For the following system of equations in echelon​ form, tell how many solutions there are in nonnegative integers.
x plus 2 y plus 3 z
equals92
3 y plus 4 z
equals36
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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
There is at least one​ solution, left parenthesis nothing comma nothing comma z right parenthesis
  
  
.
​(Use integers or fractions for any numbers in the​ expression.)
B.
There is no solution.

Let's analyze the system of equations provided in echelon form:

\[
\begin{align*}
1. & \quad x + 2y + 3z = 92 \\
2. & \quad 3y + 4z = 36
\end{align*}
\]

### Step 1: Solve the second equation for \(y\) in terms of \(z\)
The second equation is:
\[
3y + 4z = 36
\]
Rearranging for \(y\):
\[
y = \frac{36 - 4z}{3}
\]

For \(y\) to be a nonnegative integer, \(\frac{36 - 4z}{3}\) must also be a nonnegative integer. This means that \(36 - 4z\) must be divisible by 3 and the value must be nonnegative.

### Step 2: Analyzing divisibility condition
The condition \(36 - 4z\) must be divisible by 3. Let’s write this requirement mathematically:
\[
36 - 4z \equiv 0 \pmod{3}
\]
Since \(36\) is divisible by 3, we focus on \(4z \equiv 0 \pmod{3}\). This means that \(z\) itself must be divisible by 3.

Let \(z = 3k\), where \(k\) is a nonnegative integer. Substituting \(z = 3k\) into the equation for \(y\):
\[
y = \frac{36 - 4(3k)}{3} = \frac{36 - 12k}{3} = 12 - 4k
\]

For \(y\) to be nonnegative:
\[
12 - 4k \geq 0 \implies k \leq 3
\]
Thus, \(k\) can take integer values \(0, 1, 2, 3\).

### Step 3: Analyzing possible values of \(z\) and \(y\)
Substituting the values of \(k\) into \(z = 3k\) and \(y = 12 - 4k\):

- If \(k = 0\), then \(z = 0\) and \(y = 12\).
- If \(k = 1\), then \(z = 3\) and \(y = 8\).
- If \(k = 2\), then \(z = 6\) and \(y = 4\).
- If \(k = 3\), then \(z = 9\) and \(y = 0\).

### Step 4: Substitute values into the first equation to solve for \(x\)
Substituting these values of \(y\) and \(z\) into the first equation:
\[
x + 2y + 3z = 92
\]

Calculating \(x\) for each case:
- If \(z = 0\) and \(y = 12\): \(x + 2(12) + 3(0) = 92 \implies x + 24 = 92 \implies x = 68\)
- If \(z = 3\) and \(y = 8\): \(x + 2(8) + 3(3) = 92 \implies x + 16 + 9 = 92 \implies x = 67\)
- If \(z = 6\) and \(y = 4\): \(x + 2(4) + 3(6) = 92 \implies x + 8 + 18 = 92 \implies x = 66\)
- If \(z = 9\) and \(y = 0\): \(x + 2(0) + 3(9) = 92 \implies x + 0 + 27 = 92 \implies x = 65\)

### Conclusion
There are 4 sets of solutions in nonnegative integers:
- \((x, y, z) = (68, 12, 0)\)
- \((x, y, z) = (67, 8, 3)\)
- \((x, y, z) = (66, 4, 6)\)
- \((x, y, z) = (65, 0, 9)\)

The correct answer is:
**A. There is at least one solution, \((68, 12, 0)\).**

Would you like further details on how we derived these values or have any questions?

### Related Questions
1. What conditions must be met for a solution in nonnegative integers to exist in a system of equations?
2. How do you check if an expression is divisible by a specific number?
3. What is the significance of using echelon form in solving systems of equations?
4. Can there be infinitely many solutions in nonnegative integers for a linear system? Why or why not?
5. How do constraints like nonnegative integers impact the possible solutions of a system?

**Tip:** When analyzing solutions in nonnegative integers, always check that all terms meet the required divisibility and positivity conditions.

For the following system of equations in echelon​ form, tell how many solutions there are in nonnegative integers.
x + 2y + 3z = 92
3y + 4z = 36
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A. There is at least one​ solution, (x, y, z)
B. There is no solution.

Discover how to solve the system of linear equations x + 2y + 3z = 92 and 3y + 4z = 36 in echelon form. Learn the step-by-step process to find nonnegative integer solutions, including key algebraic concepts and theorems such as divisibility rules, linear combinations, and the substitution method. Suitable for students in Grades 10-12.

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