Solve (1/x-3+2/y)*(x/4+y)

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

Evaluate

$$\frac{\left(x+4y\right)\left(2x+y-6\right)}{4y\left(x-3\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-3 and y is y\left(x-3\right). Multiply \frac{1}{x-3} times \frac{y}{y}. Multiply \frac{2}{y} times \frac{x-3}{x-3}.

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

Since \frac{y}{y\left(x-3\right)} and \frac{2\left(x-3\right)}{y\left(x-3\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in y+2\left(x-3\right).

$$\frac{y+2x-6}{y\left(x-3\right)}\left(\frac{x}{4}+y\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply y times \frac{4}{4}.

$$\frac{y+2x-6}{y\left(x-3\right)}\left(\frac{x}{4}+\frac{4y}{4}\right)$$

Since \frac{x}{4} and \frac{4y}{4} have the same denominator, add them by adding their numerators.

$$\frac{y+2x-6}{y\left(x-3\right)}\times \frac{x+4y}{4}$$

Multiply \frac{y+2x-6}{y\left(x-3\right)} times \frac{x+4y}{4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(y+2x-6\right)\left(x+4y\right)}{y\left(x-3\right)\times 4}$$

Apply the distributive property by multiplying each term of y+2x-6 by each term of x+4y.

$$\frac{yx+4y^{2}+2x^{2}+8xy-6x-24y}{y\left(x-3\right)\times 4}$$

Combine yx and 8xy to get 9yx.

$$\frac{9yx+4y^{2}+2x^{2}-6x-24y}{y\left(x-3\right)\times 4}$$

Use the distributive property to multiply y by x-3.

$$\frac{9yx+4y^{2}+2x^{2}-6x-24y}{\left(yx-3y\right)\times 4}$$

Use the distributive property to multiply yx-3y by 4.

$$\frac{9yx+4y^{2}+2x^{2}-6x-24y}{4yx-12y}$$

Expand

$$\frac{2x^{2}+9xy-6x+4y^{2}-24y}{4y\left(x-3\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-3 and y is y\left(x-3\right). Multiply \frac{1}{x-3} times \frac{y}{y}. Multiply \frac{2}{y} times \frac{x-3}{x-3}.

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

Since \frac{y}{y\left(x-3\right)} and \frac{2\left(x-3\right)}{y\left(x-3\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in y+2\left(x-3\right).

$$\frac{y+2x-6}{y\left(x-3\right)}\left(\frac{x}{4}+y\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply y times \frac{4}{4}.

$$\frac{y+2x-6}{y\left(x-3\right)}\left(\frac{x}{4}+\frac{4y}{4}\right)$$

Since \frac{x}{4} and \frac{4y}{4} have the same denominator, add them by adding their numerators.

$$\frac{y+2x-6}{y\left(x-3\right)}\times \frac{x+4y}{4}$$

Multiply \frac{y+2x-6}{y\left(x-3\right)} times \frac{x+4y}{4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(y+2x-6\right)\left(x+4y\right)}{y\left(x-3\right)\times 4}$$

Apply the distributive property by multiplying each term of y+2x-6 by each term of x+4y.

$$\frac{yx+4y^{2}+2x^{2}+8xy-6x-24y}{y\left(x-3\right)\times 4}$$

Combine yx and 8xy to get 9yx.

$$\frac{9yx+4y^{2}+2x^{2}-6x-24y}{y\left(x-3\right)\times 4}$$

Use the distributive property to multiply y by x-3.

$$\frac{9yx+4y^{2}+2x^{2}-6x-24y}{\left(yx-3y\right)\times 4}$$

Use the distributive property to multiply yx-3y by 4.

$$\frac{9yx+4y^{2}+2x^{2}-6x-24y}{4yx-12y}$$