Solve z-3-z-1/9-z/frac{z{6}-2-z/z-6}

$$\frac { z - 3 - \frac { z - 1 } { 9 - z } } { \frac { z } { 6 } - \frac { 2 - z } { z - 6 } }$$

Evaluate

$$\frac{6\left(z-6\right)\left(-z^{2}+11z-26\right)}{\left(9-z\right)\left(z^{2}-12\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply z-3 times \frac{9-z}{9-z}.

$$\frac{\frac{\left(z-3\right)\left(9-z\right)}{9-z}-\frac{z-1}{9-z}}{\frac{z}{6}-\frac{2-z}{z-6}}$$

Since \frac{\left(z-3\right)\left(9-z\right)}{9-z} and \frac{z-1}{9-z} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{\left(z-3\right)\left(9-z\right)-\left(z-1\right)}{9-z}}{\frac{z}{6}-\frac{2-z}{z-6}}$$

Do the multiplications in \left(z-3\right)\left(9-z\right)-\left(z-1\right).

$$\frac{\frac{9z-z^{2}-27+3z-z+1}{9-z}}{\frac{z}{6}-\frac{2-z}{z-6}}$$

Combine like terms in 9z-z^{2}-27+3z-z+1.

$$\frac{\frac{11z-z^{2}-26}{9-z}}{\frac{z}{6}-\frac{2-z}{z-6}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 6 and z-6 is 6\left(z-6\right). Multiply \frac{z}{6} times \frac{z-6}{z-6}. Multiply \frac{2-z}{z-6} times \frac{6}{6}.

$$\frac{\frac{11z-z^{2}-26}{9-z}}{\frac{z\left(z-6\right)}{6\left(z-6\right)}-\frac{6\left(2-z\right)}{6\left(z-6\right)}}$$

Since \frac{z\left(z-6\right)}{6\left(z-6\right)} and \frac{6\left(2-z\right)}{6\left(z-6\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{11z-z^{2}-26}{9-z}}{\frac{z\left(z-6\right)-6\left(2-z\right)}{6\left(z-6\right)}}$$

Do the multiplications in z\left(z-6\right)-6\left(2-z\right).

$$\frac{\frac{11z-z^{2}-26}{9-z}}{\frac{z^{2}-6z-12+6z}{6\left(z-6\right)}}$$

Combine like terms in z^{2}-6z-12+6z.

$$\frac{\frac{11z-z^{2}-26}{9-z}}{\frac{z^{2}-12}{6\left(z-6\right)}}$$

Divide \frac{11z-z^{2}-26}{9-z} by \frac{z^{2}-12}{6\left(z-6\right)} by multiplying \frac{11z-z^{2}-26}{9-z} by the reciprocal of \frac{z^{2}-12}{6\left(z-6\right)}.

$$\frac{\left(11z-z^{2}-26\right)\times 6\left(z-6\right)}{\left(9-z\right)\left(z^{2}-12\right)}$$

Use the distributive property to multiply 11z-z^{2}-26 by 6.

$$\frac{\left(66z-6z^{2}-156\right)\left(z-6\right)}{\left(9-z\right)\left(z^{2}-12\right)}$$

Apply the distributive property by multiplying each term of 66z-6z^{2}-156 by each term of z-6.

$$\frac{66z^{2}-396z-6z^{3}+36z^{2}-156z+936}{\left(9-z\right)\left(z^{2}-12\right)}$$

Combine 66z^{2} and 36z^{2} to get 102z^{2}.

$$\frac{102z^{2}-396z-6z^{3}-156z+936}{\left(9-z\right)\left(z^{2}-12\right)}$$

Combine -396z and -156z to get -552z.

$$\frac{102z^{2}-552z-6z^{3}+936}{\left(9-z\right)\left(z^{2}-12\right)}$$

Apply the distributive property by multiplying each term of 9-z by each term of z^{2}-12.

$$\frac{102z^{2}-552z-6z^{3}+936}{9z^{2}-108-z^{3}+12z}$$

Expand

$$-\frac{6\left(z^{3}-17z^{2}+92z-156\right)}{\left(9-z\right)\left(z^{2}-12\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply z-3 times \frac{9-z}{9-z}.

$$\frac{\frac{\left(z-3\right)\left(9-z\right)}{9-z}-\frac{z-1}{9-z}}{\frac{z}{6}-\frac{2-z}{z-6}}$$

Since \frac{\left(z-3\right)\left(9-z\right)}{9-z} and \frac{z-1}{9-z} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{\left(z-3\right)\left(9-z\right)-\left(z-1\right)}{9-z}}{\frac{z}{6}-\frac{2-z}{z-6}}$$

Do the multiplications in \left(z-3\right)\left(9-z\right)-\left(z-1\right).

$$\frac{\frac{9z-z^{2}-27+3z-z+1}{9-z}}{\frac{z}{6}-\frac{2-z}{z-6}}$$

Combine like terms in 9z-z^{2}-27+3z-z+1.

$$\frac{\frac{11z-z^{2}-26}{9-z}}{\frac{z}{6}-\frac{2-z}{z-6}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 6 and z-6 is 6\left(z-6\right). Multiply \frac{z}{6} times \frac{z-6}{z-6}. Multiply \frac{2-z}{z-6} times \frac{6}{6}.

$$\frac{\frac{11z-z^{2}-26}{9-z}}{\frac{z\left(z-6\right)}{6\left(z-6\right)}-\frac{6\left(2-z\right)}{6\left(z-6\right)}}$$

Since \frac{z\left(z-6\right)}{6\left(z-6\right)} and \frac{6\left(2-z\right)}{6\left(z-6\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{11z-z^{2}-26}{9-z}}{\frac{z\left(z-6\right)-6\left(2-z\right)}{6\left(z-6\right)}}$$

Do the multiplications in z\left(z-6\right)-6\left(2-z\right).

$$\frac{\frac{11z-z^{2}-26}{9-z}}{\frac{z^{2}-6z-12+6z}{6\left(z-6\right)}}$$

Combine like terms in z^{2}-6z-12+6z.

$$\frac{\frac{11z-z^{2}-26}{9-z}}{\frac{z^{2}-12}{6\left(z-6\right)}}$$

Divide \frac{11z-z^{2}-26}{9-z} by \frac{z^{2}-12}{6\left(z-6\right)} by multiplying \frac{11z-z^{2}-26}{9-z} by the reciprocal of \frac{z^{2}-12}{6\left(z-6\right)}.

$$\frac{\left(11z-z^{2}-26\right)\times 6\left(z-6\right)}{\left(9-z\right)\left(z^{2}-12\right)}$$

Use the distributive property to multiply 11z-z^{2}-26 by 6.

$$\frac{\left(66z-6z^{2}-156\right)\left(z-6\right)}{\left(9-z\right)\left(z^{2}-12\right)}$$

Apply the distributive property by multiplying each term of 66z-6z^{2}-156 by each term of z-6.

$$\frac{66z^{2}-396z-6z^{3}+36z^{2}-156z+936}{\left(9-z\right)\left(z^{2}-12\right)}$$

Combine 66z^{2} and 36z^{2} to get 102z^{2}.

$$\frac{102z^{2}-396z-6z^{3}-156z+936}{\left(9-z\right)\left(z^{2}-12\right)}$$

Combine -396z and -156z to get -552z.

$$\frac{102z^{2}-552z-6z^{3}+936}{\left(9-z\right)\left(z^{2}-12\right)}$$

Apply the distributive property by multiplying each term of 9-z by each term of z^{2}-12.

$$\frac{102z^{2}-552z-6z^{3}+936}{9z^{2}-108-z^{3}+12z}$$