I need help isolating this variable

Question

I have to (by hand) isolate a in the following equation:

$\left( -36+6\,i \right) a=-6$

Using Maple to solve it I get $a={\frac{6}{37}+\frac{i}{37}}$

It looks so simple, but because of the complex number it is causing me some confusion. Can anyone shed some light on how Maple got this result?

Answer 1

You get $$a = \frac{-6}{-36 + 6i} = \frac{-1}{-6+i} = \frac{6+i}{(-6+i)(-6-i)}= \frac{6+i}{37}$$

Answer 2

You have $$a = {1\over 6 - i}.$$ Multiply the top and bottom by $6 + i$ and all will become clear.

Answer 3

Notice

$$ a = -6 \cdot (-36 + 6i )^{-1} $$

$$ a = -6 \cdot \frac{ -36 - 6 i }{(-36)^2 + (6i)^2 } $$

Answer 4

I think that this is the way it went -

$$ \left(-36+6i\right)a=-6 $$

Now divide both sides by $ -36+6i $

We get: $$ \frac{\left(-36+6i\right)a}{-36+6i}=\frac{-6}{-36+6i} $$

Now we simplify:

$$ a = -\frac{6}{-36+6i} = -\frac{1}{\left(-6+i\right)} $$

Now you rationalize $$ -\frac{1}{-6+i}:\quad -\frac{-6-i}{37} $$

And hence you arrive at your answer: $$ a =\frac{6}{37}+\frac{1}{37}i $$

Answer 5

I would divide through by $-6$ before doing anything else to simplify to $(6-i)a=1$

then multiply by the conjugate, $6+i$ to obtain $37a=6+i$

then divide by $37$

The rationale for this approach is (i) obvious simplifications make arithmetic easier (ii) why divide by a complex number when you can multiply and avoid fractions - easier to write and less prone to error (iii) it is always easy to divide by a rational integer.