Solve Prove that : a ) (x - 1)/(x - 3) - (x ^ 2 - x + 2)/(x ^ 2 - 2x - 3) = 1/(x + 1) | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$\left. \begin{array} { l } { \frac { x - y } { x - 3 } - \frac { x ^ { 2 } - x + 2 } { x ^ { 2 } - 2 x - 3 } = \frac { 1 } { x + 1 } } \end{array} \right.$$

Solve for x (complex solution)

$x\in \mathrm{C}\setminus -1,3$
$y=1$

Steps for Solving Linear Equation

Variable $x$ cannot be equal to any of the values $-1,3$ since division by zero is not defined. Multiply both sides of the equation by $\left(x-3\right)\left(x+1\right)$, the least common multiple of $x-3,x^{2}-2x-3,x+1$.

$$\left(x+1\right)\left(x-y\right)-\left(x^{2}-x+2\right)=x-3$$

Use the distributive property to multiply $x+1$ by $x-y$.

$$x^{2}-xy+x-y-\left(x^{2}-x+2\right)=x-3$$

To find the opposite of $x^{2}-x+2$, find the opposite of each term.

$$x^{2}-xy+x-y-x^{2}+x-2=x-3$$

Combine $x^{2}$ and $-x^{2}$ to get $0$.

$$-xy+x-y+x-2=x-3$$

Combine $x$ and $x$ to get $2x$.

$$-xy+2x-y-2=x-3$$

Subtract $x$ from both sides.

$$-xy+2x-y-2-x=-3$$

Combine $2x$ and $-x$ to get $x$.

$$-xy+x-y-2=-3$$

Add $y$ to both sides.

$$-xy+x-2=-3+y$$

Add $2$ to both sides.

$$-xy+x=-3+y+2$$

Add $-3$ and $2$ to get $-1$.

$$-xy+x=-1+y$$

Combine all terms containing $x$.

$$\left(-y+1\right)x=-1+y$$

The equation is in standard form.

$$\left(1-y\right)x=y-1$$

Divide both sides by $1-y$.

$$\frac{\left(1-y\right)x}{1-y}=\frac{y-1}{1-y}$$

Dividing by $1-y$ undoes the multiplication by $1-y$.

$$x=\frac{y-1}{1-y}$$

Divide $y-1$ by $1-y$.

$$x=-1$$

Variable $x$ cannot be equal to $-1$.

$$x\in \emptyset$$

Solve for x

$x\in \mathrm{R}\setminus 3,-1$
$y=1$

Steps for Solving Linear Equation

Variable $x$ cannot be equal to any of the values $-1,3$ since division by zero is not defined. Multiply both sides of the equation by $\left(x-3\right)\left(x+1\right)$, the least common multiple of $x-3,x^{2}-2x-3,x+1$.

$$\left(x+1\right)\left(x-y\right)-\left(x^{2}-x+2\right)=x-3$$

Use the distributive property to multiply $x+1$ by $x-y$.

$$x^{2}-xy+x-y-\left(x^{2}-x+2\right)=x-3$$

To find the opposite of $x^{2}-x+2$, find the opposite of each term.

$$x^{2}-xy+x-y-x^{2}+x-2=x-3$$

Combine $x^{2}$ and $-x^{2}$ to get $0$.

$$-xy+x-y+x-2=x-3$$

Combine $x$ and $x$ to get $2x$.

$$-xy+2x-y-2=x-3$$

Subtract $x$ from both sides.

$$-xy+2x-y-2-x=-3$$

Combine $2x$ and $-x$ to get $x$.

$$-xy+x-y-2=-3$$

Add $y$ to both sides.

$$-xy+x-2=-3+y$$

Add $2$ to both sides.

$$-xy+x=-3+y+2$$

Add $-3$ and $2$ to get $-1$.

$$-xy+x=-1+y$$

Combine all terms containing $x$.

$$\left(-y+1\right)x=-1+y$$

The equation is in standard form.

$$\left(1-y\right)x=y-1$$

Divide both sides by $1-y$.

$$\frac{\left(1-y\right)x}{1-y}=\frac{y-1}{1-y}$$

Dividing by $1-y$ undoes the multiplication by $1-y$.

$$x=\frac{y-1}{1-y}$$

Divide $y-1$ by $1-y$.

$$x=-1$$

Variable $x$ cannot be equal to $-1$.

$$x\in \emptyset$$