Math Problem Statement

Todd rewrites a system of equations of variables x, y, and z as a matrix, which he then starts to reduce. He stops when he has the result below:

1	1	0	–9

0 0 1 4 0 0 0 0

What can Todd conclude about his original system of equations? A.x = −9. B.x + y + z = 0. C.The system has no solution. D.The system has infinite solution

Solution

Let's interpret Todd's matrix to draw conclusions about the system of equations.

The given matrix is:

1 & 1 & 0 & -9 \\ 0 & 0 & 1 & 4 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix}$$ This matrix corresponds to the following system of equations: 1. From the first row: $$1x + 1y + 0z = -9$$, which simplifies to $$x + y = -9$$. 2. From the second row: $$0x + 0y + 1z = 4$$, which simplifies to $$z = 4$$. 3. The third row is $$0 = 0$$, which doesn't provide any new information. ### Step-by-step reasoning: - The third row being all zeros indicates that one equation in the system is dependent (a linear combination of others), so the system has **fewer equations than variables**, meaning the system is underdetermined. - There are two equations (from rows 1 and 2), but three variables: $$x$$, $$y$$, and $$z$$. Hence, there are **infinite solutions**. The system can be solved as follows: - From the second row, we know $$z = 4$$. - Substituting $$z = 4$$ into the first row, we get $$x + y = -9$$. - Since there are only two equations for three variables, $$x$$ and $$y$$ can take multiple values depending on each other, leading to an infinite set of solutions. ### Conclusion: Todd can conclude that the system has infinite solutions. The correct answer is **D. The system has infinite solutions**. Would you like further clarification? Here are some related questions you might find useful: 1. What are the conditions for a system of equations to have infinite solutions? 2. How can we express the solution set for this system in parametric form? 3. What does it mean if a system has no solutions, and how can we tell from the matrix? 4. How would the system change if there were three independent equations? 5. What are some common methods to reduce a matrix to row-echelon form? **Tip:** To determine the type of solutions in a system of equations (unique, none, or infinite), look for the number of independent equations relative to the number of variables.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Systems
Matrix Reduction
Row Echelon Form

Formulas

Row Reduction
Solving Linear Systems

Theorems

Rank-Nullity Theorem
Infinite Solutions in Systems of Linear Equations

Suitable Grade Level

Grades 10-12

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