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Question: If \(z=x+iy\), where \(x\) and \(y\) belongs to \(R\) and \[3x + \left( {3x - y} \right)i = 4 - 6i\]...

If z=x+iyz=x+iy, where xx and yy belongs to RR and 3x+(3x−y)i=4−6i3x + \left( {3x - y} \right)i = 4 - 6i then z=z =
(A) (43)+i10\left( {\dfrac{4}{3}} \right) + i10
(B) (43)−i10\left( {\dfrac{4}{3}} \right) - i10
(C) −(43)+i10 - \left( {\dfrac{4}{3}} \right) + i10
(D) −(43)−i10 - \left( {\dfrac{4}{3}} \right) - i10

Explanation

Solution

Here, z is the combination of both the real number and imaginary number which is a complex number. So, we will equate the real part and then the imaginary part. We will get the value of x when we equate the real part and then the value of y when we equate the imaginary part. After that, substituting the values of x and y in the given z= x+iy, we will get the final value of z.

Complete step by step solution:
Given that, 3x+(3x−y)i=4−6i3x + \left( {3x - y} \right)i = 4 - 6i
Here, we will equate the real part and imaginary parts.
First, equate the real parts, we get,
∴3x=4\therefore 3x = 4
⇒x=43\Rightarrow x = \dfrac{4}{3}
Next, equate the imaginary parts, we get,
∴3x−y=−6\therefore 3x - y = - 6
Substituting the value of x in the above equation, we will get,
⇒3(43)−y=−6\Rightarrow 3(\dfrac{4}{3}) - y = - 6
On evaluating the above equation, we will get,
⇒4−y=−6\Rightarrow 4 - y = - 6
By using the transposition method, keep only unknown term on LHS, we get,
⇒−y=−6−4\Rightarrow - y = - 6 - 4
⇒−y=−10\Rightarrow - y = - 10
⇒y=10\Rightarrow y = 10

We are given that,
∴z=x+iy\therefore z = x + iy where x and y belong to R (real).
Substituting the value of x and y in the above equation, we will get,
⇒z=43+i10\Rightarrow z = \dfrac{4}{3} + i10
Rearranging this above equation, we will get,
⇒z=43+10i\Rightarrow z = \dfrac{4}{3} + 10i

Hence, for the given 3x+(3x−y)i=4−6i3x + \left( {3x - y} \right)i = 4 - 6i the value of z=43+10iz = \dfrac{4}{3} + 10i.

Note: Complex numbers are the numbers that are expressed in the form of a+ib where a, b are real numbers and i is an imaginary number called iota. An imaginary number is usually represented by i or j, which is equal to −1\sqrt { - 1} . Thus, the square of an imaginary number is equal to a negative number (i.e. i2=−1{i^2} = - 1 ). The main application of these numbers is to represent periodic motions such as water waves, alternating current, light waves, etc.