We are tasked with determining how many solutions exist in nonnegative integers for the given system of equations:

1. $4x + 3y + 2z = 90$
2. $y - 2z = 10$

Step 1: Solve the second equation for $y$

From the second equation, we can express $y$ in terms of $z$:

$y = 2z + 10$

Step 2: Substitute into the first equation

Now, substitute $y = 2z + 10$ into the first equation $4x + 3y + 2z = 90$:

$4x + 3(2z + 10) + 2z = 90$

Simplify the expression:

$4x + 6z + 30 + 2z = 90$ $4x + 8z + 30 = 90$ $4x + 8z = 60$ $x + 2z = 15$

Step 3: Solve for $x$ in terms of $z$

$x = 15 - 2z$

Step 4: Analyze the constraints

We now have the following conditions:

• $x = 15 - 2z$
• $y = 2z + 10$

For $x$ and $y$ to be nonnegative integers, we need to ensure that:

1. $x \geq 0$, meaning $15 - 2z \geq 0$, or $z \leq 7.5$. Since $z$ must be a nonnegative integer, the possible values for $z$ are $z = 0, 1, 2, 3, 4, 5, 6, 7$.

2. $y \geq 0$, meaning $2z + 10 \geq 0$, which is always true since $z$ is nonnegative.

Step 5: Find the integer solutions

Now, let's verify the integer solutions for each $z$:

• $z = 0$: $x = 15 - 2(0) = 15$, $y = 2(0) + 10 = 10$
• $z = 1$: $x = 15 - 2(1) = 13$, $y = 2(1) + 10 = 12$
• $z = 2$: $x = 15 - 2(2) = 11$, $y = 2(2) + 10 = 14$
• $z = 3$: $x = 15 - 2(3) = 9$, $y = 2(3) + 10 = 16$
• $z = 4$: $x = 15 - 2(4) = 7$, $y = 2(4) + 10 = 18$
• $z = 5$: $x = 15 - 2(5) = 5$, $y = 2(5) + 10 = 20$
• $z = 6$: $x = 15 - 2(6) = 3$, $y = 2(6) + 10 = 22$
• $z = 7$: $x = 15 - 2(7) = 1$, $y = 2(7) + 10 = 24$

Step 6: Conclusion

There are 8 solutions for $(x, y, z)$ in nonnegative integers.

Thus, the correct answer is:

A. There is at least one solution, $(x, y, z) = (15 - 2z, 2z + 10, z)$, where $z = 0, 1, 2, 3, 4, 5, 6, 7$.

Would you like more details on the steps or have any questions?

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Relative Questions:

1. What is the significance of transforming equations into echelon form?
2. Can a system of equations have an infinite number of nonnegative integer solutions?
3. How does the nature of coefficients affect the existence of integer solutions?
4. What methods can be used to solve systems of linear Diophantine equations?
5. How do constraints like nonnegative integers influence solution sets?

Tip: Always simplify one equation first when solving systems—it reduces complexity in substitution steps.