Solve (2/3x+3/2y)*(3/5x-y)-(5/4x-frac{1}{2}y)*(frac{1}{3}x+frac{2}{5}y)

$$( \frac { 2 } { 3 } x + \frac { 3 } { 2 } y ) \cdot ( \frac { 3 } { 5 } x - y ) - ( \frac { 5 } { 4 } x - \frac { 1 } { 2 } y ) \cdot ( \frac { 1 } { 3 } x + \frac { 2 } { 5 } y )$$

Evaluate

$$-\frac{xy}{10}-\frac{13y^{2}}{10}-\frac{x^{2}}{60}$$

Solution Steps

Apply the distributive property by multiplying each term of \frac{2}{3}x+\frac{3}{2}y by each term of \frac{3}{5}x-y.

$$\frac{2}{3}x\times \frac{3}{5}x+\frac{2}{3}x\left(-1\right)y+\frac{3}{2}y\times \frac{3}{5}x+\frac{3}{2}y\left(-1\right)y-\left(\frac{5}{4}x-\frac{1}{2}y\right)\left(\frac{1}{3}x+\frac{2}{5}y\right)$$

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

$$\frac{2}{3}x^{2}\times \frac{3}{5}+\frac{2}{3}x\left(-1\right)y+\frac{3}{2}y\times \frac{3}{5}x+\frac{3}{2}y\left(-1\right)y-\left(\frac{5}{4}x-\frac{1}{2}y\right)\left(\frac{1}{3}x+\frac{2}{5}y\right)$$

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

$$\frac{2}{3}x^{2}\times \frac{3}{5}+\frac{2}{3}x\left(-1\right)y+\frac{3}{2}y\times \frac{3}{5}x+\frac{3}{2}y^{2}\left(-1\right)-\left(\frac{5}{4}x-\frac{1}{2}y\right)\left(\frac{1}{3}x+\frac{2}{5}y\right)$$

Multiply \frac{2}{3} times \frac{3}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2\times 3}{3\times 5}x^{2}+\frac{2}{3}x\left(-1\right)y+\frac{3}{2}y\times \frac{3}{5}x+\frac{3}{2}y^{2}\left(-1\right)-\left(\frac{5}{4}x-\frac{1}{2}y\right)\left(\frac{1}{3}x+\frac{2}{5}y\right)$$

Cancel out 3 in both numerator and denominator.

$$\frac{2}{5}x^{2}+\frac{2}{3}x\left(-1\right)y+\frac{3}{2}y\times \frac{3}{5}x+\frac{3}{2}y^{2}\left(-1\right)-\left(\frac{5}{4}x-\frac{1}{2}y\right)\left(\frac{1}{3}x+\frac{2}{5}y\right)$$

Multiply \frac{2}{3} and -1 to get -\frac{2}{3}.

$$\frac{2}{5}x^{2}-\frac{2}{3}xy+\frac{3}{2}y\times \frac{3}{5}x+\frac{3}{2}y^{2}\left(-1\right)-\left(\frac{5}{4}x-\frac{1}{2}y\right)\left(\frac{1}{3}x+\frac{2}{5}y\right)$$

Multiply \frac{3}{2} times \frac{3}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2}{5}x^{2}-\frac{2}{3}xy+\frac{3\times 3}{2\times 5}yx+\frac{3}{2}y^{2}\left(-1\right)-\left(\frac{5}{4}x-\frac{1}{2}y\right)\left(\frac{1}{3}x+\frac{2}{5}y\right)$$

Do the multiplications in the fraction \frac{3\times 3}{2\times 5}.

$$\frac{2}{5}x^{2}-\frac{2}{3}xy+\frac{9}{10}yx+\frac{3}{2}y^{2}\left(-1\right)-\left(\frac{5}{4}x-\frac{1}{2}y\right)\left(\frac{1}{3}x+\frac{2}{5}y\right)$$

Combine -\frac{2}{3}xy and \frac{9}{10}yx to get \frac{7}{30}xy.

$$\frac{2}{5}x^{2}+\frac{7}{30}xy+\frac{3}{2}y^{2}\left(-1\right)-\left(\frac{5}{4}x-\frac{1}{2}y\right)\left(\frac{1}{3}x+\frac{2}{5}y\right)$$

Multiply \frac{3}{2} and -1 to get -\frac{3}{2}.

$$\frac{2}{5}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\left(\frac{5}{4}x-\frac{1}{2}y\right)\left(\frac{1}{3}x+\frac{2}{5}y\right)$$

Apply the distributive property by multiplying each term of \frac{5}{4}x-\frac{1}{2}y by each term of \frac{1}{3}x+\frac{2}{5}y.

$$\frac{2}{5}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\left(\frac{5}{4}x\times \frac{1}{3}x+\frac{5}{4}x\times \frac{2}{5}y-\frac{1}{2}y\times \frac{1}{3}x-\frac{1}{2}y\times \frac{2}{5}y\right)$$

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

$$\frac{2}{5}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\left(\frac{5}{4}x^{2}\times \frac{1}{3}+\frac{5}{4}x\times \frac{2}{5}y-\frac{1}{2}y\times \frac{1}{3}x-\frac{1}{2}y\times \frac{2}{5}y\right)$$

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

$$\frac{2}{5}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\left(\frac{5}{4}x^{2}\times \frac{1}{3}+\frac{5}{4}x\times \frac{2}{5}y-\frac{1}{2}y\times \frac{1}{3}x-\frac{1}{2}y^{2}\times \frac{2}{5}\right)$$

Multiply \frac{5}{4} times \frac{1}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2}{5}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\left(\frac{5\times 1}{4\times 3}x^{2}+\frac{5}{4}x\times \frac{2}{5}y-\frac{1}{2}y\times \frac{1}{3}x-\frac{1}{2}y^{2}\times \frac{2}{5}\right)$$

Do the multiplications in the fraction \frac{5\times 1}{4\times 3}.

$$\frac{2}{5}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\left(\frac{5}{12}x^{2}+\frac{5}{4}x\times \frac{2}{5}y-\frac{1}{2}y\times \frac{1}{3}x-\frac{1}{2}y^{2}\times \frac{2}{5}\right)$$

Multiply \frac{5}{4} times \frac{2}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2}{5}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\left(\frac{5}{12}x^{2}+\frac{5\times 2}{4\times 5}xy-\frac{1}{2}y\times \frac{1}{3}x-\frac{1}{2}y^{2}\times \frac{2}{5}\right)$$

Cancel out 5 in both numerator and denominator.

$$\frac{2}{5}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\left(\frac{5}{12}x^{2}+\frac{2}{4}xy-\frac{1}{2}y\times \frac{1}{3}x-\frac{1}{2}y^{2}\times \frac{2}{5}\right)$$

Reduce the fraction \frac{2}{4} to lowest terms by extracting and canceling out 2.

$$\frac{2}{5}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\left(\frac{5}{12}x^{2}+\frac{1}{2}xy-\frac{1}{2}y\times \frac{1}{3}x-\frac{1}{2}y^{2}\times \frac{2}{5}\right)$$

Multiply -\frac{1}{2} times \frac{1}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2}{5}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\left(\frac{5}{12}x^{2}+\frac{1}{2}xy+\frac{-1}{2\times 3}yx-\frac{1}{2}y^{2}\times \frac{2}{5}\right)$$

Do the multiplications in the fraction \frac{-1}{2\times 3}.

$$\frac{2}{5}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\left(\frac{5}{12}x^{2}+\frac{1}{2}xy+\frac{-1}{6}yx-\frac{1}{2}y^{2}\times \frac{2}{5}\right)$$

Fraction \frac{-1}{6} can be rewritten as -\frac{1}{6} by extracting the negative sign.

$$\frac{2}{5}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\left(\frac{5}{12}x^{2}+\frac{1}{2}xy-\frac{1}{6}yx-\frac{1}{2}y^{2}\times \frac{2}{5}\right)$$

Combine \frac{1}{2}xy and -\frac{1}{6}yx to get \frac{1}{3}xy.

$$\frac{2}{5}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\left(\frac{5}{12}x^{2}+\frac{1}{3}xy-\frac{1}{2}y^{2}\times \frac{2}{5}\right)$$

Multiply -\frac{1}{2} times \frac{2}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2}{5}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\left(\frac{5}{12}x^{2}+\frac{1}{3}xy+\frac{-2}{2\times 5}y^{2}\right)$$

Cancel out 2 in both numerator and denominator.

$$\frac{2}{5}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\left(\frac{5}{12}x^{2}+\frac{1}{3}xy+\frac{-1}{5}y^{2}\right)$$

Fraction \frac{-1}{5} can be rewritten as -\frac{1}{5} by extracting the negative sign.

$$\frac{2}{5}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\left(\frac{5}{12}x^{2}+\frac{1}{3}xy-\frac{1}{5}y^{2}\right)$$

To find the opposite of \frac{5}{12}x^{2}+\frac{1}{3}xy-\frac{1}{5}y^{2}, find the opposite of each term.

$$\frac{2}{5}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\frac{5}{12}x^{2}-\frac{1}{3}xy-\left(-\frac{1}{5}y^{2}\right)$$

The opposite of -\frac{1}{5}y^{2} is \frac{1}{5}y^{2}.

$$\frac{2}{5}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\frac{5}{12}x^{2}-\frac{1}{3}xy+\frac{1}{5}y^{2}$$

Combine \frac{2}{5}x^{2} and -\frac{5}{12}x^{2} to get -\frac{1}{60}x^{2}.

$$-\frac{1}{60}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\frac{1}{3}xy+\frac{1}{5}y^{2}$$

Combine \frac{7}{30}xy and -\frac{1}{3}xy to get -\frac{1}{10}xy.

$$-\frac{1}{60}x^{2}-\frac{1}{10}xy-\frac{3}{2}y^{2}+\frac{1}{5}y^{2}$$

Combine -\frac{3}{2}y^{2} and \frac{1}{5}y^{2} to get -\frac{13}{10}y^{2}.

$$-\frac{1}{60}x^{2}-\frac{1}{10}xy-\frac{13}{10}y^{2}$$

Expand

$$-\frac{xy}{10}-\frac{13y^{2}}{10}-\frac{x^{2}}{60}$$

Solution Steps

Apply the distributive property by multiplying each term of \frac{2}{3}x+\frac{3}{2}y by each term of \frac{3}{5}x-y.

$$\frac{2}{3}x\times \frac{3}{5}x+\frac{2}{3}x\left(-1\right)y+\frac{3}{2}y\times \frac{3}{5}x+\frac{3}{2}y\left(-1\right)y-\left(\frac{5}{4}x-\frac{1}{2}y\right)\left(\frac{1}{3}x+\frac{2}{5}y\right)$$

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

$$\frac{2}{3}x^{2}\times \frac{3}{5}+\frac{2}{3}x\left(-1\right)y+\frac{3}{2}y\times \frac{3}{5}x+\frac{3}{2}y\left(-1\right)y-\left(\frac{5}{4}x-\frac{1}{2}y\right)\left(\frac{1}{3}x+\frac{2}{5}y\right)$$

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

$$\frac{2}{3}x^{2}\times \frac{3}{5}+\frac{2}{3}x\left(-1\right)y+\frac{3}{2}y\times \frac{3}{5}x+\frac{3}{2}y^{2}\left(-1\right)-\left(\frac{5}{4}x-\frac{1}{2}y\right)\left(\frac{1}{3}x+\frac{2}{5}y\right)$$

Multiply \frac{2}{3} times \frac{3}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2\times 3}{3\times 5}x^{2}+\frac{2}{3}x\left(-1\right)y+\frac{3}{2}y\times \frac{3}{5}x+\frac{3}{2}y^{2}\left(-1\right)-\left(\frac{5}{4}x-\frac{1}{2}y\right)\left(\frac{1}{3}x+\frac{2}{5}y\right)$$

Cancel out 3 in both numerator and denominator.

$$\frac{2}{5}x^{2}+\frac{2}{3}x\left(-1\right)y+\frac{3}{2}y\times \frac{3}{5}x+\frac{3}{2}y^{2}\left(-1\right)-\left(\frac{5}{4}x-\frac{1}{2}y\right)\left(\frac{1}{3}x+\frac{2}{5}y\right)$$

Multiply \frac{2}{3} and -1 to get -\frac{2}{3}.

$$\frac{2}{5}x^{2}-\frac{2}{3}xy+\frac{3}{2}y\times \frac{3}{5}x+\frac{3}{2}y^{2}\left(-1\right)-\left(\frac{5}{4}x-\frac{1}{2}y\right)\left(\frac{1}{3}x+\frac{2}{5}y\right)$$

Multiply \frac{3}{2} times \frac{3}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2}{5}x^{2}-\frac{2}{3}xy+\frac{3\times 3}{2\times 5}yx+\frac{3}{2}y^{2}\left(-1\right)-\left(\frac{5}{4}x-\frac{1}{2}y\right)\left(\frac{1}{3}x+\frac{2}{5}y\right)$$

Do the multiplications in the fraction \frac{3\times 3}{2\times 5}.

$$\frac{2}{5}x^{2}-\frac{2}{3}xy+\frac{9}{10}yx+\frac{3}{2}y^{2}\left(-1\right)-\left(\frac{5}{4}x-\frac{1}{2}y\right)\left(\frac{1}{3}x+\frac{2}{5}y\right)$$

Combine -\frac{2}{3}xy and \frac{9}{10}yx to get \frac{7}{30}xy.

$$\frac{2}{5}x^{2}+\frac{7}{30}xy+\frac{3}{2}y^{2}\left(-1\right)-\left(\frac{5}{4}x-\frac{1}{2}y\right)\left(\frac{1}{3}x+\frac{2}{5}y\right)$$

Multiply \frac{3}{2} and -1 to get -\frac{3}{2}.

$$\frac{2}{5}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\left(\frac{5}{4}x-\frac{1}{2}y\right)\left(\frac{1}{3}x+\frac{2}{5}y\right)$$

Apply the distributive property by multiplying each term of \frac{5}{4}x-\frac{1}{2}y by each term of \frac{1}{3}x+\frac{2}{5}y.

$$\frac{2}{5}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\left(\frac{5}{4}x\times \frac{1}{3}x+\frac{5}{4}x\times \frac{2}{5}y-\frac{1}{2}y\times \frac{1}{3}x-\frac{1}{2}y\times \frac{2}{5}y\right)$$

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

$$\frac{2}{5}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\left(\frac{5}{4}x^{2}\times \frac{1}{3}+\frac{5}{4}x\times \frac{2}{5}y-\frac{1}{2}y\times \frac{1}{3}x-\frac{1}{2}y\times \frac{2}{5}y\right)$$

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

$$\frac{2}{5}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\left(\frac{5}{4}x^{2}\times \frac{1}{3}+\frac{5}{4}x\times \frac{2}{5}y-\frac{1}{2}y\times \frac{1}{3}x-\frac{1}{2}y^{2}\times \frac{2}{5}\right)$$

Multiply \frac{5}{4} times \frac{1}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2}{5}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\left(\frac{5\times 1}{4\times 3}x^{2}+\frac{5}{4}x\times \frac{2}{5}y-\frac{1}{2}y\times \frac{1}{3}x-\frac{1}{2}y^{2}\times \frac{2}{5}\right)$$

Do the multiplications in the fraction \frac{5\times 1}{4\times 3}.

$$\frac{2}{5}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\left(\frac{5}{12}x^{2}+\frac{5}{4}x\times \frac{2}{5}y-\frac{1}{2}y\times \frac{1}{3}x-\frac{1}{2}y^{2}\times \frac{2}{5}\right)$$

Multiply \frac{5}{4} times \frac{2}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2}{5}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\left(\frac{5}{12}x^{2}+\frac{5\times 2}{4\times 5}xy-\frac{1}{2}y\times \frac{1}{3}x-\frac{1}{2}y^{2}\times \frac{2}{5}\right)$$

Cancel out 5 in both numerator and denominator.

$$\frac{2}{5}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\left(\frac{5}{12}x^{2}+\frac{2}{4}xy-\frac{1}{2}y\times \frac{1}{3}x-\frac{1}{2}y^{2}\times \frac{2}{5}\right)$$

Reduce the fraction \frac{2}{4} to lowest terms by extracting and canceling out 2.

$$\frac{2}{5}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\left(\frac{5}{12}x^{2}+\frac{1}{2}xy-\frac{1}{2}y\times \frac{1}{3}x-\frac{1}{2}y^{2}\times \frac{2}{5}\right)$$

Multiply -\frac{1}{2} times \frac{1}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2}{5}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\left(\frac{5}{12}x^{2}+\frac{1}{2}xy+\frac{-1}{2\times 3}yx-\frac{1}{2}y^{2}\times \frac{2}{5}\right)$$

Do the multiplications in the fraction \frac{-1}{2\times 3}.

$$\frac{2}{5}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\left(\frac{5}{12}x^{2}+\frac{1}{2}xy+\frac{-1}{6}yx-\frac{1}{2}y^{2}\times \frac{2}{5}\right)$$

Fraction \frac{-1}{6} can be rewritten as -\frac{1}{6} by extracting the negative sign.

$$\frac{2}{5}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\left(\frac{5}{12}x^{2}+\frac{1}{2}xy-\frac{1}{6}yx-\frac{1}{2}y^{2}\times \frac{2}{5}\right)$$

Combine \frac{1}{2}xy and -\frac{1}{6}yx to get \frac{1}{3}xy.

$$\frac{2}{5}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\left(\frac{5}{12}x^{2}+\frac{1}{3}xy-\frac{1}{2}y^{2}\times \frac{2}{5}\right)$$

Multiply -\frac{1}{2} times \frac{2}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2}{5}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\left(\frac{5}{12}x^{2}+\frac{1}{3}xy+\frac{-2}{2\times 5}y^{2}\right)$$

Cancel out 2 in both numerator and denominator.

$$\frac{2}{5}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\left(\frac{5}{12}x^{2}+\frac{1}{3}xy+\frac{-1}{5}y^{2}\right)$$

Fraction \frac{-1}{5} can be rewritten as -\frac{1}{5} by extracting the negative sign.

$$\frac{2}{5}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\left(\frac{5}{12}x^{2}+\frac{1}{3}xy-\frac{1}{5}y^{2}\right)$$

To find the opposite of \frac{5}{12}x^{2}+\frac{1}{3}xy-\frac{1}{5}y^{2}, find the opposite of each term.

$$\frac{2}{5}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\frac{5}{12}x^{2}-\frac{1}{3}xy-\left(-\frac{1}{5}y^{2}\right)$$

The opposite of -\frac{1}{5}y^{2} is \frac{1}{5}y^{2}.

$$\frac{2}{5}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\frac{5}{12}x^{2}-\frac{1}{3}xy+\frac{1}{5}y^{2}$$

Combine \frac{2}{5}x^{2} and -\frac{5}{12}x^{2} to get -\frac{1}{60}x^{2}.

$$-\frac{1}{60}x^{2}+\frac{7}{30}xy-\frac{3}{2}y^{2}-\frac{1}{3}xy+\frac{1}{5}y^{2}$$

Combine \frac{7}{30}xy and -\frac{1}{3}xy to get -\frac{1}{10}xy.

$$-\frac{1}{60}x^{2}-\frac{1}{10}xy-\frac{3}{2}y^{2}+\frac{1}{5}y^{2}$$

Combine -\frac{3}{2}y^{2} and \frac{1}{5}y^{2} to get -\frac{13}{10}y^{2}.

$$-\frac{1}{60}x^{2}-\frac{1}{10}xy-\frac{13}{10}y^{2}$$