Math Problem Statement
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 Question content area bottom 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.
Solution
Let's analyze the system of equations provided in echelon form:
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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Systems of Linear Equations
Echelon Form
Nonnegative Integer Solutions
Formulas
x + 2y + 3z = 92
3y + 4z = 36
y = (36 - 4z) / 3
z = 3k
y = 12 - 4k
Theorems
Divisibility Rules
Linear Combination
Substitution Method
Suitable Grade Level
Grades 10-12
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