaluate

$$\frac{4y\left(5x-3y\right)}{3}$$

Solution Steps

Apply the distributive property by multiplying each term of x+y by each term of 2x-2y.

$$2x^{2}-2xy+2yx-2y^{2}+\left(3y-2x\right)\left(\frac{3}{2}x-\frac{1}{3}y\right)-\left(x+2y\right)\left(\frac{1}{2}y-x\right)$$

Combine -2xy and 2yx to get 0.

$$2x^{2}-2y^{2}+\left(3y-2x\right)\left(\frac{3}{2}x-\frac{1}{3}y\right)-\left(x+2y\right)\left(\frac{1}{2}y-x\right)$$

Apply the distributive property by multiplying each term of 3y-2x by each term of \frac{3}{2}x-\frac{1}{3}y.

$$2x^{2}-2y^{2}+3y\times \frac{3}{2}x+3y\left(-\frac{1}{3}\right)y-2x\times \frac{3}{2}x-2x\left(-\frac{1}{3}\right)y-\left(x+2y\right)\left(\frac{1}{2}y-x\right)$$

Multiply y and y to get y^{2}.

$$2x^{2}-2y^{2}+3y\times \frac{3}{2}x+3y^{2}\left(-\frac{1}{3}\right)-2x\times \frac{3}{2}x-2x\left(-\frac{1}{3}\right)y-\left(x+2y\right)\left(\frac{1}{2}y-x\right)$$

Multiply x and x to get x^{2}.

$$2x^{2}-2y^{2}+3y\times \frac{3}{2}x+3y^{2}\left(-\frac{1}{3}\right)-2x^{2}\times \frac{3}{2}-2x\left(-\frac{1}{3}\right)y-\left(x+2y\right)\left(\frac{1}{2}y-x\right)$$

Express 3\times \frac{3}{2} as a single fraction.

$$2x^{2}-2y^{2}+\frac{3\times 3}{2}yx+3y^{2}\left(-\frac{1}{3}\right)-2x^{2}\times \frac{3}{2}-2x\left(-\frac{1}{3}\right)y-\left(x+2y\right)\left(\frac{1}{2}y-x\right)$$

Multiply 3 and 3 to get 9.

$$2x^{2}-2y^{2}+\frac{9}{2}yx+3y^{2}\left(-\frac{1}{3}\right)-2x^{2}\times \frac{3}{2}-2x\left(-\frac{1}{3}\right)y-\left(x+2y\right)\left(\frac{1}{2}y-x\right)$$

Cancel out 3 and 3.

$$2x^{2}-2y^{2}+\frac{9}{2}yx-y^{2}-2x^{2}\times \frac{3}{2}-2x\left(-\frac{1}{3}\right)y-\left(x+2y\right)\left(\frac{1}{2}y-x\right)$$

Multiply -2 times \frac{3}{2}.

$$2x^{2}-2y^{2}+\frac{9}{2}yx-y^{2}-3x^{2}-2x\left(-\frac{1}{3}\right)y-\left(x+2y\right)\left(\frac{1}{2}y-x\right)$$

Express -2\left(-\frac{1}{3}\right) as a single fraction.

$$2x^{2}-2y^{2}+\frac{9}{2}yx-y^{2}-3x^{2}+\frac{-2\left(-1\right)}{3}xy-\left(x+2y\right)\left(\frac{1}{2}y-x\right)$$

Multiply -2 and -1 to get 2.

$$2x^{2}-2y^{2}+\frac{9}{2}yx-y^{2}-3x^{2}+\frac{2}{3}xy-\left(x+2y\right)\left(\frac{1}{2}y-x\right)$$

Combine \frac{9}{2}yx and \frac{2}{3}xy to get \frac{31}{6}yx.

$$2x^{2}-2y^{2}+\frac{31}{6}yx-y^{2}-3x^{2}-\left(x+2y\right)\left(\frac{1}{2}y-x\right)$$

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

$$2x^{2}-3y^{2}+\frac{31}{6}yx-3x^{2}-\left(x+2y\right)\left(\frac{1}{2}y-x\right)$$

Combine 2x^{2} and -3x^{2} to get -x^{2}.

$$-x^{2}-3y^{2}+\frac{31}{6}yx-\left(x+2y\right)\left(\frac{1}{2}y-x\right)$$

Apply the distributive property by multiplying each term of x+2y by each term of \frac{1}{2}y-x.

$$-x^{2}-3y^{2}+\frac{31}{6}yx-\left(x\times \frac{1}{2}y-x^{2}+2y\times \frac{1}{2}y-2yx\right)$$

Multiply y and y to get y^{2}.

$$-x^{2}-3y^{2}+\frac{31}{6}yx-\left(x\times \frac{1}{2}y-x^{2}+2y^{2}\times \frac{1}{2}-2yx\right)$$

Cancel out 2 and 2.

$$-x^{2}-3y^{2}+\frac{31}{6}yx-\left(x\times \frac{1}{2}y-x^{2}+y^{2}-2yx\right)$$

Combine x\times \frac{1}{2}y and -2yx to get -\frac{3}{2}xy.

$$-x^{2}-3y^{2}+\frac{31}{6}yx-\left(-\frac{3}{2}xy-x^{2}+y^{2}\right)$$

To find the opposite of -\frac{3}{2}xy-x^{2}+y^{2}, find the opposite of each term.

$$-x^{2}-3y^{2}+\frac{31}{6}yx-\left(-\frac{3}{2}xy\right)-\left(-x^{2}\right)-y^{2}$$

The opposite of -\frac{3}{2}xy is \frac{3}{2}xy.

$$-x^{2}-3y^{2}+\frac{31}{6}yx+\frac{3}{2}xy-\left(-x^{2}\right)-y^{2}$$

The opposite of -x^{2} is x^{2}.

$$-x^{2}-3y^{2}+\frac{31}{6}yx+\frac{3}{2}xy+x^{2}-y^{2}$$

Combine \frac{31}{6}yx and \frac{3}{2}xy to get \frac{20}{3}yx.

$$-x^{2}-3y^{2}+\frac{20}{3}yx+x^{2}-y^{2}$$

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

$$-3y^{2}+\frac{20}{3}yx-y^{2}$$

Combine -3y^{2} and -y^{2} to get -4y^{2}.

$$-4y^{2}+\frac{20}{3}yx$$

Expand

$$\frac{20xy}{3}-4y^{2}$$

Solution Steps

Apply the distributive property by multiplying each term of x+y by each term of 2x-2y.

$$2x^{2}-2xy+2yx-2y^{2}+\left(3y-2x\right)\left(\frac{3}{2}x-\frac{1}{3}y\right)-\left(x+2y\right)\left(\frac{1}{2}y-x\right)$$

Combine -2xy and 2yx to get 0.

$$2x^{2}-2y^{2}+\left(3y-2x\right)\left(\frac{3}{2}x-\frac{1}{3}y\right)-\left(x+2y\right)\left(\frac{1}{2}y-x\right)$$

Apply the distributive property by multiplying each term of 3y-2x by each term of \frac{3}{2}x-\frac{1}{3}y.

$$2x^{2}-2y^{2}+3y\times \frac{3}{2}x+3y\left(-\frac{1}{3}\right)y-2x\times \frac{3}{2}x-2x\left(-\frac{1}{3}\right)y-\left(x+2y\right)\left(\frac{1}{2}y-x\right)$$

Multiply y and y to get y^{2}.

$$2x^{2}-2y^{2}+3y\times \frac{3}{2}x+3y^{2}\left(-\frac{1}{3}\right)-2x\times \frac{3}{2}x-2x\left(-\frac{1}{3}\right)y-\left(x+2y\right)\left(\frac{1}{2}y-x\right)$$

Multiply x and x to get x^{2}.

$$2x^{2}-2y^{2}+3y\times \frac{3}{2}x+3y^{2}\left(-\frac{1}{3}\right)-2x^{2}\times \frac{3}{2}-2x\left(-\frac{1}{3}\right)y-\left(x+2y\right)\left(\frac{1}{2}y-x\right)$$

Express 3\times \frac{3}{2} as a single fraction.

$$2x^{2}-2y^{2}+\frac{3\times 3}{2}yx+3y^{2}\left(-\frac{1}{3}\right)-2x^{2}\times \frac{3}{2}-2x\left(-\frac{1}{3}\right)y-\left(x+2y\right)\left(\frac{1}{2}y-x\right)$$

Multiply 3 and 3 to get 9.

$$2x^{2}-2y^{2}+\frac{9}{2}yx+3y^{2}\left(-\frac{1}{3}\right)-2x^{2}\times \frac{3}{2}-2x\left(-\frac{1}{3}\right)y-\left(x+2y\right)\left(\frac{1}{2}y-x\right)$$

Cancel out 3 and 3.

$$2x^{2}-2y^{2}+\frac{9}{2}yx-y^{2}-2x^{2}\times \frac{3}{2}-2x\left(-\frac{1}{3}\right)y-\left(x+2y\right)\left(\frac{1}{2}y-x\right)$$

Multiply -2 times \frac{3}{2}.

$$2x^{2}-2y^{2}+\frac{9}{2}yx-y^{2}-3x^{2}-2x\left(-\frac{1}{3}\right)y-\left(x+2y\right)\left(\frac{1}{2}y-x\right)$$

Express -2\left(-\frac{1}{3}\right) as a single fraction.

$$2x^{2}-2y^{2}+\frac{9}{2}yx-y^{2}-3x^{2}+\frac{-2\left(-1\right)}{3}xy-\left(x+2y\right)\left(\frac{1}{2}y-x\right)$$

Multiply -2 and -1 to get 2.

$$2x^{2}-2y^{2}+\frac{9}{2}yx-y^{2}-3x^{2}+\frac{2}{3}xy-\left(x+2y\right)\left(\frac{1}{2}y-x\right)$$

Combine \frac{9}{2}yx and \frac{2}{3}xy to get \frac{31}{6}yx.

$$2x^{2}-2y^{2}+\frac{31}{6}yx-y^{2}-3x^{2}-\left(x+2y\right)\left(\frac{1}{2}y-x\right)$$

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

$$2x^{2}-3y^{2}+\frac{31}{6}yx-3x^{2}-\left(x+2y\right)\left(\frac{1}{2}y-x\right)$$

Combine 2x^{2} and -3x^{2} to get -x^{2}.

$$-x^{2}-3y^{2}+\frac{31}{6}yx-\left(x+2y\right)\left(\frac{1}{2}y-x\right)$$

Apply the distributive property by multiplying each term of x+2y by each term of \frac{1}{2}y-x.

$$-x^{2}-3y^{2}+\frac{31}{6}yx-\left(x\times \frac{1}{2}y-x^{2}+2y\times \frac{1}{2}y-2yx\right)$$

Multiply y and y to get y^{2}.

$$-x^{2}-3y^{2}+\frac{31}{6}yx-\left(x\times \frac{1}{2}y-x^{2}+2y^{2}\times \frac{1}{2}-2yx\right)$$

Cancel out 2 and 2.

$$-x^{2}-3y^{2}+\frac{31}{6}yx-\left(x\times \frac{1}{2}y-x^{2}+y^{2}-2yx\right)$$

Combine x\times \frac{1}{2}y and -2yx to get -\frac{3}{2}xy.

$$-x^{2}-3y^{2}+\frac{31}{6}yx-\left(-\frac{3}{2}xy-x^{2}+y^{2}\right)$$

To find the opposite of -\frac{3}{2}xy-x^{2}+y^{2}, find the opposite of each term.

$$-x^{2}-3y^{2}+\frac{31}{6}yx-\left(-\frac{3}{2}xy\right)-\left(-x^{2}\right)-y^{2}$$

The opposite of -\frac{3}{2}xy is \frac{3}{2}xy.

$$-x^{2}-3y^{2}+\frac{31}{6}yx+\frac{3}{2}xy-\left(-x^{2}\right)-y^{2}$$

The opposite of -x^{2} is x^{2}.

$$-x^{2}-3y^{2}+\frac{31}{6}yx+\frac{3}{2}xy+x^{2}-y^{2}$$

Combine \frac{31}{6}yx and \frac{3}{2}xy to get \frac{20}{3}yx.

$$-x^{2}-3y^{2}+\frac{20}{3}yx+x^{2}-y^{2}$$

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

$$-3y^{2}+\frac{20}{3}yx-y^{2}$$

Combine -3y^{2} and -y^{2} to get -4y^{2}.

$$-4y^{2}+\frac{20}{3}yx$$