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

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

Evaluate

$$\frac{\left(6x-1\right)\left(3x+2y\right)}{12}$$

Solution Steps

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

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

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

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

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

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

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

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

Do the multiplications in the fraction \frac{1\left(-1\right)}{2\times 2}.

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

Fraction \frac{-1}{4} can be rewritten as -\frac{1}{4} by extracting the negative sign.

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

Cancel out 3 and 3.

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

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

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

Do the multiplications in the fraction \frac{1\left(-1\right)}{3\times 2}.

$$\frac{3}{2}x^{2}-\frac{1}{4}x+yx+\frac{-1}{6}y$$

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

$$\frac{3}{2}x^{2}-\frac{1}{4}x+yx-\frac{1}{6}y$$

Expand

$$xy+\frac{3x^{2}}{2}-\frac{x}{4}-\frac{y}{6}$$

Solution Steps

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

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

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

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

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

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

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

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

Do the multiplications in the fraction \frac{1\left(-1\right)}{2\times 2}.

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

Fraction \frac{-1}{4} can be rewritten as -\frac{1}{4} by extracting the negative sign.

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

Cancel out 3 and 3.

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

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

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

Do the multiplications in the fraction \frac{1\left(-1\right)}{3\times 2}.

$$\frac{3}{2}x^{2}-\frac{1}{4}x+yx+\frac{-1}{6}y$$

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

$$\frac{3}{2}x^{2}-\frac{1}{4}x+yx-\frac{1}{6}y$$