Question

Решить систему уравнений: $\bf \Huge \displaystyle \left \{ {{\bigg(\log_\frac{3}{2} (x+y)\bigg)^2+\bigg(\log_\frac{3}{2}(x-y)\bigg)^2-\bigg(\log_\frac{3}{2}(2x)\bigg)^2=1 } \atop {\log_\frac{2}{3}(x+y)\cdot \log_\frac{2}{3} (x-y)-\log_\frac{2}{3} (2x)=0}} \right. .$

Answer 1

Ответ:

(1;1/2)

Объяснение:

{ (log 3/2 (x+y) )² + (log 3/2 (x-y))²- (log 3/2 (2x))² =1,

{ log 2/3 (x+y) * log 2/3 (x-y) - log 2/3 (2x) = 0;

ОДР: х+у>0, х-у>0, 2х>0

пусть log 3/2 (x+y) = а, log 3/2 (x-у) = b, log 3/2 (2x) =c,

тогда log 2/3 (x+y) = log (3/2)^(-1) (x+y) = 1/(-1) log 3/2 (x+y) = - log 3/2 (x+y) = -а,

аналогично, log 2/3 (x-y) = -b, log 2/3 (2x) = -c.

то есть

{ а² + b² - c² = 1,

{ -a*(-b) - (-c) = 0.

{ а² + b² - c² = 1,

{ ab + c = 0.

{ а² + b² - c² = 1,

{ c = -ab;

a² + b² - (-ab)² = 1

a² + b² - a²b² - 1 = 0

a²(1-b²) - (1-b²) = 0

(a²-1)(1-b²)=0

a²=1 или b²=1

a=±1 или b=±1

1) a = 1

аb + c = 0, 1*b+c=0, b=-c

log 3/2 (x+y) = 1 → x+y = 3/2 → у = 3/2 - х

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

log 3/2 (x-y) + log 3/2 (2x) = 0

log 3/2 ((x-y) * (2x)) = 0

(x-y) * (2x) = 1

(х - (3/2-х)) * 2х = 1

(2х - 3/2) * 2х = 1

4х² -3х -1 = 0, D = (-3)²-4*4*(-1)=9+16=25=5²

x1=(-(-3)+5)/(2*4)=8/8=1 > 0, y = 3/2-1=1/2,

x+y=3/2>0 x-y=1/2> 0 — подходит (проверка одр),

х2 = (-(-3)-5)/(2*4)=-2/8=-1/4 < 0 — не подходит по одр

2) а = -1

аb + c = 0, -1*b+c=0, -b=-c, b=c

log 3/2 (x+y) = -1 → x+y = 2/3

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

так как 3/2>1, то х-у = 2х, у = -х

х+у=0 ≠ 2/3 → нет решений при а = -1

3) b = 1

аb + c = 0, 1*a+c=0, a=-c

log 3/2 (x-y) = 1 → x-y = 3/2 → y = x -3/2

log 3/2 (x+y) = - log 3/2 (2x)

log 3/2 (x+y) + log 3/2 (2x) = 0

log 3/2 ((x+y) * (2x)) = 0

(x+y) * (2x) = 1

(х - (x-3/2)) * 2х = 1

(x - x + 3/2) * 2x = 1

(3/2) * 2х = 1

3x = 1

x = 1/3 > 0

y = 1/3 - 3/2 = 2/6 - 9/6 = -7/6

x+y = 1/3+(-7/6) = 2/6-7/6=-5/6<0 — не подходит по одр → нет решений при b = 1

4) b = -1

аb + c = 0, -1*a+c=0, -a=-c, a=c

log 3/2 (x-y) = -1 → x-y = 2/3

log 3/2 (x+y) = log 3/2 (2x)

так как 3/2>1, то х+у = 2х, у = x

х-у=0 ≠ 2/3 → нет решений при b = -1