other and even shorter solution if xy = 35, then y = 35/x. replace y in the first equation: x² -(35/x)² = 24 --> x^4 -24 x² -1225 = 0 this is a biquadratic equation, which can be solved by substitution z = x², the quadratic equation z² - 24 z -1225 = 0 has two solutions z1 = 49 and z2 = -25 resubstitution x² = z gives us the solutions x1 = 7 and x2 = -7 and the according y1 = 5 and y2 = -5 if needed: there are the complex solutions x3 = + 5i and y3 = -7i or x4 = -7i and y4 = 5i Done in about 5 minutes on a paper of the size of a postcard btw. this awesome music makes me extremely nervous
Une mendova nje rruge tjeter. Nxjerri x=35 /y ose te kunderten. Nxjerrim y=35/×. Bejme zevendesimin ne emuacionin e pare. Perftojme nje ekuacion me nje panjohur. Zgjidhja juaj eshte me komode se thjeshon ca shumzime dhe pjestime. Flm per kenaqesine qe na dhe.
А где результат и обоснование, что найдены все решения и других решение нет? And where is the result and the justification that all solutions have been found and there are no other solutions?
24 is 42 reverse & 35 is 53 reverse 53 -42=11 what is 11 ? else one "1" first clue ! 24*2 (cause there is 2 lines !) =48 (but dont forget the clue)= 49 !!! (we progress) we have a 9 and a 4 and in 1st line we have "-" so 9-4=5 yes we found it !!!! and also cause 2 lines 5+2=7 yes one again !!!!!!! i'm good or not ??? XD PRANK
If x₁; y₁) is the solution of the original system, then (-x₁; -y₁) is the solution to this system. There may be no other solutions, because xy = 35 > 0, which means that x and y are either positive or negative. Since x²-y²=24 phrases x²> y² phrases |x|>|y|, first it is enough to find a solution tone (x₁; y₁), where x₁> y₁>0. x²-y²=24⇔(x+y)(x-y)=24⇨(x+y)²(x-y)²=576⇔(x²+2xy+y²)( x²-+2xy+y²)=576⇔( x²+y²+70)( x²+y²-70)=576⇔( x²+y²)²-4900=576⇔( x²+y²)²=5476⇨( x²+y²)=74. The counts are x²+2xy+y²=74+70=144 and x+y=12, x²-2xy+yx=74-70=4 and x-y=2. Then 2x=14 and x=7, and y=x-2=5. Finally: (x₁; y₁)=(7; 5) , (x₂; y₂)=(-7; -5).
x^2 - y^2 = 24, ie. ( x + y ) ( x - y ) = 24 xy = 35 ( x - y )^2 = ( x + y )^2 - 4xy = ( x + y )^2 - 140 let ( x + y )^2 = t ( x - y )^2 = t - 140 ( ( x + y ) ( x - y ) )^2 = ( x + y )^2 * ( x - y )^2 = t * ( t - 140 ) = 24^2 = 576 t^2 - 140t - 576 = 0 t = ( 140 ± ( 140^2 + 4*1*576 )^1/2 ) / 2 140^2 + 4 * 576 = 19600 + 2304 = 21904 1 4 8 ************************ 2 19 04 1 ----------------- ( 1*20 = 20 ) 20 119 - 80 ( 20 * 4 = 80 ) - 16 ------------------( 14 * 20 = 280 ) 280 23 04 -22 40 ( 280 * 8 = 2240 ) - 64 =========== 0 ( 21904 ) ^1/2 = 148 t = ( 140 ± 148 ) / 2 = 144, -4 t = ( x + y )^2 > 0, so t = 144 t = ( x + y )^2 = 144, => x + y = ±12 ( x - y )^2 = ( x + y )^2 - 140 = 4, => x + y = ±2 ( x + y )^2 = x^2 - y^2 = 24 > 0 x + y = 12, x - y = 2, then x = 7, y = 5 # x + y = -12, x - y = -2, then x = -7, y = -5 #
Даже не считая понял что х=7 и у=5
X^2-y^2=24
Xy=35
Sol=> 35=7*5
X^2-y^2=24
1. X=5, y=7 )-> 25-49=-24 < erro
2. x=7, y=5 )-> 49-25=24 < fin
So, x=7; y=5
Решил в уме за 30секунд
7 и 5
Чего думать? 7 и 5!!!! Плюс минус, четыре пары!
Ну видно же сразу , что х= 7
y =5
other and even shorter solution
if xy = 35, then y = 35/x.
replace y in the first equation:
x² -(35/x)² = 24 --> x^4 -24 x² -1225 = 0
this is a biquadratic equation, which can be solved by substitution z = x²,
the quadratic equation z² - 24 z -1225 = 0 has two solutions z1 = 49 and z2 = -25
resubstitution x² = z gives us the solutions
x1 = 7 and x2 = -7 and the according
y1 = 5 and y2 = -5
if needed: there are the complex solutions
x3 = + 5i and y3 = -7i or x4 = -7i and y4 = 5i
Done in about 5 minutes on a paper of the size of a postcard
btw. this awesome music makes me extremely nervous
Correct, he dropped the imaginary solutions.
XX - YY = 24 , XY = 35 .
35 = 1 x 35 = 5 x 7
7 x 7 - 5 x 5 = 49 - 25 = 24 = XX - YY .
X = 7 , Y = 5 ./.
1) x = 7, y = 5
2) x = -7, y = -5
В уме решается.
X^2-Y^2 =(x+y)(x-y) = 24
xy=35 > 7*5
(7+5)*(7-5) > 12*2=24
X=7; Y=5
X^2-Y^2 = 24
X*Y=35 Y^2=35^2/X^2
X^2-35^2/X^2 = 24
(X^2)^2-35^2 = 24*X^2
Z^2-24*Z-35^2=0
Z=24+-sq(24^2-4*1*(-35^2)) =
=24+-2*sq(12^2+35^2)
=24+-2*37=24+-74= -50 и 98
Z=X^2 x=+-sqr (-50 и 98)
（X^2-Y^2）/（XY）=24/35，35X^2-24XY-35Y^2=0，（5X-7Y）（7X+5Y)=0，XY=35>0, 7X+5Y0, 5X-7Y=0, Let X=7K,Y=5K, XY=35K^2=35,K^2=1, K=-+1, X=-+7,Y=-+5
(-5; -7), (5; 7), (-7; -5), (7; 5)
Let x/y=t solve for t we get t=49/35. Then solve for y get y=+-5 final solve for x we say answer x=+-7. We have two solutions for problems
7x5=35
×=7
y=5
😊
XY=35 x=_5or5, ×=_7or7 x^- y^=24. X=_7or7 y=_5or5
x=7 and y=5 is the answer
7² -5² = 49-25=24
,7 x 5= 35
X=7, y=5.
X=7,,,Y=5
this can be done so much more easily
Is type se to student exam me fail ho jayega Kyoko ek ghanta laga diya while it was very easy to solve
X=7. Y=5
Это видно без решения -5,-7 и 5,7
This isn’t an Olympiad problem - it’s trivial
X-7 y-5
Une mendova nje rruge tjeter. Nxjerri x=35 /y ose te kunderten. Nxjerrim y=35/×. Bejme zevendesimin ne emuacionin e pare. Perftojme nje ekuacion me nje panjohur. Zgjidhja juaj eshte me komode se thjeshon ca shumzime dhe pjestime. Flm per kenaqesine qe na dhe.
Nunca lo había visto sin hablar y le pueda uno comprender❤
x=7, y=5
Answer at a glance in 10 seconds. It is Asian math power, but missing solution process. Different culture.
X = 7 ; Y = 5.
Методом подбора - за 10 секунд...
А где результат и обоснование, что найдены все решения и других решение нет? And where is the result and the justification that all solutions have been found and there are no other solutions?
24 is 42 reverse & 35 is 53 reverse 53 -42=11 what is 11 ? else one "1" first clue !
24*2 (cause there is 2 lines !) =48 (but dont forget the clue)= 49 !!! (we progress)
we have a 9 and a 4 and in 1st line we have "-" so 9-4=5 yes we found it !!!!
and also cause 2 lines 5+2=7 yes one again !!!!!!! i'm good or not ??? XD PRANK
7,5
Х = 7 у = 5
x=7 and y=5
If x₁; y₁) is the solution of the original system, then (-x₁; -y₁) is the solution to this system. There may be no other solutions, because xy = 35 > 0, which means that x and y are either positive or negative. Since x²-y²=24 phrases x²> y² phrases |x|>|y|, first it is enough to find a solution tone (x₁; y₁), where x₁> y₁>0.
x²-y²=24⇔(x+y)(x-y)=24⇨(x+y)²(x-y)²=576⇔(x²+2xy+y²)( x²-+2xy+y²)=576⇔( x²+y²+70)( x²+y²-70)=576⇔( x²+y²)²-4900=576⇔( x²+y²)²=5476⇨( x²+y²)=74. The counts are x²+2xy+y²=74+70=144 and x+y=12, x²-2xy+yx=74-70=4 and x-y=2. Then 2x=14 and x=7, and y=x-2=5. Finally: (x₁; y₁)=(7; 5) , (x₂; y₂)=(-7; -5).
Note yx replace with y^2
We can easily to see (x^2+y^2)^2=(x^2-y^2)^2+(2xy)^2=24^2+(70)^2=
576+4900.
Х=7... У=5
Итог
No need to calculation, x=7 & y=5
x/y = 7/5 and you cross-multiplied? 😩
Красивое решение!
Ahhhhh7/5......ahhhhh
인수분해..?
This disgusting music vould not be change?
9:55 from this ratio, x=7 and y=5, no ? The rest should be a question of sign.
No.....only cross multiply.
x^2 - y^2 = 24, ie. ( x + y ) ( x - y ) = 24
xy = 35
( x - y )^2 = ( x + y )^2 - 4xy = ( x + y )^2 - 140
let ( x + y )^2 = t
( x - y )^2 = t - 140
( ( x + y ) ( x - y ) )^2
= ( x + y )^2 * ( x - y )^2
= t * ( t - 140 )
= 24^2 = 576
t^2 - 140t - 576 = 0
t = ( 140 ± ( 140^2 + 4*1*576 )^1/2 ) / 2
140^2 + 4 * 576 = 19600 + 2304 = 21904
1 4 8
************************
2 19 04
1
----------------- ( 1*20 = 20 )
20 119
- 80 ( 20 * 4 = 80 )
- 16
------------------( 14 * 20 = 280 )
280 23 04
-22 40 ( 280 * 8 = 2240 )
- 64
===========
0
( 21904 ) ^1/2 = 148
t = ( 140 ± 148 ) / 2 = 144, -4
t = ( x + y )^2 > 0, so t = 144
t = ( x + y )^2 = 144, => x + y = ±12
( x - y )^2 = ( x + y )^2 - 140 = 4, => x + y = ±2
( x + y )^2 = x^2 - y^2 = 24 > 0
x + y = 12, x - y = 2, then x = 7, y = 5 #
x + y = -12, x - y = -2, then x = -7, y = -5 #
7,5