Maths Qs -- equations @ 10points

Suppose the numbers are x and y.

If the sum of these two numbers is 1, then we have x + y = 1

If the difference between their reciprocals is 3/2, we have 1/x -1/y = 3/2

From the first equation y = 1 - x
Substituting into the second equation, we have

1/x - 1/(1-x) = 3/2
(1-x-x)/[x(1-x)] = 3/2
(1-2x)/(x - x^2) = 3/2
2 - 4x = 3x - 3x^2
3x^2 - 7x + 2 = 0
(3x-1)(x-2) =0
(3x-1)=0 or (x-2)=0
x=1/3 or x = 2

when x = 1/3, from the first equation, y = 1-x = 2/3, therefore the larger of the two numbers is 2/3
similarly when x = 2, y = 1- x = -1, therefore the larger of the two numbers is 2.

Ans: The larger of the two numbers is 2/3 or 2

x+y=1 ==> y = 1 - x
|1/y - 1/x| = 3/2 ( | | means absolute value)
=> 1/y - 1/x = +3/2 or 1/y - 1/x = -3/2

consider 1/(1-x) - 1/x = +3/2
[x-(1-x)]/[x(1-x)] = 3/2
(2x-1)/[x(1-x)] =3/2
4x-2 = 3x - 3x^2
3x^2+x-2 = 0
(3x+2)(x-1) = 0
x=-2/3 or 1
if x = 1 y=0 (rej, coz 1/y = 0 undefined)
if x = -2/3, y=1-x=5/3 (rej, coz |1/y - 1/x| =/= 3/2)

then consider 1/(1-x) - 1/x = -3/2
[x-(1-x)]/[x(1-x)] = -3/2
(2x-1)/[x(1-x)] =-3/2
4x-2 = -3x + 3x^2
3x^2-7x+2 = 0
(3x-1)(x-2) = 0
x=2 or 1/3
if x=2, y =-1
if x=1/3, y=2/3

therefore, the larger of the two numbers is 1 or 2/3

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Remarks: you should consider the diff as both +-3/2 to avoid missing out any possible values

also
3y - 3y^2 = 4y -2
3y^2 - y + 2 = 0 (wrong)

3y^2 + y - 2 = 0 (correct)

please further check your steps

Let the two numbers be x and y, while x is larger
x+y=1 ---- 1 and 1/x -1/y = 3/2 ----- 2
from 1, x=1-y

sub x = 1-y into 2,
1/1-y - 1/y = 3/2
y-(1-y)/y(1-y) = 3/2
2y-1/y-y^2 = 3/2
3y - 3y^2 = 4y -2
3y^2 - y + 2 = 0
(3y-2) (y+1) = 0
y = 2/3 or -1

計唔到 y = 2 喎 !
點解呢 ........

x+y=1
1/x-1/y=3/2
(y-x)/xy=3/2
y-x=3xy/2
2y-2x=3xy
1/(1-y)-1/y=3/2
通分母
(y-1+y)/(1-y)y=3/2
2y-1/(y-y^2)=3/2
4y-2=3y-3y^2
3y^2+y-2=0
(3y-2)(y+1)=0
y=2/3 or y=-1
x=1/3 or y=2

Let x,y be the two numbers:
---- x+y=1____(1)
----1/x-1/y=3/2____(2)
By (2),
y=1-x______(3)
Sub (3)into (2),
1/x-1/(1-x)=3/2
2(1-2x)=3(1-x)x
3x^2 - 7x+ 2=0
i.e. x=2 or x=1/3.......(4)
=>if x=2,y= -1,
=>if x=1/3, y= 2/3
i.e. The two larger numbers are 2 and 2/3