Уровнение 1/x+1/y= 1/15 , 23 1/3 + 10/y =1 Надо решить

Предыдущий
вопрос Следующий
вопрос

1/x + 1/y = 1/15

Multiplying both sides by 15xy to get rid of the denominators:

15y + 15x = xy

Rearranging the equation:

xy - 15x - 15y = 0

Using Simon's Favorite Factoring Trick by adding and subtracting 225 on the left side to factorize the equation:

xy - 15x - 15y + 225 = 225
(x - 15)(y - 15) = 225

Since 225 = 335*5, the factors of 225 can be written as pairs: (1, 225), (3, 75), (5, 45), (9, 25), (15, 15).

So, the possible solutions for x and y are:

x - 15 = 1, y - 15 = 225 --> x = 16, y = 240
x - 15 = 3, y - 15 = 75 --> x = 18, y = 90
x - 15 = 5, y - 15 = 45 --> x = 20, y = 60
x - 15 = 9, y - 15 = 25 --> x = 24, y = 40
x - 15 = 15, y - 15 = 15 --> x = 30, y = 30

Now, we can check each pair of values in the second equation:

23 1/3 + 10/y = 1

x = 16, y = 240:
23 1/3 + 10/240 = 1
23 1/3 + 1/24 = 1
This equation is not satisfied.

x = 18, y = 90:
23 1/3 + 10/90 = 1
23 1/3 + 1/9 = 1
This equation is not satisfied.

x = 20, y = 60:
23 1/3 + 10/60 = 1
23 1/3 + 1/6 = 1
This equation is satisfied.

Therefore, the solution to the system of equations is x = 20, y = 60.