Question

If z = x + iy, $x,y\in R$ and $3x+(3x-y)i=4-6i$ then find the value of z.

Answer 1

Hint: To find the value of z first we have to find the value of x and y, to find that we will compare the real part and imaginary part of the given equation and then we will solve the two equations to find the value of x and y.

Complete step-by-step answer:
$3x+(3x-y)i=4-6i$
This is an equation in which we have to compare the real part and imaginary part.
( Imaginary part is the coefficient of i and the real part is which does not contain i )
So from comparing we get,
$\begin{align}
  & 3x=4\text{ }............\text{ (1)} \\
 & \text{3x}-y=-6\text{ }.............\text{ (2)} \\
 & \\
\end{align}$
Now we are going to solve these two equations to find the value of x and y.
We must try to find the shortest way to solve these two equations in two variables such that it takes a minimum amount of time and effort.
There are many methods to solve this but we will use this one,
From equation (1) we find the value of x which comes out to be:
$x=\dfrac{4}{3}$ .
Now using the value of x in equation (2) we get,
$\begin{align}
  & 4-y=-6 \\
 & \Rightarrow y=10 \\
\end{align}$
Now we have the value of x and y so we will find the value of z,
$z=\dfrac{4}{3}+10i$ .
Hence, the answer to this question is $z=\dfrac{4}{3}+10i$.

Note: One should also know how to distinguish between the real and imaginary part, and there is a chance of calculation mistake while solving two equations and two variables, we can also directly put the value of 3x in equation (2) which will help us to solve this question faster.