Solve ax+b-3x+2ab/3=1/2

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

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\frac{3\left(-2x+2b-1\right)}{2\left(3x-2b\right)}\text{, }&x\neq \frac{2b}{3}\\a\in \mathrm{C}\text{, }&x=1\text{ and }b=\frac{3}{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$6ax+6b-2\left(3x+2ab\right)=3$$

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

$$6ax+6b-6x-4ab=3$$

Subtract 6b from both sides.

$$6ax-6x-4ab=3-6b$$

Add 6x to both sides.

$$6ax-4ab=3-6b+6x$$

Combine all terms containing a.

$$\left(6x-4b\right)a=3-6b+6x$$

The equation is in standard form.

$$\left(6x-4b\right)a=6x-6b+3$$

Divide both sides by 6x-4b.

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

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

$$a=\frac{6x-6b+3}{6x-4b}$$

Divide 3-6b+6x by 6x-4b.

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

Solve for b (complex solution)

$$\left\{\begin{matrix}b=-\frac{3\left(2ax-2x-1\right)}{2\left(3-2a\right)}\text{, }&a\neq \frac{3}{2}\\b\in \mathrm{C}\text{, }&x=1\text{ and }a=\frac{3}{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$6ax+6b-2\left(3x+2ab\right)=3$$

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

$$6ax+6b-6x-4ab=3$$

Subtract 6ax from both sides.

$$6b-6x-4ab=3-6ax$$

Add 6x to both sides.

$$6b-4ab=3-6ax+6x$$

Combine all terms containing b.

$$\left(6-4a\right)b=3-6ax+6x$$

The equation is in standard form.

$$\left(6-4a\right)b=3+6x-6ax$$

Divide both sides by 6-4a.

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

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

$$b=\frac{3+6x-6ax}{6-4a}$$

Divide 3-6ax+6x by 6-4a.

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

Solve for a

$$\left\{\begin{matrix}a=-\frac{3\left(-2x+2b-1\right)}{2\left(3x-2b\right)}\text{, }&x\neq \frac{2b}{3}\\a\in \mathrm{R}\text{, }&x=1\text{ and }b=\frac{3}{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$6ax+6b-2\left(3x+2ab\right)=3$$

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

$$6ax+6b-6x-4ab=3$$

Subtract 6b from both sides.

$$6ax-6x-4ab=3-6b$$

Add 6x to both sides.

$$6ax-4ab=3-6b+6x$$

Combine all terms containing a.

$$\left(6x-4b\right)a=3-6b+6x$$

The equation is in standard form.

$$\left(6x-4b\right)a=6x-6b+3$$

Divide both sides by 6x-4b.

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

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

$$a=\frac{6x-6b+3}{6x-4b}$$

Divide 3-6b+6x by 6x-4b.

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

Solve for b

$$\left\{\begin{matrix}b=-\frac{3\left(2ax-2x-1\right)}{2\left(3-2a\right)}\text{, }&a\neq \frac{3}{2}\\b\in \mathrm{R}\text{, }&x=1\text{ and }a=\frac{3}{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$6ax+6b-2\left(3x+2ab\right)=3$$

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

$$6ax+6b-6x-4ab=3$$

Subtract 6ax from both sides.

$$6b-6x-4ab=3-6ax$$

Add 6x to both sides.

$$6b-4ab=3-6ax+6x$$

Combine all terms containing b.

$$\left(6-4a\right)b=3-6ax+6x$$

The equation is in standard form.

$$\left(6-4a\right)b=3+6x-6ax$$

Divide both sides by 6-4a.

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

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

$$b=\frac{3+6x-6ax}{6-4a}$$

Divide 3-6ax+6x by 6-4a.

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