Solve 4ax-a^2/2+a/6=ax-2x+1/3

$$\frac { 4 a x - a ^ { 2 } } { 2 } + \frac { a } { 6 } = a x - \frac { 2 x + 1 } { 3 }$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=\frac{a-1}{2}\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&a=-\frac{2}{3}\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 6, the least common multiple of 2,6,3.

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

Use the distributive property to multiply 3 by 4ax-a^{2}.

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

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

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

Subtract 6ax from both sides.

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

Combine 12xa and -6ax to get 6xa.

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

Add 4x to both sides.

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

Add 3a^{2} to both sides.

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

Subtract a from both sides.

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

Combine all terms containing x.

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

The equation is in standard form.

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

Divide both sides by 6a+4.

$$\frac{\left(6a+4\right)x}{6a+4}=\frac{\left(a-1\right)\left(3a+2\right)}{6a+4}$$

Dividing by 6a+4 undoes the multiplication by 6a+4.

$$x=\frac{\left(a-1\right)\left(3a+2\right)}{6a+4}$$

Divide \left(-1+a\right)\left(2+3a\right) by 6a+4.

$$x=\frac{a-1}{2}$$

Solve for x

$$\left\{\begin{matrix}\\x=\frac{a-1}{2}\text{, }&\text{unconditionally}\\x\in \mathrm{R}\text{, }&a=-\frac{2}{3}\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 6, the least common multiple of 2,6,3.

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

Use the distributive property to multiply 3 by 4ax-a^{2}.

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

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

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

Subtract 6ax from both sides.

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

Combine 12xa and -6ax to get 6xa.

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

Add 4x to both sides.

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

Add 3a^{2} to both sides.

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

Subtract a from both sides.

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

Combine all terms containing x.

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

The equation is in standard form.

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

Divide both sides by 6a+4.

$$\frac{\left(6a+4\right)x}{6a+4}=\frac{\left(a-1\right)\left(3a+2\right)}{6a+4}$$

Dividing by 6a+4 undoes the multiplication by 6a+4.

$$x=\frac{\left(a-1\right)\left(3a+2\right)}{6a+4}$$

Divide \left(-1+a\right)\left(2+3a\right) by 6a+4.

$$x=\frac{a-1}{2}$$

Solve for a

$$a=-\frac{2}{3}\approx -0.666666667$$

$$a=2x+1$$