Solve a/1-i+b/3i=i

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

Solve for a

$$a=\left(\frac{1}{3}+\frac{1}{3}i\right)b+\left(1+i\right)$$

Steps for Solving Linear Equation

Subtract \frac{b}{3i} from both sides.

$$\frac{a}{1-i}=i-\frac{b}{3i}$$

The equation is in standard form.

$$\left(\frac{1}{2}+\frac{1}{2}i\right)a=\frac{ib}{3}+i$$

Divide both sides by \frac{1}{2}+\frac{1}{2}i.

$$\frac{\left(\frac{1}{2}+\frac{1}{2}i\right)a}{\frac{1}{2}+\frac{1}{2}i}=\frac{\frac{ib}{3}+i}{\frac{1}{2}+\frac{1}{2}i}$$

Dividing by \frac{1}{2}+\frac{1}{2}i undoes the multiplication by \frac{1}{2}+\frac{1}{2}i.

$$a=\frac{\frac{ib}{3}+i}{\frac{1}{2}+\frac{1}{2}i}$$

Divide i+\frac{ib}{3} by \frac{1}{2}+\frac{1}{2}i.

$$a=\left(\frac{1}{3}+\frac{1}{3}i\right)b+\left(1+i\right)$$

Solve for b

$$b=\left(\frac{3}{2}-\frac{3}{2}i\right)a-3$$

Steps for Solving Linear Equation

Subtract \frac{a}{1-i} from both sides.

$$\frac{b}{3i}=i-\frac{a}{1-i}$$

The equation is in standard form.

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

Divide both sides by -\frac{1}{3}i.

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

Dividing by -\frac{1}{3}i undoes the multiplication by -\frac{1}{3}i.

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

Divide i+\left(-\frac{1}{2}-\frac{1}{2}i\right)a by -\frac{1}{3}i.

$$b=\left(\frac{3}{2}-\frac{3}{2}i\right)a-3$$