Thread: Solution of a Complex Equation

1. Solution of a Complex Equation

$\displaystyle (3-i)(z+4-2i)=10+20i$

My attempt:

$\displaystyle \begin{array}{rcrcrc} (3-i)(z+4-2i)=10+20i\\ 3z + 12-6i-zi-4i+2i^{2}=10+20i\\ 3z+10-10i-zi=10+20i\\ 3z-zi=30i \end{array}$

I have no idea where to go from there.

2. Originally Posted by Viral
$\displaystyle (3-i)(z+4-2i)=10+20i$

My attempt:

$\displaystyle \begin{array}{rcrcrc} (3-i)(z+4-2i)=10+20i\\ 3z + 12-6i-zi-4i+2i^{2}=10+20i\\ 3z+10-10i-zi=10+20i\\ 3z-zi=30i \end{array}$

I have no idea where to go from there.
$\displaystyle 3z - zi = 30i$

Let $\displaystyle z = x + yi \Rightarrow 3z - zi = 3(x+yi) - i(x+yi) = 3x + 3yi - xi + y = (3x + y) + (3y - x)i$

So:

$\displaystyle (3x+y) + (3y-x)i = 0 + 30i$

Can you solve from here?

3. I didn't understand how you worked the following out:

$\displaystyle 3x + 3yi - xi + y = (3x + y) + (3y - x)i$

4. Originally Posted by Viral
I didn't understand how you worked the following out:

$\displaystyle 3x + 3yi - xi + y = (3x + y) + (3y - x)i$

$\displaystyle 3x + 3yi - xi + y = 3x+y+3yi-xi = 3x + y + i(3y) - i(x) = 3x + y + i(3y-x)$

5. Ehh, this is confusing. I've understood everything so far. So from that I'm guessing $\displaystyle 3(x+y)=0, (3y+x)i=30i$ But I just don't see how I change that into $\displaystyle z=x+yi$.

6. Originally Posted by Viral
Ehh, this is confusing. I've understood everything so far. So from that I'm guessing $\displaystyle 3(x+y)=0, (3y+x)i=30i$ But I just don't see how I change that into $\displaystyle z=x+yi$.
You are left with 2 equations that you need to solve for x,y:

$\displaystyle 3x+y = 0 \Rightarrow x = \frac{-y}{3}$
$\displaystyle (3y-x)i = (30)i \Rightarrow 3y - x = 30$

Find the possible values for x,y then substitute them into $\displaystyle z = x + yi$ and get your values of z.

Spoiler:
substitute $\displaystyle x = -y/3$ into the second equation to get:
$\displaystyle 3y +y/3 = 30 \Rightarrow \frac{10}{3} \cdot y = 30 \Rightarrow y = 9$
From the first equation we know $\displaystyle x = -y/3 \Rightarrow x = -3$

Finally, this leaves us with:
$\displaystyle x = -3, y = 9 \Rightarrow z = x + yi = -3 + 9i$

7. How did you get $\displaystyle x=-y$?

8. Originally Posted by Viral
$\displaystyle (3-i)(z+4-2i)=10+20i$
You both are making this far too complicated.
Solve it like an elementary algebra problem: divide and then subtract.

$\displaystyle \frac{{10 + 20i}}{{3 - i}} = \frac{{(10 + 20i)(3 + i)}}{{10}} = 1 + 7i$.

So $\displaystyle z=(1+7i)-(4-2i)$

9. $\displaystyle \frac{10+20i}{3-i}=(z+4-2i)$.

But then how did you get 1+7i from there?

10. Originally Posted by Viral
$\displaystyle \frac{10+20i}{3-i}=(z+4-2i)$.

But then how did you get 1+7i from there?
$\displaystyle \frac{10+20i}{3-i} = \frac{10+20i}{3-i} \cdot \frac{3+i}{3+i} =$ $\displaystyle \frac{(10+20i)(3+i)}{(3-i)(3+i)} = \frac{30 + 10i + 60i - 20}{3^2 + 1^2} =$ $\displaystyle \frac{10 + 70i}{10} = 1 + 7i$