1 2 Equality And Square Roots – Class 11TH Math

1.2 Equality of Two Complex Numbers

The two complex numbers $z_1 = a + bi$ and $z_2 = c + di$ are said to be equal if and only if their real and imaginary parts are equal:

a + bi = c + di \Leftrightarrow a = c \land b = d

If $(3 + 2i)(x + iy) = 5 + 12i$ , where $x, y \in \mathbb{R}$ , then find the values of $x$ and $y$ .

Solution

\begin{align*} (3 + 2i)(x + iy) &= 5 + 12i \\ 3x + 3iy + 2ix + 2i^2 y &= 5 + 12i \\ (3x - 2y) + (2x + 3y)i &= 5 + 12i \end{align*}

Comparing real and imaginary parts:

\begin{align*} 3x - 2y &= 5 \tag{i} \\ 2x + 3y &= 12 \tag{ii} \end{align*}

Multiply (i) by $3$ and (ii) by $2$ :

\begin{align*} 9x - 6y &= 15 \\ 4x + 6y &= 24 \end{align*}

Adding:

13x = 39 \implies x = 3

Substitute $x = 3$ in (i):

3(3) - 2y = 5 \implies 9 - 2y = 5 \implies y = 2

Thus $x = 3$ , $y = 2$ .