Question

For the following system of equations in echelon form, tell how many solutions there are in nonnegative integers.

x + 3y +z = 80
7y + 2z=28
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
OA. There are nonnegative solutions.
B. There are infinitely many solutions.
OC. There is no solution.

PLEASE HELPP I DONT UNDERSTAND THIS

Answer 1

A. There are 3 non-negative solutions.

Explanation:

You want to know the number of nonnegative integer solutions there are to the system of equations ...

• x +3y +z = 80
• 7y +2z = 28

Solutions

Solving for y, we have ...

7y = 28 -2z

y = 4 - 2/7z . . . . . . divide by 7

Substituting this into the first equation, we can solve for x:

x = 80 -3y -z

x = 80 -3(4 -2/7z) -z = 68 -1/7z

Non-negative integers

In order to have x, y, and z be integers, we require ...

z = 7n . . . . . . for n ≥ 0

y = 4 -(2/7)(7n) = 4 -2n

x = 80 -3y -z = 80 -3(4 -2n) -7n = 68 -n

In summary:

(x, y, z) = (68 -n, 4 -2n, 7n) . . . . . for integer n

Restrictions

In order for y to be a nonnegative integer, we require ...

4 -2n ≥ 0

n ≤ 2 . . . . . . . . . . divide by 2, add n

In order for x to be a nonnegative integer, we require ...

68 -n ≥ 0

n ≤ 68

The requirement for y ≥ 0 is more restrictive, so the possible values of n are ...

0 ≤ n ≤ 2

There are 3 integers in this range: {0, 1, 2}.

There are 3 nonnegative solutions:

(x, y, z) ∈ {(68, 4, 0), (67, 2, 7), (66, 0, 14)}

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Answer 2

A

Explanation:

are you sure this is the full problem description ? you did not skip anything ?

did you mean for answer option B to be :

there are infinitely many non-negative integer solutions.

what you gave us here is

x + 3y + z = 80

7y + 2z = 28

to narrow this down we can multiply the first equation by 2 and then subtract the second from the first :

2x + 6y + 2z = 160

- 7y + 2z = 28

-----------------------------

2x - y 0 = 132

so, looking at

2x - y = 132

y = 2x - 132

this is a line, where for x >= 66

x and y are positive (non-negative) numbers.

and z can have any fitting value incl. non-negative ones.

then we can use that e.g. in the second equation

7(2x - 132) + 2z = 28

14x - 924 + 2z = 28

14x + 2z = 952

7x + z = 476

z = -7x + 476

for z to be non-negative

7x <= 476

x <= 68

so overall, for non-negative solutions

66 <= x <= 68

for integer solutions that is

66, 67, 68

that means that the possible integer solutions for y are

0, 2, 4

and the possible integer solutions for z are

462, 469, 476

as we can see, there are limited ("some") solutions with non-negative integers.

so, answer A is correct.

BUT we know that one equation with 2 variables or 2 equations with 3 variables has/have infinitely many solutions.

and clearly, some of them are non-negative (x, y and z are positive numbers).

so, if B is really phrased like you wrote it, then many greetings to your teacher, and I know what he/she meant, but B is formally correct too.

if it is phrased more like I wrote it above, then also formally only A is correct.