Solve 6-2e/3x-2=1-a/x+3+12/3x^2+7x-6

$$\frac { 6 - 2 e } { 3 x - 2 } = \frac { 1 - a } { x + 3 } + \frac { 12 } { 3 x ^ { 2 } + 7 x - 6 }$$

Solve for a (complex solution)

$$a=-\frac{-2ex+3x+8-6e}{3x-2}$$

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

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(3x-2\right)\left(x+3\right), the least common multiple of 3x-2,x+3,3x^{2}+7x-6.

$$\left(x+3\right)\left(6-2e\right)=\left(3x-2\right)\left(1-a\right)+12$$

Use the distributive property to multiply x+3 by 6-2e.

$$6x-2xe+18-6e=\left(3x-2\right)\left(1-a\right)+12$$

Use the distributive property to multiply 3x-2 by 1-a.

$$6x-2xe+18-6e=3x-3ax-2+2a+12$$

Add -2 and 12 to get 10.

$$6x-2xe+18-6e=3x-3ax+10+2a$$

Swap sides so that all variable terms are on the left hand side.

$$3x-3ax+10+2a=6x-2xe+18-6e$$

Subtract 3x from both sides.

$$-3ax+10+2a=6x-2xe+18-6e-3x$$

Combine 6x and -3x to get 3x.

$$-3ax+10+2a=3x-2xe+18-6e$$

Subtract 10 from both sides.

$$-3ax+2a=3x-2xe+18-6e-10$$

Subtract 10 from 18 to get 8.

$$-3ax+2a=3x-2xe+8-6e$$

Combine all terms containing a.

$$\left(-3x+2\right)a=3x-2xe+8-6e$$

The equation is in standard form.

$$\left(2-3x\right)a=-2ex+3x+8-6e$$

Divide both sides by 2-3x.

$$\frac{\left(2-3x\right)a}{2-3x}=\frac{-2ex+3x+8-6e}{2-3x}$$

Dividing by 2-3x undoes the multiplication by 2-3x.

$$a=\frac{-2ex+3x+8-6e}{2-3x}$$

Solve for a

$$a=-\frac{-2ex+3x+8-6e}{3x-2}$$

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

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(3x-2\right)\left(x+3\right), the least common multiple of 3x-2,x+3,3x^{2}+7x-6.

$$\left(x+3\right)\left(6-2e\right)=\left(3x-2\right)\left(1-a\right)+12$$

Use the distributive property to multiply x+3 by 6-2e.

$$6x-2xe+18-6e=\left(3x-2\right)\left(1-a\right)+12$$

Use the distributive property to multiply 3x-2 by 1-a.

$$6x-2xe+18-6e=3x-3ax-2+2a+12$$

Add -2 and 12 to get 10.

$$6x-2xe+18-6e=3x-3ax+10+2a$$

Swap sides so that all variable terms are on the left hand side.

$$3x-3ax+10+2a=6x-2xe+18-6e$$

Subtract 3x from both sides.

$$-3ax+10+2a=6x-2xe+18-6e-3x$$

Combine 6x and -3x to get 3x.

$$-3ax+10+2a=3x-2xe+18-6e$$

Subtract 10 from both sides.

$$-3ax+2a=3x-2xe+18-6e-10$$

Subtract 10 from 18 to get 8.

$$-3ax+2a=3x-2xe+8-6e$$

Combine all terms containing a.

$$\left(-3x+2\right)a=3x-2xe+8-6e$$

The equation is in standard form.

$$\left(2-3x\right)a=-2ex+3x+8-6e$$

Divide both sides by -3x+2.

$$\frac{\left(2-3x\right)a}{2-3x}=\frac{-2ex+3x+8-6e}{2-3x}$$

Dividing by -3x+2 undoes the multiplication by -3x+2.

$$a=\frac{-2ex+3x+8-6e}{2-3x}$$

Solve for x

$$x=-\frac{2\left(-a+4-3e\right)}{3a+3-2e}$$

$$a\neq -\frac{1}{11}\text{ and }a\neq \frac{2e}{3}-1$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values -3,\frac{2}{3} since division by zero is not defined. Multiply both sides of the equation by \left(3x-2\right)\left(x+3\right), the least common multiple of 3x-2,x+3,3x^{2}+7x-6.

$$\left(x+3\right)\left(6-2e\right)=\left(3x-2\right)\left(1-a\right)+12$$

Use the distributive property to multiply x+3 by 6-2e.

$$6x-2xe+18-6e=\left(3x-2\right)\left(1-a\right)+12$$

Use the distributive property to multiply 3x-2 by 1-a.

$$6x-2xe+18-6e=3x-3xa-2+2a+12$$

Add -2 and 12 to get 10.

$$6x-2xe+18-6e=3x-3xa+10+2a$$

Subtract 3x from both sides.

$$6x-2xe+18-6e-3x=-3xa+10+2a$$

Combine 6x and -3x to get 3x.

$$3x-2xe+18-6e=-3xa+10+2a$$

Add 3xa to both sides.

$$3x-2xe+18-6e+3xa=10+2a$$

Subtract 18 from both sides.

$$3x-2xe-6e+3xa=10+2a-18$$

Subtract 18 from 10 to get -8.

$$3x-2xe-6e+3xa=-8+2a$$

Add 6e to both sides.

$$3x-2xe+3xa=-8+2a+6e$$

Combine all terms containing x.

$$\left(3-2e+3a\right)x=-8+2a+6e$$

The equation is in standard form.

$$\left(3a+3-2e\right)x=2a+6e-8$$

Divide both sides by 3+3a-2e.

$$\frac{\left(3a+3-2e\right)x}{3a+3-2e}=\frac{2a+6e-8}{3a+3-2e}$$

Dividing by 3+3a-2e undoes the multiplication by 3+3a-2e.

$$x=\frac{2a+6e-8}{3a+3-2e}$$

Divide -8+2a+6e by 3+3a-2e.

$$x=\frac{2\left(a+3e-4\right)}{3a+3-2e}$$

Variable x cannot be equal to any of the values -3,\frac{2}{3}.

$$x=\frac{2\left(a+3e-4\right)}{3a+3-2e}\text{, }x\neq -3\text{ and }x\neq \frac{2}{3}$$