Question

find the real number x and y such that x/1+2¡+y/3+2¡=5+6¡/-1 +8¡​

Answer 1

$\huge\fbox\color{aqua}{Answer:–}$

x=1

y=2

Step-by-step explanation

$We are given that \frac{x}{1+2i} +\frac{y}{3+2i} =\frac{5+6i}{-1+8}$

Now, rationalizing the denominator og all the terms, we get

$⇒\begin{gathered}\frac{x(1-2i)}{5} +\frac{y(3-2i)}{13}= \frac{(5+6i)(-1-8i)}{65} \\\end{gathered}$

Separating the real and imaginary parts from the above equation, we get,

$⇒[[\frac{x}{5}+\frac{3y}{13} ]-[\frac{2x}{5}+\frac{2y}{13} ]i=\frac{43}{65} -\frac{46}{65}[/ted]</p><p></p><p>Hence, comparing real and imaginary parts of the above equation we get,</p><p></p><p>[tex]\frac{x}{5} +\frac{3y}{13} =\frac{43}{65}$ ...... (1)

Multiplying 2 in the both sides we get,

$\tt{\frac{2x}{5} +\frac{6y}{13} =\frac{86}{65}}[tex]</p><p>...... (2)</p><p></p><p>[tex]\color{purple}{Again, \frac{2x}{5} +\frac{2y}{13}=\frac{46}{65}}$

Now, solving equations (2) and (3) we get,

4y/13=40/65

⇒ y=2

Again from equation (1), we get, x/5+6/13=43/65

⇒ x/5 =13/65

⇒ x=1 .

Hence, the values of the real numbers x and y are 1 and 2 respectively. (Answer)