Math Problem Statement
Two augmented matrices for two linear systems in the variables x, y, and z are given below. The augmented matrices are in reduced row-echelon form. For each system, choose the best description of its solution. If applicable, give the solution.
Solution
Solution to the Given Problems
(a) The Augmented Matrix:
1 & 0 & -2 & | & -7 \\ 0 & 1 & 1 & | & 3 \\ 0 & 0 & 0 & | & 0 \end{bmatrix}$$ - The last row, $$0 = 0$$, does not introduce any inconsistency, meaning the system is consistent. - The number of pivot columns is **2**, corresponding to $$x$$ and $$y$$, while $$z$$ is a free variable. - Hence, the system has **infinitely many solutions**. The solution is parameterized as follows: $$x = -7 + 2z, \quad y = 3 - z, \quad z = z$$ Thus, the general solution is: $$(x, y, z) = (-7 + 2z, 3 - z, z), \quad z \in \mathbb{R}.$$ Correct answer for (a): - **The system has infinitely many solutions: $$(x, y, z) = (-7 + 2z, 3 - z, z)$$**. --- #### (b) The Augmented Matrix: $$\begin{bmatrix} 1 & 0 & 0 & | & -2 \\ 0 & 1 & 0 & | & 5 \\ 0 & 0 & 0 & | & 3 \end{bmatrix}$$ - The last row, $$0 = 3$$, is a contradiction. This makes the system **inconsistent**. - Hence, the system has **no solution**. Correct answer for (b): - **The system has no solution.** --- ### Let me know if you want a detailed explanation of any step or have questions! Here are 5 related questions for further exploration: 1. What does reduced row-echelon form mean, and how is it used to solve systems of equations? 2. How do free variables influence the number of solutions in a system? 3. What are the steps to convert a matrix to reduced row-echelon form? 4. How can we interpret inconsistent systems geometrically? 5. What happens when the augmented matrix has more equations than variables? **Tip:** Always check for inconsistencies in the augmented column when analyzing reduced row-echelon form matrices.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Matrix Operations
Reduced Row-Echelon Form
Formulas
Reduced row-echelon form definition
Consistency of linear systems (free variables, pivot columns)
Theorems
Fundamental Theorem of Linear Algebra
Consistency and Inconsistency of Linear Systems
Suitable Grade Level
College level or advanced high school