Question

Find the values of x and y

1-2i/ 2+i + 4-i/3+2i = x+iy

Answer 1

`x=(10)/(13)` and
`y=-(24)/(13)`

Explanation:

The given complex number equation is:

`(1-2i)/(2+i)+(4-i)/(3+2i)=x+yi`

We simplify the LHS and compare with the RHS

We collect LCD on the left to get:

`((1-2i)(3+2i)+(4-i)(2+i))/((2+i)(3+2i))=x+yi`

`(3+2i-6i+4+8+4i-2i+1)/(6+4i+3i-2)=x+yi`

Simplify to get:

`(16-2i)/(4+7i)=x+yi`

Rationalize the LHS:

`((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.

`(64-112i-8i-14)/(16+49)=x+yi`

Simplify to get:

`(50-120i)/(65)=x+yi`

`(10-24i)/(13)=x+yi`

`(10)/(13)-(24)/(13)i=x+yi`

By comparing real parts and imaginary parts; we have;

`x=(10)/(13)` and
`y=-(24)/(13)`