Solve two equations of Linear Inequality involving three different variables through code - c++

Question

I have two linear inequality equations with me, I am trying to understand how can I solve them using the code.
2x + z <= A
3y + z <= B
I know the values of A and B. I am trying to find all the possible positive values of x , y and z that will satisfy the given equation.
I want to solve the above equation using code but I haven't been able to come up with anything.
Any pointers are helpful. Thanks.

Answer 1

Linear inequalities can be represented as boolean expressions. Either the inequality is true, or false. We can use a function in C++ just like a function in mathematics here.
bool equation_one(double x, double z, double A)
{
return ((2 * x) + z <= A);
}
bool equation_two(double y, double z, double A)
{
return ((3 * y) + z <= A);
}
Then, if we wanted to see if any values satisfies these equations, we can just try values.
NOTE: This is horribly inefficient. We are looking at cubic time bounds. But, this is much more straight forward then using a linear algebra technique.
for(double x = 0; x <= A/2; x++)
for(double y = 0; y <= B/3; y++)
for(double z = 0; z <= A && z <= B; z++)
if(equation_one(x, z, A) && equation_two(y, z, B))
//These are the values we want!
EDIT: My bad, you said positive values only. Fixed.
EDIT 2: Thought I'd provide some output. For A = 20 and B = 20, these are all possible positive integer values that satisfy the equations with x, y, and z respectively.
0 0 0
0 0 1
0 0 2
0 0 3
0 0 4
0 0 5
0 0 6
0 0 7
0 0 8
0 0 9
0 0 10
0 0 11
0 0 12
0 0 13
0 0 14
0 0 15
0 0 16
0 0 17
0 0 18
0 0 19
0 0 20
0 1 0
0 1 1
0 1 2
0 1 3
0 1 4
0 1 5
0 1 6
0 1 7
0 1 8
0 1 9
0 1 10
0 1 11
0 1 12
0 1 13
0 1 14
0 1 15
0 1 16
0 1 17
0 2 0
0 2 1
0 2 2
0 2 3
0 2 4
0 2 5
0 2 6
0 2 7
0 2 8
0 2 9
0 2 10
0 2 11
0 2 12
0 2 13
0 2 14
0 3 0
0 3 1
0 3 2
0 3 3
0 3 4
0 3 5
0 3 6
0 3 7
0 3 8
0 3 9
0 3 10
0 3 11
0 4 0
0 4 1
0 4 2
0 4 3
0 4 4
0 4 5
0 4 6
0 4 7
0 4 8
0 5 0
0 5 1
0 5 2
0 5 3
0 5 4
0 5 5
0 6 0
0 6 1
0 6 2
1 0 0
1 0 1
1 0 2
1 0 3
1 0 4
1 0 5
1 0 6
1 0 7
1 0 8
1 0 9
1 0 10
1 0 11
1 0 12
1 0 13
1 0 14
1 0 15
1 0 16
1 0 17
1 0 18
1 1 0
1 1 1
1 1 2
1 1 3
1 1 4
1 1 5
1 1 6
1 1 7
1 1 8
1 1 9
1 1 10
1 1 11
1 1 12
1 1 13
1 1 14
1 1 15
1 1 16
1 1 17
1 2 0
1 2 1
1 2 2
1 2 3
1 2 4
1 2 5
1 2 6
1 2 7
1 2 8
1 2 9
1 2 10
1 2 11
1 2 12
1 2 13
1 2 14
1 3 0
1 3 1
1 3 2
1 3 3
1 3 4
1 3 5
1 3 6
1 3 7
1 3 8
1 3 9
1 3 10
1 3 11
1 4 0
1 4 1
1 4 2
1 4 3
1 4 4
1 4 5
1 4 6
1 4 7
1 4 8
1 5 0
1 5 1
1 5 2
1 5 3
1 5 4
1 5 5
1 6 0
1 6 1
1 6 2
2 0 0
2 0 1
2 0 2
2 0 3
2 0 4
2 0 5
2 0 6
2 0 7
2 0 8
2 0 9
2 0 10
2 0 11
2 0 12
2 0 13
2 0 14
2 0 15
2 0 16
2 1 0
2 1 1
2 1 2
2 1 3
2 1 4
2 1 5
2 1 6
2 1 7
2 1 8
2 1 9
2 1 10
2 1 11
2 1 12
2 1 13
2 1 14
2 1 15
2 1 16
2 2 0
2 2 1
2 2 2
2 2 3
2 2 4
2 2 5
2 2 6
2 2 7
2 2 8
2 2 9
2 2 10
2 2 11
2 2 12
2 2 13
2 2 14
2 3 0
2 3 1
2 3 2
2 3 3
2 3 4
2 3 5
2 3 6
2 3 7
2 3 8
2 3 9
2 3 10
2 3 11
2 4 0
2 4 1
2 4 2
2 4 3
2 4 4
2 4 5
2 4 6
2 4 7
2 4 8
2 5 0
2 5 1
2 5 2
2 5 3
2 5 4
2 5 5
2 6 0
2 6 1
2 6 2
3 0 0
3 0 1
3 0 2
3 0 3
3 0 4
3 0 5
3 0 6
3 0 7
3 0 8
3 0 9
3 0 10
3 0 11
3 0 12
3 0 13
3 0 14
3 1 0
3 1 1
3 1 2
3 1 3
3 1 4
3 1 5
3 1 6
3 1 7
3 1 8
3 1 9
3 1 10
3 1 11
3 1 12
3 1 13
3 1 14
3 2 0
3 2 1
3 2 2
3 2 3
3 2 4
3 2 5
3 2 6
3 2 7
3 2 8
3 2 9
3 2 10
3 2 11
3 2 12
3 2 13
3 2 14
3 3 0
3 3 1
3 3 2
3 3 3
3 3 4
3 3 5
3 3 6
3 3 7
3 3 8
3 3 9
3 3 10
3 3 11
3 4 0
3 4 1
3 4 2
3 4 3
3 4 4
3 4 5
3 4 6
3 4 7
3 4 8
3 5 0
3 5 1
3 5 2
3 5 3
3 5 4
3 5 5
3 6 0
3 6 1
3 6 2
4 0 0
4 0 1
4 0 2
4 0 3
4 0 4
4 0 5
4 0 6
4 0 7
4 0 8
4 0 9
4 0 10
4 0 11
4 0 12
4 1 0
4 1 1
4 1 2
4 1 3
4 1 4
4 1 5
4 1 6
4 1 7
4 1 8
4 1 9
4 1 10
4 1 11
4 1 12
4 2 0
4 2 1
4 2 2
4 2 3
4 2 4
4 2 5
4 2 6
4 2 7
4 2 8
4 2 9
4 2 10
4 2 11
4 2 12
4 3 0
4 3 1
4 3 2
4 3 3
4 3 4
4 3 5
4 3 6
4 3 7
4 3 8
4 3 9
4 3 10
4 3 11
4 4 0
4 4 1
4 4 2
4 4 3
4 4 4
4 4 5
4 4 6
4 4 7
4 4 8
4 5 0
4 5 1
4 5 2
4 5 3
4 5 4
4 5 5
4 6 0
4 6 1
4 6 2
5 0 0
5 0 1
5 0 2
5 0 3
5 0 4
5 0 5
5 0 6
5 0 7
5 0 8
5 0 9
5 0 10
5 1 0
5 1 1
5 1 2
5 1 3
5 1 4
5 1 5
5 1 6
5 1 7
5 1 8
5 1 9
5 1 10
5 2 0
5 2 1
5 2 2
5 2 3
5 2 4
5 2 5
5 2 6
5 2 7
5 2 8
5 2 9
5 2 10
5 3 0
5 3 1
5 3 2
5 3 3
5 3 4
5 3 5
5 3 6
5 3 7
5 3 8
5 3 9
5 3 10
5 4 0
5 4 1
5 4 2
5 4 3
5 4 4
5 4 5
5 4 6
5 4 7
5 4 8
5 5 0
5 5 1
5 5 2
5 5 3
5 5 4
5 5 5
5 6 0
5 6 1
5 6 2
6 0 0
6 0 1
6 0 2
6 0 3
6 0 4
6 0 5
6 0 6
6 0 7
6 0 8
6 1 0
6 1 1
6 1 2
6 1 3
6 1 4
6 1 5
6 1 6
6 1 7
6 1 8
6 2 0
6 2 1
6 2 2
6 2 3
6 2 4
6 2 5
6 2 6
6 2 7
6 2 8
6 3 0
6 3 1
6 3 2
6 3 3
6 3 4
6 3 5
6 3 6
6 3 7
6 3 8
6 4 0
6 4 1
6 4 2
6 4 3
6 4 4
6 4 5
6 4 6
6 4 7
6 4 8
6 5 0
6 5 1
6 5 2
6 5 3
6 5 4
6 5 5
6 6 0
6 6 1
6 6 2
7 0 0
7 0 1
7 0 2
7 0 3
7 0 4
7 0 5
7 0 6
7 1 0
7 1 1
7 1 2
7 1 3
7 1 4
7 1 5
7 1 6
7 2 0
7 2 1
7 2 2
7 2 3
7 2 4
7 2 5
7 2 6
7 3 0
7 3 1
7 3 2
7 3 3
7 3 4
7 3 5
7 3 6
7 4 0
7 4 1
7 4 2
7 4 3
7 4 4
7 4 5
7 4 6
7 5 0
7 5 1
7 5 2
7 5 3
7 5 4
7 5 5
7 6 0
7 6 1
7 6 2
8 0 0
8 0 1
8 0 2
8 0 3
8 0 4
8 1 0
8 1 1
8 1 2
8 1 3
8 1 4
8 2 0
8 2 1
8 2 2
8 2 3
8 2 4
8 3 0
8 3 1
8 3 2
8 3 3
8 3 4
8 4 0
8 4 1
8 4 2
8 4 3
8 4 4
8 5 0
8 5 1
8 5 2
8 5 3
8 5 4
8 6 0
8 6 1
8 6 2
9 0 0
9 0 1
9 0 2
9 1 0
9 1 1
9 1 2
9 2 0
9 2 1
9 2 2
9 3 0
9 3 1
9 3 2
9 4 0
9 4 1
9 4 2
9 5 0
9 5 1
9 5 2
9 6 0
9 6 1
9 6 2
10 0 0
10 1 0
10 2 0
10 3 0
10 4 0
10 5 0
10 6 0