# Integer solutions of system of equations

• MHB
• anemone
In summary, integer solutions in a system of equations are whole number values that satisfy all the equations in the system. These can be found using the method of substitution, where one equation is solved for a variable and then substituted into the other equations. It is possible for a system of equations to have more than one set of integer solutions, and if there are no integer solutions, the system is said to be inconsistent. While there are some techniques that can make finding integer solutions easier, there is no one universal shortcut.
anemone
Gold Member
MHB
POTW Director
Find all integer solutions of the system of equations $x+y+z=3$ and $x^3+y^3+z^3=3$.

we are given
$x+y+z = 3 \cdots(1)$
and
$x^3+y^3+z^3 = 3\cdots(2)$
from (1)
$x+y = 3 - z\cdots(3)$
and from (2)
$x^3+y^3 = 3 - z^3\cdots(4)$
From (3) and (4)
because $x+y$ divides $x^3+y^3$ so $x+y$ divides $3-z^3$ or $3-z$ divides $3-z^3$
so $z-3$ divides $z^3- 3$
as $z-3$ divides $z^3-3^3$ or $z^3 - 27$
so $z-3$ divides $(z^3-3) - (z^3- 27) = 24$
further if we have mod 9 then
$x^3 = 0\, or 1\,or\, -1$
$y^3 = 0\, or 1\,or\, -1$
$z^3 = 0\, or 1\,or\, -1$
as we have $x^3+y^3+z^3 = 3$ so we have $x^3=y^3=z^3 = 1$ mod 9
so $x \equiv y \equiv z \equiv 1\pmod 3$
so we need to take x-3 such that they are 1 mod 3 and factor of 24
they are ${ -8, -2, 1, 4}$
This gives choices for x as $(-5, 1, 4, 7)$
same for y and z and we can checking the sets get $x=y=z=1$

Last edited:

anemone said:
May be. I would like to know the correct answer

There was a typo error in first line and I corrected the same. Otherwise I do not find error if any. This may be pointed

uppose $(x,\,y,\,z)$ is the set of solution.

From the identity $(x+y+z)^3-(x^3+y^3+z^3)=3(x+y)(y+z)(z+x)$, we get $8=(3-z)(3-x)(3-y)$. Since $6=(3-z)+(3-x)+(3-y)$, checking the factorization of 8, we see that the solutions are $(1,\,1,\,1)$, $(-5,\,4,\,4)$, $(4,\,-5,\,4)$ and $(4,\,4,\,-5)$.

## What are integer solutions of a system of equations?

Integer solutions of a system of equations refer to the values of the variables in the equations that result in whole number solutions. These solutions can be represented as ordered pairs or triplets, depending on the number of variables in the system.

## How do you find integer solutions of a system of equations?

To find integer solutions, you can use various methods such as substitution, elimination, or graphing. These methods involve manipulating the equations to isolate the variables and then plugging in different integer values to see which ones satisfy all of the equations in the system.

## Can a system of equations have more than one integer solution?

Yes, a system of equations can have multiple integer solutions. This occurs when there are more equations than variables, resulting in an infinite number of solutions. It can also happen when there are equal or parallel lines in the system, resulting in an infinite number of solutions.

## What happens if a system of equations has no integer solutions?

If a system of equations has no integer solutions, it means that there are no values of the variables that satisfy all of the equations in the system. This could happen if the equations are inconsistent or if they represent parallel lines that never intersect.

## Can a system of equations with integer solutions have non-integer solutions as well?

Yes, a system of equations with integer solutions can also have non-integer solutions. This can occur when the equations involve fractions or decimals, resulting in solutions that are not whole numbers. It is important to specify whether the solutions should be integers or not when solving a system of equations.

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