The title is very vague, apologies. I'm taking A Level Further Mathematics, and we've come to the end of our topic on Argand Diagrams.
There is a question (which I'm not looking for the answer to), which states this:
The real and imaginary parts of the complex number z = x + iy satisfy the equation (4 - 3i)x - (1 + 6i)y - 3 = 0. Find the value of x and the value of y
Based on the method at the back of the textbook, I understand that to solve x and y, you start by substituting the real parts of the two equations in parentheses into one expression (i.e. 4x - y = 3) and the imaginary parts into another expression (-3x - 6y = 3).
My question is, why does this work? Can somebody give me an explanation as to why it is possible to take the real and imaginary parts of each expression and put them separately into the equation?
Apologies if this is really vague or put in a weird manner, I'm not exactly sure how I should format my question. If there's any confusion to the wording of my question please let me know and I'll try and re-explain.
Also sorry for the tags, I tried Argand-diagram and imaginary-number but it wouldn't allow me.
Thanks!
- R
If reals $a,\,b,\,c,\,d$ satisfy $a+bi=c+di$, then $a-c=(d-b)i$ and $(a-c)^2=-(d-b)^2$. The left-hand side is $\ge 0$ while the right-hand side is $\le 0$, so both are $0$ for equality. Thus $a=c,\,b=d$.