Sum of the numerator and denominator of the fraction is 4 more than twice the numerator . If the numerator and denominator is deceased by 3.they are in the ratio 2:3. Determine …

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asked Sep 8, 2021 207k views

1 Answer

Gyre · Answer 1 · 2021-09-12T20:54:09+0000

The system is:

(x)/(y)=2x + 4

(x-3)/(y-3)=(2)/(3) (y musn't be 3).

Solving for y in the first equation gives you:

y=(x)/(2x+4)

Simplify the second equation to:

2(y-3)=3(x-3)

Insert y you solved in first equation to second equation to get:

(x)/(x+2)-6=3x-3

Further simplification gets you to quadratic equation:

$x=(3x+3)(x+2)\implies x=3x^2+9x+6$
3x^2+8x+6=0 .

Solving quadratic equation using quadratic formula gives you two solutions of x in complex plane:

$x_1=\boxed{-(8)/(6)+(√(8))/(6)i}$

$x_2=\boxed{-(8)/(6)-(√(8))/(6)i}$

Insert the found x-es into first equation to find the y-s:

$y_1=\boxed{-(1)/(2)+(√(2))/(2)i}$

$y_2=\boxed{-(1)/(2)-(√(2))/(2)i}$

So there are two solutions (points)
P_1(x_1,y_1),P_2(x_2,y_2) both in
$\mathbb{C}$ plane.

And the two fractions are:

(-(8)/(6)+(√(8))/(6)i)/(-(1)/(2)+(√(2))/(2)i) .

(-(8)/(6)-(√(8))/(6)i)/(-(1)/(2)-(√(2))/(2)i) .

Hope this helps.