Solve z/z-2-7/z+5=z^2/z^2+3z-10

$$\frac { z } { z - 2 } - \frac { 7 } { z + 5 } = \frac { z ^ { 2 } } { z ^ { 2 } + 3 z - 10 }$$

Solve for z

$$z=7$$

Steps for Solving Linear Equation

Variable z cannot be equal to any of the values -5,2 since division by zero is not defined. Multiply both sides of the equation by \left(z-2\right)\left(z+5\right), the least common multiple of z-2,z+5,z^{2}+3z-10.

$$\left(z+5\right)z-\left(z-2\right)\times 7=z^{2}$$

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

$$z^{2}+5z-\left(z-2\right)\times 7=z^{2}$$

Use the distributive property to multiply z-2 by 7.

$$z^{2}+5z-\left(7z-14\right)=z^{2}$$

To find the opposite of 7z-14, find the opposite of each term.

$$z^{2}+5z-7z+14=z^{2}$$

Combine 5z and -7z to get -2z.

$$z^{2}-2z+14=z^{2}$$

Subtract z^{2} from both sides.

$$z^{2}-2z+14-z^{2}=0$$

Combine z^{2} and -z^{2} to get 0.

$$-2z+14=0$$

Subtract 14 from both sides. Anything subtracted from zero gives its negation.

$$-2z=-14$$

Divide both sides by -2.

$$z=\frac{-14}{-2}$$

Divide -14 by -2 to get 7.

$$z=7$$