find all nonnegative integer solutions of the system
x+y<8
x+t<8
t+y<8
z+t<8
if t=0, we have
x<8
y<8
z<8
x+y<8
so, 0<x<8, y<8-x, z<8
List them all.
If t=1, change the 8's to 7's in the first 3 conditions.
And so on, up to t=7:
x<1
y<1
z<1
x+y<8
(x,y,z,t) = (0,0,0,7)
Don't see the point of the exercise, but there it is.
sory a mistake, the corect one problem is find all nonnegative integer solutions of the system
x+y<8
x+Z<8
t+y<8
z+t<8
To find all nonnegative integer solutions of the system, we will solve each inequality separately and then combine the solutions.
1) x + y < 8:
Let's start by assigning values to x and y.
If we let x = 0, then the inequality becomes 0 + y < 8, which simplifies to y < 8.
Therefore, when x = 0, any value of y less than 8 satisfies the inequality.
Repeating the same process for x = 1, 2, 3, ..., we find that when x is any nonnegative integer, y must be less than 8 to satisfy the inequality.
So, the solution for the inequality x + y < 8 is:
0 ≤ x < ∞
0 ≤ y < 8
2) x + t < 8:
Using a similar approach, let's assign values to x and t.
If we let x = 0, then the inequality becomes 0 + t < 8, simplifying to t < 8.
Again, any nonnegative integer value of x would satisfy the inequality.
Therefore, the solution for the inequality x + t < 8 is:
0 ≤ x < ∞
0 ≤ t < 8
3) t + y < 8:
Assigning values to t and y:
If we let t = 0, then the inequality becomes 0 + y < 8, simplifying to y < 8.
Just like before, any nonnegative integer value for t satisfies the inequality.
Therefore, the solution for the inequality t + y < 8 is:
0 ≤ t < ∞
0 ≤ y < 8
4) z + t < 8:
Similar to the previous inequalities:
If we let z = 0, then the inequality becomes 0 + t < 8, simplifying to t < 8.
In this case, any nonnegative integer value for z satisfies the inequality.
Therefore, the solution for the inequality z + t < 8 is:
0 ≤ z < ∞
0 ≤ t < 8
Now, to find all the solutions of the system, we need to find the intersection of the solutions for each inequality.
Combining all the solutions for x, y, t, and z, we get:
0 ≤ x, y, z < ∞
0 ≤ t < 8
Therefore, the nonnegative integer solutions for the system are:
0 ≤ x, y, z < ∞
0 ≤ t < 8