Найти целочисленные решения уравнения y^2+xy-y+2x-8=0

Выразим из этого всего х: х=-(у²-у-8)/(у+2)= числитель делим на знаменатель=-(у-3)+2/(у+2) дробная часть имеет целое значение при у=-4,-3,-1,0 при у=-4 х=7-1=6 при у=-3 х=6-2=4 при у=-1 х=4+2=6 при у=0 х=3+1=4

Simplifying

4x3 + -81x = 0

Reorder the terms:

-81x + 4x3 = 0

Solving

-81x + 4x3 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), 'x'.

x(-81 + 4x2) = 0

Factor a difference between two squares.

x((9 + 2x)(-9 + 2x)) = 0

Subproblem 1

Set the factor 'x' equal to zero and attempt to solve:

Simplifying

x = 0

Solving

x = 0

Move all terms containing x to the left, all other terms to the right.

Simplifying

x = 0

Subproblem 2

Set the factor '(9 + 2x)' equal to zero and attempt to solve:

Simplifying

9 + 2x = 0

Solving

9 + 2x = 0

Move all terms containing x to the left, all other terms to the right.

Add '-9' to each side of the equation.

9 + -9 + 2x = 0 + -9

Combine like terms: 9 + -9 = 0

0 + 2x = 0 + -9

2x = 0 + -9

Combine like terms: 0 + -9 = -9

2x = -9

Divide each side by '2'.

x = -4.5

Simplifying

x = -4.5

Subproblem 3

Set the factor '(-9 + 2x)' equal to zero and attempt to solve:

Simplifying

-9 + 2x = 0

Solving

-9 + 2x = 0

Move all terms containing x to the left, all other terms to the right.

Add '9' to each side of the equation.

-9 + 9 + 2x = 0 + 9

Combine like terms: -9 + 9 = 0

0 + 2x = 0 + 9

2x = 0 + 9

Combine like terms: 0 + 9 = 9

2x = 9

Divide each side by '2'.

x = 4.5

Simplifying

x = 4.5

Solution

x = {0, -4.5, 4.5}

x + 1 / y = y + 1 / z преобразуем в: x - y = 1 / z - 1 / y = (y - z) / (y * z)  y + 1 / z = z + 1 / x преобразуем в: y - z = 1 / x - 1 / z = (z - x) / (x * z)  z + 1 / x = x + 1 / y преобразуем в: z - x = 1 / y - 1 / x = (x - y) / (x * y)  умножим полученные 3 равенства:   (x - y) * (y - z) * (z - x) = (y - z) / (y * z) * (z - x) / (x * z) * (x - y) / (x * y)  (x - y) * (y - z) * (z - x) = (x - y) * (y - z) * (z - x) / (x * y * z)^2  отсюда следует, что или (x * y * z)^2 = 1,  или скажем x = y, но тогда и y = z (следует из: x + 1 / y = y + 1 / z).