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 given system of inequalities, we can go through each variable one by one and determine the range of values for which the inequality holds true.
Let's start with the variable x:
From the inequality x + y < 8, we know that x < 8 - y.
Next, let's consider the variable y:
From the inequality t + y < 8, we know that y < 8 - t.
Moving on to the variable z:
From the inequality x + z < 8, we know that z < 8 - x.
Finally, let's consider the variable t:
From the inequality z + t < 8, we know that t < 8 - z.
Now, let's analyze the given information to find the range of values for each variable:
1. Since x + y < 8 and x < 8 - y, we can set up a table to list the possible values of x and y:
x | y
-----------------
0 | 7
1 | 6
2 | 5
3 | 4
4 | 3
5 | 2
6 | 1
7 | 0
2. Since t + y < 8 and y < 8 - t, the possible values of t and y can be listed:
t | y
-----------------
0 | 7
1 | 6
2 | 5
3 | 4
4 | 3
5 | 2
6 | 1
7 | 0
3. From the inequality x + z < 8 and z < 8 - x, the possible values of x and z are as follows:
x | z
-----------------
0 | 7
1 | 6
2 | 5
3 | 4
4 | 3
5 | 2
6 | 1
7 | 0
4. Considering the inequality z + t < 8 and t < 8 - z, the possible values of z and t are:
z | t
-----------------
0 | 7
1 | 6
2 | 5
3 | 4
4 | 3
5 | 2
6 | 1
7 | 0
Now we can find the common nonnegative integer solutions by comparing the values of x, y, z, and t from the four tables. These solutions are:
(0, 7, 7, 0)
(1, 6, 6, 1)
(2, 5, 5, 2)
(3, 4, 4, 3)
(4, 3, 3, 4)
(5, 2, 2, 5)
(6, 1, 1, 6)
(7, 0, 0, 7)
These are all the nonnegative integer solutions for the given system of inequalities.