Solve 3x-2a/a+6x+1/2a+1=3x-a

$$\frac { 3 x - 2 a } { a } + \frac { 6 x + 1 } { 2 a + 1 } = 3 x - a$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{a}{3}\text{, }&a\neq 0\text{ and }a\neq -\frac{1}{2}\\x\in \mathrm{C}\text{, }&a=\frac{\sqrt{17}+3}{4}\text{ or }a=\frac{3-\sqrt{17}}{4}\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by a\left(2a+1\right), the least common multiple of a,2a+1.

$$\left(2a+1\right)\left(3x-2a\right)+a\left(6x+1\right)=3xa\left(2a+1\right)-aa\left(2a+1\right)$$

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

$$6ax-4a^{2}+3x-2a+a\left(6x+1\right)=3xa\left(2a+1\right)-aa\left(2a+1\right)$$

Use the distributive property to multiply a by 6x+1.

$$6ax-4a^{2}+3x-2a+6ax+a=3xa\left(2a+1\right)-aa\left(2a+1\right)$$

Combine 6ax and 6ax to get 12ax.

$$12ax-4a^{2}+3x-2a+a=3xa\left(2a+1\right)-aa\left(2a+1\right)$$

Combine -2a and a to get -a.

$$12ax-4a^{2}+3x-a=3xa\left(2a+1\right)-aa\left(2a+1\right)$$

Multiply a and a to get a^{2}.

$$12ax-4a^{2}+3x-a=3xa\left(2a+1\right)-a^{2}\left(2a+1\right)$$

Use the distributive property to multiply 3xa by 2a+1.

$$12ax-4a^{2}+3x-a=6xa^{2}+3xa-a^{2}\left(2a+1\right)$$

Use the distributive property to multiply -a^{2} by 2a+1.

$$12ax-4a^{2}+3x-a=6xa^{2}+3xa-2a^{3}-a^{2}$$

Subtract 6xa^{2} from both sides.

$$12ax-4a^{2}+3x-a-6xa^{2}=3xa-2a^{3}-a^{2}$$

Subtract 3xa from both sides.

$$12ax-4a^{2}+3x-a-6xa^{2}-3xa=-2a^{3}-a^{2}$$

Combine 12ax and -3xa to get 9ax.

$$9ax-4a^{2}+3x-a-6xa^{2}=-2a^{3}-a^{2}$$

Add 4a^{2} to both sides.

$$9ax+3x-a-6xa^{2}=-2a^{3}-a^{2}+4a^{2}$$

Combine -a^{2} and 4a^{2} to get 3a^{2}.

$$9ax+3x-a-6xa^{2}=-2a^{3}+3a^{2}$$

Add a to both sides.

$$9ax+3x-6xa^{2}=-2a^{3}+3a^{2}+a$$

Combine all terms containing x.

$$\left(9a+3-6a^{2}\right)x=-2a^{3}+3a^{2}+a$$

The equation is in standard form.

$$\left(3+9a-6a^{2}\right)x=a+3a^{2}-2a^{3}$$

Divide both sides by 9a+3-6a^{2}.

$$\frac{\left(3+9a-6a^{2}\right)x}{3+9a-6a^{2}}=\frac{a\left(1+3a-2a^{2}\right)}{3+9a-6a^{2}}$$

Dividing by 9a+3-6a^{2} undoes the multiplication by 9a+3-6a^{2}.

$$x=\frac{a\left(1+3a-2a^{2}\right)}{3+9a-6a^{2}}$$

Divide a\left(-2a^{2}+3a+1\right) by 9a+3-6a^{2}.

$$x=\frac{a}{3}$$

Solve for x

$$\left\{\begin{matrix}x=\frac{a}{3}\text{, }&a\neq 0\text{ and }a\neq -\frac{1}{2}\\x\in \mathrm{R}\text{, }&a=\frac{\sqrt{17}+3}{4}\text{ or }a=\frac{3-\sqrt{17}}{4}\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by a\left(2a+1\right), the least common multiple of a,2a+1.

$$\left(2a+1\right)\left(3x-2a\right)+a\left(6x+1\right)=3xa\left(2a+1\right)-aa\left(2a+1\right)$$

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

$$6ax-4a^{2}+3x-2a+a\left(6x+1\right)=3xa\left(2a+1\right)-aa\left(2a+1\right)$$

Use the distributive property to multiply a by 6x+1.

$$6ax-4a^{2}+3x-2a+6ax+a=3xa\left(2a+1\right)-aa\left(2a+1\right)$$

Combine 6ax and 6ax to get 12ax.

$$12ax-4a^{2}+3x-2a+a=3xa\left(2a+1\right)-aa\left(2a+1\right)$$

Combine -2a and a to get -a.

$$12ax-4a^{2}+3x-a=3xa\left(2a+1\right)-aa\left(2a+1\right)$$

Multiply a and a to get a^{2}.

$$12ax-4a^{2}+3x-a=3xa\left(2a+1\right)-a^{2}\left(2a+1\right)$$

Use the distributive property to multiply 3xa by 2a+1.

$$12ax-4a^{2}+3x-a=6xa^{2}+3xa-a^{2}\left(2a+1\right)$$

Use the distributive property to multiply -a^{2} by 2a+1.

$$12ax-4a^{2}+3x-a=6xa^{2}+3xa-2a^{3}-a^{2}$$

Subtract 6xa^{2} from both sides.

$$12ax-4a^{2}+3x-a-6xa^{2}=3xa-2a^{3}-a^{2}$$

Subtract 3xa from both sides.

$$12ax-4a^{2}+3x-a-6xa^{2}-3xa=-2a^{3}-a^{2}$$

Combine 12ax and -3xa to get 9ax.

$$9ax-4a^{2}+3x-a-6xa^{2}=-2a^{3}-a^{2}$$

Add 4a^{2} to both sides.

$$9ax+3x-a-6xa^{2}=-2a^{3}-a^{2}+4a^{2}$$

Combine -a^{2} and 4a^{2} to get 3a^{2}.

$$9ax+3x-a-6xa^{2}=-2a^{3}+3a^{2}$$

Add a to both sides.

$$9ax+3x-6xa^{2}=-2a^{3}+3a^{2}+a$$

Combine all terms containing x.

$$\left(9a+3-6a^{2}\right)x=-2a^{3}+3a^{2}+a$$

The equation is in standard form.

$$\left(3+9a-6a^{2}\right)x=a+3a^{2}-2a^{3}$$

Divide both sides by 9a+3-6a^{2}.

$$\frac{\left(3+9a-6a^{2}\right)x}{3+9a-6a^{2}}=\frac{a\left(1+3a-2a^{2}\right)}{3+9a-6a^{2}}$$

Dividing by 9a+3-6a^{2} undoes the multiplication by 9a+3-6a^{2}.

$$x=\frac{a\left(1+3a-2a^{2}\right)}{3+9a-6a^{2}}$$

Divide a\left(-2a^{2}+3a+1\right) by 9a+3-6a^{2}.

$$x=\frac{a}{3}$$

Solve for a

$$\left\{\begin{matrix}\\a=\frac{\sqrt{17}+3}{4}\approx 1.780776406\text{; }a=\frac{3-\sqrt{17}}{4}\approx -0.280776406\text{, }&\text{unconditionally}\\a=3x\text{, }&x\neq 0\text{ and }x\neq -\frac{1}{6}\end{matrix}\right.$$