Question

Given that x, y$\in $ R, solve (x + iy) + (y – 50) = 9 + 4i.

Answer 1

Hint: Given equation is of the form a + ib, where a is real no, and b is the imaginary part. Separate the real part and imaginary part from the given equation. Frame 2 equation from it. Solve it and get the values of x and y.

Complete step-by-step answer:
We know that a complex number is a number that can be expressed in the form a +bi, where a and b are real numbers. Here ‘i’ is a solution of the equation ${{x}^{2}}=-1$. No real no: can satisfy the equation. So ‘i’ is called an imaginary number.
Now from the complex no: a + bi, a is called real part and b is called the imaginary pest. If speaking geometrically, complex no: extends the concept of 1D number line to 2D complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part.
Now we have been given (x + iy) + (7 – 5i) = 9 + 4i
Let us open their brackets and separate the real part and the imaginary part.
x + iy + 7 – 5i = 9 + 4i
(x + 7) + (iy – 5i) = 9 + 4i
(x + y) + i(y – 5) = 9 + 4i.
From the above x + 7 = 9 represents the real part and i(y – 5) = 4i represents the imaginary part
x + 7 = 9…………………… (i)
i(y – 5) = 4i, imaginary pest and i on LHS and RHS
y – 5 = 4…………………….. (ii)
from (i), x + 7 = 9
$\therefore $ x = 9 – 7 = 2 $\Rightarrow $ x = 2
From (ii), y – 5 = 4 $\Rightarrow $ y = 5 + 4 = 9
Thus we solved the given equation and we got value of x and y as
x = 2 and y = 9

Note: From the given expression, we should be able to classify the real part and the imaginary part. Thus equate both real and imaginary part from both the given LHS and RHS. Solve for x and y.