Solve (2)/((x)^(2)-1)-(1)/(y-1)=(2)/(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

$$\frac{ 2 }{ { x }^{ 2 } -1 } - \frac{ 1 }{ y-1 } = \frac{ 2 }{ 1 }$$

Solve for y (complex solution)

$y=-\frac{x^{2}-3}{2\left(2-x^{2}\right)}$
$x\neq -1\text{ and }x\neq 1\text{ and }x\neq -\sqrt{2}\text{ and }x\neq \sqrt{2}$

Steps for Solving Linear Equation

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

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

Use the distributive property to multiply $y-1$ by $2$.

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

To find the opposite of $x^{2}-1$, find the opposite of each term.

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

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

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

Use the distributive property to multiply $y-1$ by $x^{2}-1$.

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

Use the distributive property to multiply $yx^{2}-y-x^{2}+1$ by $2$.

$$2y-1-x^{2}=2yx^{2}-2y-2x^{2}+2$$

Subtract $2yx^{2}$ from both sides.

$$2y-1-x^{2}-2yx^{2}=-2y-2x^{2}+2$$

Add $2y$ to both sides.

$$2y-1-x^{2}-2yx^{2}+2y=-2x^{2}+2$$

Combine $2y$ and $2y$ to get $4y$.

$$4y-1-x^{2}-2yx^{2}=-2x^{2}+2$$

Add $1$ to both sides.

$$4y-x^{2}-2yx^{2}=-2x^{2}+2+1$$

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

$$4y-x^{2}-2yx^{2}=-2x^{2}+3$$

Add $x^{2}$ to both sides.

$$4y-2yx^{2}=-2x^{2}+3+x^{2}$$

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

$$4y-2yx^{2}=-x^{2}+3$$

Combine all terms containing $y$.

$$\left(4-2x^{2}\right)y=-x^{2}+3$$

The equation is in standard form.

$$\left(4-2x^{2}\right)y=3-x^{2}$$

Divide both sides by $-2x^{2}+4$.

$$\frac{\left(4-2x^{2}\right)y}{4-2x^{2}}=\frac{3-x^{2}}{4-2x^{2}}$$

Dividing by $-2x^{2}+4$ undoes the multiplication by $-2x^{2}+4$.

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

Divide $3-x^{2}$ by $-2x^{2}+4$.

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

Variable $y$ cannot be equal to $1$.

$$y=\frac{3-x^{2}}{2\left(2-x^{2}\right)}\text{, }y\neq 1$$

Solve for y

$y=-\frac{x^{2}-3}{2\left(2-x^{2}\right)}$
$|x|\neq \sqrt{2}\text{ and }|x|\neq 1$

Steps for Solving Linear Equation

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

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

Use the distributive property to multiply $y-1$ by $2$.

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

To find the opposite of $x^{2}-1$, find the opposite of each term.

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

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

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

Use the distributive property to multiply $y-1$ by $x^{2}-1$.

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

Use the distributive property to multiply $yx^{2}-y-x^{2}+1$ by $2$.

$$2y-1-x^{2}=2yx^{2}-2y-2x^{2}+2$$

Subtract $2yx^{2}$ from both sides.

$$2y-1-x^{2}-2yx^{2}=-2y-2x^{2}+2$$

Add $2y$ to both sides.

$$2y-1-x^{2}-2yx^{2}+2y=-2x^{2}+2$$

Combine $2y$ and $2y$ to get $4y$.

$$4y-1-x^{2}-2yx^{2}=-2x^{2}+2$$

Add $1$ to both sides.

$$4y-x^{2}-2yx^{2}=-2x^{2}+2+1$$

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

$$4y-x^{2}-2yx^{2}=-2x^{2}+3$$

Add $x^{2}$ to both sides.

$$4y-2yx^{2}=-2x^{2}+3+x^{2}$$

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

$$4y-2yx^{2}=-x^{2}+3$$

Combine all terms containing $y$.

$$\left(4-2x^{2}\right)y=-x^{2}+3$$

The equation is in standard form.

$$\left(4-2x^{2}\right)y=3-x^{2}$$

Divide both sides by $-2x^{2}+4$.

$$\frac{\left(4-2x^{2}\right)y}{4-2x^{2}}=\frac{3-x^{2}}{4-2x^{2}}$$

Dividing by $-2x^{2}+4$ undoes the multiplication by $-2x^{2}+4$.

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

Divide $-x^{2}+3$ by $-2x^{2}+4$.

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

Variable $y$ cannot be equal to $1$.

$$y=\frac{3-x^{2}}{2\left(2-x^{2}\right)}\text{, }y\neq 1$$

Solve for x (complex solution)

$x=-i\left(1-2y\right)^{-\frac{1}{2}}\sqrt{4y-3}$
$x=i\left(1-2y\right)^{-\frac{1}{2}}\sqrt{4y-3}\text{, }y\neq 1\text{ and }y\neq \frac{1}{2}$

Solve for x

$x=\sqrt{-\frac{4y-3}{1-2y}}$
$x=-\sqrt{-\frac{4y-3}{1-2y}}\text{, }y<\frac{1}{2}\text{ or }\left(y\neq 1\text{ and }y\geq \frac{3}{4}\right)$