Solve 4/z-2-9/2z+9+5/z-2=0

$$\frac { 4 } { z - 2 } - \frac { 9 } { 2 z + 9 } + \frac { 5 } { z - 2 } = 0$$

Solve for z

$$z=-11$$

Steps for Solving Linear Equation

Variable z cannot be equal to any of the values -\frac{9}{2},2 since division by zero is not defined. Multiply both sides of the equation by \left(z-2\right)\left(2z+9\right), the least common multiple of z-2,2z+9.

$$\left(2z+9\right)\times 4-\left(z-2\right)\times 9+\left(2z+9\right)\times 5=0$$

Use the distributive property to multiply 2z+9 by 4.

$$8z+36-\left(z-2\right)\times 9+\left(2z+9\right)\times 5=0$$

Use the distributive property to multiply z-2 by 9.

$$8z+36-\left(9z-18\right)+\left(2z+9\right)\times 5=0$$

To find the opposite of 9z-18, find the opposite of each term.

$$8z+36-9z+18+\left(2z+9\right)\times 5=0$$

Combine 8z and -9z to get -z.

$$-z+36+18+\left(2z+9\right)\times 5=0$$

Add 36 and 18 to get 54.

$$-z+54+\left(2z+9\right)\times 5=0$$

Use the distributive property to multiply 2z+9 by 5.

$$-z+54+10z+45=0$$

Combine -z and 10z to get 9z.

$$9z+54+45=0$$

Add 54 and 45 to get 99.

$$9z+99=0$$

Subtract 99 from both sides. Anything subtracted from zero gives its negation.

$$9z=-99$$

Divide both sides by 9.

$$z=\frac{-99}{9}$$

Divide -99 by 9 to get -11.

$$z=-11$$