Final answer:
To determine the number of solutions in nonnegative integers for the given system of equations, we need to analyze the equations in echelon form and solve for the variables. The system has a solution in nonnegative integers if z is greater than or equal to 40.
Step-by-step explanation:
To determine the number of solutions in nonnegative integers for the given system of equations, we need to analyze the equations in echelon form:
x + 2y + z = 8
5y + 2z = 40
From the second equation, we can solve for y in terms of z: y = (40 - 2z)/5.
Substituting this into the first equation, we get x + 2((40 - 2z)/5) + z = 8.
Simplifying, we have x + (80 - 4z)/5 + z = 8.
Multiplying both sides by 5, we get 5x + 80 - 4z + 5z = 40.
Combining like terms, we have 5x - z + 80 = 40.
Now, we can solve for x in terms of z: x = (40 - 80 + z)/5.
Simplifying, we have x = (z - 40)/5.
Since we are looking for nonnegative integers, both x and z must be greater than or equal to 0. Therefore, z - 40 ≥ 0, which implies z ≥ 40.
So, the system of equations has a solution in nonnegative integers if z ≥ 40.