Solve 5x-y/12x+3y+8xy+2y^2/48x^2+24xy+3y^2-y+x/3y+12

$$\frac { 5 x - y } { 12 x + 3 y } + \frac { 8 x y + 2 y ^ { 2 } } { 48 x ^ { 2 } + 24 x y + 3 y ^ { 2 } } - \frac { y + x } { 3 y + 12 }$$

Evaluate

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{8xy+2y^{2}}{48x^{2}+24xy+3y^{2}}.

$$\frac{5x-y}{12x+3y}+\frac{2y\left(4x+y\right)}{3\left(4x+y\right)^{2}}-\frac{y+x}{3y+12}$$

Cancel out 4x+y in both numerator and denominator.

$$\frac{5x-y}{12x+3y}+\frac{2y}{3\left(4x+y\right)}-\frac{y+x}{3y+12}$$

Factor 12x+3y.

$$\frac{5x-y}{3\left(4x+y\right)}+\frac{2y}{3\left(4x+y\right)}-\frac{y+x}{3y+12}$$

Since \frac{5x-y}{3\left(4x+y\right)} and \frac{2y}{3\left(4x+y\right)} have the same denominator, add them by adding their numerators.

$$\frac{5x-y+2y}{3\left(4x+y\right)}-\frac{y+x}{3y+12}$$

Combine like terms in 5x-y+2y.

$$\frac{5x+y}{3\left(4x+y\right)}-\frac{y+x}{3y+12}$$

Factor 3y+12.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3\left(4x+y\right) and 3\left(y+4\right) is 3\left(y+4\right)\left(4x+y\right). Multiply \frac{5x+y}{3\left(4x+y\right)} times \frac{y+4}{y+4}. Multiply \frac{y+x}{3\left(y+4\right)} times \frac{4x+y}{4x+y}.

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

Since \frac{\left(5x+y\right)\left(y+4\right)}{3\left(y+4\right)\left(4x+y\right)} and \frac{\left(y+x\right)\left(4x+y\right)}{3\left(y+4\right)\left(4x+y\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(5x+y\right)\left(y+4\right)-\left(y+x\right)\left(4x+y\right).

$$\frac{5xy+20x+y^{2}+4y-4yx-y^{2}-4x^{2}-xy}{3\left(y+4\right)\left(4x+y\right)}$$

Combine like terms in 5xy+20x+y^{2}+4y-4yx-y^{2}-4x^{2}-xy.

$$\frac{20x+4y-4x^{2}}{3\left(y+4\right)\left(4x+y\right)}$$

Expand 3\left(y+4\right)\left(4x+y\right).

$$\frac{20x+4y-4x^{2}}{12xy+48x+3y^{2}+12y}$$

Expand

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{8xy+2y^{2}}{48x^{2}+24xy+3y^{2}}.

$$\frac{5x-y}{12x+3y}+\frac{2y\left(4x+y\right)}{3\left(4x+y\right)^{2}}-\frac{y+x}{3y+12}$$

Cancel out 4x+y in both numerator and denominator.

$$\frac{5x-y}{12x+3y}+\frac{2y}{3\left(4x+y\right)}-\frac{y+x}{3y+12}$$

Factor 12x+3y.

$$\frac{5x-y}{3\left(4x+y\right)}+\frac{2y}{3\left(4x+y\right)}-\frac{y+x}{3y+12}$$

Since \frac{5x-y}{3\left(4x+y\right)} and \frac{2y}{3\left(4x+y\right)} have the same denominator, add them by adding their numerators.

$$\frac{5x-y+2y}{3\left(4x+y\right)}-\frac{y+x}{3y+12}$$

Combine like terms in 5x-y+2y.

$$\frac{5x+y}{3\left(4x+y\right)}-\frac{y+x}{3y+12}$$

Factor 3y+12.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3\left(4x+y\right) and 3\left(y+4\right) is 3\left(y+4\right)\left(4x+y\right). Multiply \frac{5x+y}{3\left(4x+y\right)} times \frac{y+4}{y+4}. Multiply \frac{y+x}{3\left(y+4\right)} times \frac{4x+y}{4x+y}.

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

Since \frac{\left(5x+y\right)\left(y+4\right)}{3\left(y+4\right)\left(4x+y\right)} and \frac{\left(y+x\right)\left(4x+y\right)}{3\left(y+4\right)\left(4x+y\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(5x+y\right)\left(y+4\right)-\left(y+x\right)\left(4x+y\right).

$$\frac{5xy+20x+y^{2}+4y-4yx-y^{2}-4x^{2}-xy}{3\left(y+4\right)\left(4x+y\right)}$$

Combine like terms in 5xy+20x+y^{2}+4y-4yx-y^{2}-4x^{2}-xy.

$$\frac{20x+4y-4x^{2}}{3\left(y+4\right)\left(4x+y\right)}$$

Expand 3\left(y+4\right)\left(4x+y\right).

$$\frac{20x+4y-4x^{2}}{12xy+48x+3y^{2}+12y}$$