Solve 5/5z-11+4/2z-3:-3/5-z

$$\frac { 5 } { 5 z - 11 } + \frac { 4 } { 2 z - 3 } : \frac { - 3 } { 5 - z }$$

Evaluate

$$\frac{20z^{2}-114z+175}{3\left(2z-3\right)\left(5z-11\right)}$$

Short Solution Steps

Divide \frac{4}{2z-3} by \frac{-3}{5-z} by multiplying \frac{4}{2z-3} by the reciprocal of \frac{-3}{5-z}.

$$\frac{5}{5z-11}+\frac{4\left(5-z\right)}{\left(2z-3\right)\left(-3\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5z-11 and \left(2z-3\right)\left(-3\right) is -3\left(2z-3\right)\left(5z-11\right). Multiply \frac{5}{5z-11} times \frac{-3\left(2z-3\right)}{-3\left(2z-3\right)}. Multiply \frac{4\left(5-z\right)}{\left(2z-3\right)\left(-3\right)} times \frac{5z-11}{5z-11}.

$$\frac{5\left(-3\right)\left(2z-3\right)}{-3\left(2z-3\right)\left(5z-11\right)}+\frac{4\left(5-z\right)\left(5z-11\right)}{-3\left(2z-3\right)\left(5z-11\right)}$$

Since \frac{5\left(-3\right)\left(2z-3\right)}{-3\left(2z-3\right)\left(5z-11\right)} and \frac{4\left(5-z\right)\left(5z-11\right)}{-3\left(2z-3\right)\left(5z-11\right)} have the same denominator, add them by adding their numerators.

$$\frac{5\left(-3\right)\left(2z-3\right)+4\left(5-z\right)\left(5z-11\right)}{-3\left(2z-3\right)\left(5z-11\right)}$$

Do the multiplications in 5\left(-3\right)\left(2z-3\right)+4\left(5-z\right)\left(5z-11\right).

$$\frac{-30z+45+100z-220-20z^{2}+44z}{-3\left(2z-3\right)\left(5z-11\right)}$$

Combine like terms in -30z+45+100z-220-20z^{2}+44z.

$$\frac{114z-175-20z^{2}}{-3\left(2z-3\right)\left(5z-11\right)}$$

Expand -3\left(2z-3\right)\left(5z-11\right).

$$\frac{114z-175-20z^{2}}{-30z^{2}+111z-99}$$

Factor

$$\frac{20z^{2}-114z+175}{3\left(2z-3\right)\left(5z-11\right)}$$