Solve 2x-a/3a+x+1/2a+1=x/9

$$\frac { 2 x - a } { 3 a } + \frac { x + 1 } { 2 a + 1 } = \frac { x } { 9 }$$

Solve for x

$$x=-\frac{3a\left(1-a\right)}{3+10a-a^{2}}$$

$$a\neq 2\sqrt{7}+5\text{ and }a\neq 5-2\sqrt{7}\text{ and }a\neq 0\text{ and }a\neq -\frac{1}{2}$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 9a\left(2a+1\right), the least common multiple of 3a,2a+1,9.

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

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

$$12ax-6a^{2}+6x-3a+9a\left(x+1\right)=a\left(2a+1\right)x$$

Use the distributive property to multiply 9a by x+1.

$$12ax-6a^{2}+6x-3a+9ax+9a=a\left(2a+1\right)x$$

Combine 12ax and 9ax to get 21ax.

$$21ax-6a^{2}+6x-3a+9a=a\left(2a+1\right)x$$

Combine -3a and 9a to get 6a.

$$21ax-6a^{2}+6x+6a=a\left(2a+1\right)x$$

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

$$21ax-6a^{2}+6x+6a=\left(2a^{2}+a\right)x$$

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

$$21ax-6a^{2}+6x+6a=2a^{2}x+ax$$

Subtract 2a^{2}x from both sides.

$$21ax-6a^{2}+6x+6a-2a^{2}x=ax$$

Subtract ax from both sides.

$$21ax-6a^{2}+6x+6a-2a^{2}x-ax=0$$

Combine 21ax and -ax to get 20ax.

$$20ax-6a^{2}+6x+6a-2a^{2}x=0$$

Add 6a^{2} to both sides. Anything plus zero gives itself.

$$20ax+6x+6a-2a^{2}x=6a^{2}$$

Subtract 6a from both sides.

$$20ax+6x-2a^{2}x=6a^{2}-6a$$

Combine all terms containing x.

$$\left(20a+6-2a^{2}\right)x=6a^{2}-6a$$

The equation is in standard form.

$$\left(6+20a-2a^{2}\right)x=6a^{2}-6a$$

Divide both sides by 20a+6-2a^{2}.

$$\frac{\left(6+20a-2a^{2}\right)x}{6+20a-2a^{2}}=\frac{6a\left(a-1\right)}{6+20a-2a^{2}}$$

Dividing by 20a+6-2a^{2} undoes the multiplication by 20a+6-2a^{2}.

$$x=\frac{6a\left(a-1\right)}{6+20a-2a^{2}}$$

Divide 6a\left(-1+a\right) by 20a+6-2a^{2}.

$$x=\frac{3a\left(a-1\right)}{3+10a-a^{2}}$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=\frac{\sqrt{\left(4x+3\right)\left(28x+3\right)}+10x+3}{2\left(x+3\right)}\text{, }&x\neq -3\text{ and }x\neq -1\\a=-\frac{\sqrt{\left(4x+3\right)\left(28x+3\right)}-10x-3}{2\left(x+3\right)}\text{, }&x\neq 0\text{ and }x\neq -3\\a=-\frac{1}{3}\approx -0.333333333\text{, }&x=-3\end{matrix}\right.$$

Solve for a

$$\left\{\begin{matrix}a=\frac{\sqrt{\left(4x+3\right)\left(28x+3\right)}+10x+3}{2\left(x+3\right)}\text{, }&x\geq -\frac{3}{28}\text{ or }\left(x\neq -3\text{ and }x\leq -\frac{3}{4}\text{ and }x\neq -1\right)\\a=-\frac{\sqrt{\left(4x+3\right)\left(28x+3\right)}-10x-3}{2\left(x+3\right)}\text{, }&\left(x\neq -3\text{ and }x\leq -\frac{3}{4}\right)\text{ or }\left(x\neq 0\text{ and }x\geq -\frac{3}{28}\right)\\a=-\frac{1}{3}\approx -0.333333333\text{, }&x=-3\end{matrix}\right.$$