Solve 2/x-1-1/y+2=4

$$\frac { 2 } { x - 1 } - \frac { 1 } { y + 2 } = 4$$

Solve for x

$$x=\frac{6y+13}{4y+9}$$

$$y\neq -2\text{ and }y\neq -\frac{9}{4}$$

Steps for Solving Linear Equation

Variable x 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+2\right), the least common multiple of x-1,y+2.

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

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

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

To find the opposite of x-1, find the opposite of each term.

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

Add 4 and 1 to get 5.

$$2y+5-x=4\left(x-1\right)\left(y+2\right)$$

Use the distributive property to multiply 4 by x-1.

$$2y+5-x=\left(4x-4\right)\left(y+2\right)$$

Use the distributive property to multiply 4x-4 by y+2.

$$2y+5-x=4xy+8x-4y-8$$

Subtract 4xy from both sides.

$$2y+5-x-4xy=8x-4y-8$$

Subtract 8x from both sides.

$$2y+5-x-4xy-8x=-4y-8$$

Combine -x and -8x to get -9x.

$$2y+5-9x-4xy=-4y-8$$

Subtract 2y from both sides.

$$5-9x-4xy=-4y-8-2y$$

Combine -4y and -2y to get -6y.

$$5-9x-4xy=-6y-8$$

Subtract 5 from both sides.

$$-9x-4xy=-6y-8-5$$

Subtract 5 from -8 to get -13.

$$-9x-4xy=-6y-13$$

Combine all terms containing x.

$$\left(-9-4y\right)x=-6y-13$$

The equation is in standard form.

$$\left(-4y-9\right)x=-6y-13$$

Divide both sides by -4y-9.

$$\frac{\left(-4y-9\right)x}{-4y-9}=\frac{-6y-13}{-4y-9}$$

Dividing by -4y-9 undoes the multiplication by -4y-9.

$$x=\frac{-6y-13}{-4y-9}$$

Divide -6y-13 by -4y-9.

$$x=\frac{6y+13}{4y+9}$$

Variable x cannot be equal to 1.

$$x=\frac{6y+13}{4y+9}\text{, }x\neq 1$$

Solve for y

$$y=-\frac{13-9x}{2\left(3-2x\right)}$$

$$x\neq 1\text{ and }x\neq \frac{3}{2}$$

Steps for Solving Linear Equation

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

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

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

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

To find the opposite of x-1, find the opposite of each term.

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

Add 4 and 1 to get 5.

$$2y+5-x=4\left(x-1\right)\left(y+2\right)$$

Use the distributive property to multiply 4 by x-1.

$$2y+5-x=\left(4x-4\right)\left(y+2\right)$$

Use the distributive property to multiply 4x-4 by y+2.

$$2y+5-x=4xy+8x-4y-8$$

Subtract 4xy from both sides.

$$2y+5-x-4xy=8x-4y-8$$

Add 4y to both sides.

$$2y+5-x-4xy+4y=8x-8$$

Combine 2y and 4y to get 6y.

$$6y+5-x-4xy=8x-8$$

Subtract 5 from both sides.

$$6y-x-4xy=8x-8-5$$

Subtract 5 from -8 to get -13.

$$6y-x-4xy=8x-13$$

Add x to both sides.

$$6y-4xy=8x-13+x$$

Combine 8x and x to get 9x.

$$6y-4xy=9x-13$$

Combine all terms containing y.

$$\left(6-4x\right)y=9x-13$$

Divide both sides by -4x+6.

$$\frac{\left(6-4x\right)y}{6-4x}=\frac{9x-13}{6-4x}$$

Dividing by -4x+6 undoes the multiplication by -4x+6.

$$y=\frac{9x-13}{6-4x}$$

Divide 9x-13 by -4x+6.

$$y=\frac{9x-13}{2\left(3-2x\right)}$$

Variable y cannot be equal to -2.

$$y=\frac{9x-13}{2\left(3-2x\right)}\text{, }y\neq -2$$