Solve x+2/y-2=z-3/y^2-4}

$$\frac { x + 2 } { y - 2 } = \frac { z - 3 } { y ^ { 2 } - 4 } \}$$

Solve for x

$$x=-\frac{2y-z+7}{y+2}$$

$$|y|\neq 2$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(y-2\right)\left(y+2\right), the least common multiple of y-2,y^{2}-4.

$$\left(y+2\right)\left(x+2\right)=z-3$$

Use the distributive property to multiply y+2 by x+2.

$$yx+2y+2x+4=z-3$$

Subtract 2y from both sides.

$$yx+2x+4=z-3-2y$$

Subtract 4 from both sides.

$$yx+2x=z-3-2y-4$$

Subtract 4 from -3 to get -7.

$$yx+2x=z-7-2y$$

Combine all terms containing x.

$$\left(y+2\right)x=z-7-2y$$

The equation is in standard form.

$$\left(y+2\right)x=-2y+z-7$$

Divide both sides by y+2.

$$\frac{\left(y+2\right)x}{y+2}=\frac{-2y+z-7}{y+2}$$

Dividing by y+2 undoes the multiplication by y+2.

$$x=\frac{-2y+z-7}{y+2}$$

Solve for y

$$\left\{\begin{matrix}y=-\frac{2x-z+7}{x+2}\text{, }&x\neq \frac{z-11}{4}\text{ and }x\neq -2\text{ and }z\neq 3\\y\in \mathrm{R}\setminus 2,-2\text{, }&x=-2\text{ and }z=3\end{matrix}\right.$$

Steps for Solving Linear Equation

Variable y cannot be equal to any of the values -2,2 since division by zero is not defined. Multiply both sides of the equation by \left(y-2\right)\left(y+2\right), the least common multiple of y-2,y^{2}-4.

$$\left(y+2\right)\left(x+2\right)=z-3$$

Use the distributive property to multiply y+2 by x+2.

$$yx+2y+2x+4=z-3$$

Subtract 2x from both sides.

$$yx+2y+4=z-3-2x$$

Subtract 4 from both sides.

$$yx+2y=z-3-2x-4$$

Subtract 4 from -3 to get -7.

$$yx+2y=z-7-2x$$

Combine all terms containing y.

$$\left(x+2\right)y=z-7-2x$$

The equation is in standard form.

$$\left(x+2\right)y=-2x+z-7$$

Divide both sides by x+2.

$$\frac{\left(x+2\right)y}{x+2}=\frac{-2x+z-7}{x+2}$$

Dividing by x+2 undoes the multiplication by x+2.

$$y=\frac{-2x+z-7}{x+2}$$

Variable y cannot be equal to any of the values -2,2.

$$y=\frac{-2x+z-7}{x+2}\text{, }y\neq -2\text{ and }y\neq 2$$