Solve 2/x-2-3/y-1=1

$$\frac { 2 } { x - 2 } - \frac { 3 } { y - 1 } = 1$$

Solve for x

$$x=\frac{2\left(2y+1\right)}{y+2}$$

$$y\neq 1\text{ and }y\neq -2$$

Steps for Solving Linear Equation

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

$$\left(y-1\right)\times 2-\left(x-2\right)\times 3=\left(x-2\right)\left(y-1\right)$$

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

$$2y-2-\left(x-2\right)\times 3=\left(x-2\right)\left(y-1\right)$$

Use the distributive property to multiply x-2 by 3.

$$2y-2-\left(3x-6\right)=\left(x-2\right)\left(y-1\right)$$

To find the opposite of 3x-6, find the opposite of each term.

$$2y-2-3x+6=\left(x-2\right)\left(y-1\right)$$

Add -2 and 6 to get 4.

$$2y+4-3x=\left(x-2\right)\left(y-1\right)$$

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

$$2y+4-3x=xy-x-2y+2$$

Subtract xy from both sides.

$$2y+4-3x-xy=-x-2y+2$$

Add x to both sides.

$$2y+4-3x-xy+x=-2y+2$$

Combine -3x and x to get -2x.

$$2y+4-2x-xy=-2y+2$$

Subtract 2y from both sides.

$$4-2x-xy=-2y+2-2y$$

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

$$4-2x-xy=-4y+2$$

Subtract 4 from both sides.

$$-2x-xy=-4y+2-4$$

Subtract 4 from 2 to get -2.

$$-2x-xy=-4y-2$$

Combine all terms containing x.

$$\left(-2-y\right)x=-4y-2$$

The equation is in standard form.

$$\left(-y-2\right)x=-4y-2$$

Divide both sides by -y-2.

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

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

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

Divide -4y-2 by -y-2.

$$x=\frac{2\left(2y+1\right)}{y+2}$$

Variable x cannot be equal to 2.

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

Solve for y

$$y=-\frac{2\left(1-x\right)}{4-x}$$

$$x\neq 2\text{ and }x\neq 4$$

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-2\right)\left(y-1\right), the least common multiple of x-2,y-1.

$$\left(y-1\right)\times 2-\left(x-2\right)\times 3=\left(x-2\right)\left(y-1\right)$$

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

$$2y-2-\left(x-2\right)\times 3=\left(x-2\right)\left(y-1\right)$$

Use the distributive property to multiply x-2 by 3.

$$2y-2-\left(3x-6\right)=\left(x-2\right)\left(y-1\right)$$

To find the opposite of 3x-6, find the opposite of each term.

$$2y-2-3x+6=\left(x-2\right)\left(y-1\right)$$

Add -2 and 6 to get 4.

$$2y+4-3x=\left(x-2\right)\left(y-1\right)$$

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

$$2y+4-3x=xy-x-2y+2$$

Subtract xy from both sides.

$$2y+4-3x-xy=-x-2y+2$$

Add 2y to both sides.

$$2y+4-3x-xy+2y=-x+2$$

Combine 2y and 2y to get 4y.

$$4y+4-3x-xy=-x+2$$

Subtract 4 from both sides.

$$4y-3x-xy=-x+2-4$$

Subtract 4 from 2 to get -2.

$$4y-3x-xy=-x-2$$

Add 3x to both sides.

$$4y-xy=-x-2+3x$$

Combine -x and 3x to get 2x.

$$4y-xy=2x-2$$

Combine all terms containing y.

$$\left(4-x\right)y=2x-2$$

Divide both sides by -x+4.

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

Dividing by -x+4 undoes the multiplication by -x+4.

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

Divide -2+2x by -x+4.

$$y=\frac{2\left(x-1\right)}{4-x}$$

Variable y cannot be equal to 1.

$$y=\frac{2\left(x-1\right)}{4-x}\text{, }y\neq 1$$