Question

Find the values of x and y 1-2i/ 2+i + 4-i/3+2i = x+iy

Answer 1

Answer:

$x=\frac{10}{13}$ and $y=-\frac{24}{13}$

Step-by-step explanation:

The given complex number equation is:

$\frac{1-2i}{2+i}+\frac{4-i}{3+2i}=x+yi$

We simplify the LHS and compare with the RHS

We collect LCD on the left to get:

$\frac{(1-2i)(3+2i)+(4-i)(2+i)}{(2+i)(3+2i)}=x+yi$

$\frac{3+2i-6i+4+8+4i-2i+1}{6+4i+3i-2}=x+yi$

Simplify to get:

$\frac{16-2i}{4+7i}=x+yi$

Rationalize the LHS:

$\frac{(16-2i)(4-7i)}{(4+7i)(4-7i)}=x+yi$

Expand the numerator using the distributive property and the denominator using difference  of two squares.

$\frac{64-112i-8i-14}{16+49}=x+yi$

Simplify to get:

$\frac{50-120i}{65}=x+yi$

$\frac{10-24i}{13}=x+yi$

$\frac{10}{13}-\frac{24}{13}i=x+yi$

By comparing real parts and imaginary parts; we have;

$x=\frac{10}{13}$ and $y=-\frac{24}{13}$