Step-by-step explanation:
[tex]\textsf{\large{\underline{Solution}:}}[/tex]
Given That:
[tex] \rm: \longmapsto (3 - 4i)(x + iy) = 1 + 0i[/tex]
[tex] \rm: \longmapsto 3(x + iy) - 4i(x + iy)= 1 + 0i[/tex]
[tex] \rm: \longmapsto 3x + 3iy - 4ix +4y= 1 + 0i[/tex]
On rearranging the terms, we get:
[tex] \rm: \longmapsto (3x + 4y)+ (3y - 4x)i = 1 + 0i[/tex]
Comparing both sides, we get:
[tex] \rm: \longmapsto 3x + 4y = 1 --- (i)[/tex]
[tex] \rm: \longmapsto 3y - 4x = 0 --- (ii)[/tex]
Multiplying (i) by 4, we get:
[tex] \rm: \longmapsto 12x + 16y = 4--- (iii)[/tex]
Multiplying (ii) by 3, we get:
[tex] \rm: \longmapsto 9y - 12x = 0 --- (iv)[/tex]
Adding equations (iii) and (iv), we get:
[tex] \rm: \longmapsto25y = 4[/tex]
[tex] \rm: \longmapsto y =\dfrac{4}{25} [/tex]
From (ii), we get:
[tex] \rm: \longmapsto 3y - 4x = 0 [/tex]
[tex] \rm: \longmapsto 3y = 4x [/tex]
[tex] \rm: \longmapsto x = \dfrac{3}{4} y[/tex]
[tex] \rm: \longmapsto x = \dfrac{3}{4} \times \dfrac{4}{25} [/tex]
[tex] \rm: \longmapsto x = \dfrac{3}{25} [/tex]
Therefore:
[tex] \rm: \longmapsto (x,y)= \bigg( \dfrac{3}{25}, \dfrac{4}{25} \bigg)[/tex]
★ Which is our required answer.
[tex]\textsf{\large{\underline{More To Know}:}}[/tex]
[tex]\rm1.\ i^{4n} = 1[/tex]
[tex]\rm2. \ i^{4n+1} = i[/tex]
[tex]\rm3.\ i^{4n+2} = -1[/tex]
[tex]\rm4.\ i^{4n+3} = -i[/tex]
