Explanation:
Sure, let's create a system of three equations in three variables with integer solutions. Let's use the variables x, y, and z. We'll create a system of equations and find a solution:
Here's the system of equations:
2x + 3y - z = 8
4x - 2y + 2z = 12
x + 5y + 3z = 18
Now, let's solve this system of equations using the method of substitution or elimination.
First, let's isolate z in the first equation:
2x + 3y - z = 8
-z = 8 - 2x - 3y
z = -8 + 2x + 3y
Now, substitute this expression for z into the other two equations:
4x - 2y + 2z = 12
4x - 2y + 2(-8 + 2x + 3y) = 12
Simplify:
4x - 2y - 16 + 4x + 6y = 12
Combine like terms:
8x + 4y - 16 = 12
Add 16 to both sides:
8x + 4y = 28
Now, let's isolate z in the third equation:
x + 5y + 3z = 18
3z = 18 - x - 5y
z = 6 - (1/3)x - (5/3)y
Now, substitute this expression for z into the equation we derived:
8x + 4y = 28
8x + 4y = 28
This equation does not contain z, which means it holds true for any values of x and y. Therefore, there are infinitely many solutions to this system of equations, and we can choose any integer values for x and y to find corresponding integer values for z.
For example, let's choose x = 2 and y = 3:
8(2) + 4(3) = 16 + 12 = 28
So, x = 2, y = 3, and z can be calculated using the expressions we derived earlier:
z = -8 + 2x + 3y = -8 + 2(2) + 3(3) = -8 + 4 + 9 = 5
Therefore, one solution to the system of equations is x = 2, y = 3, and z = 5, all of which are integers, and there are infinitely many such solutions.