Math Problem Statement

Given the following augmented matrix in row-reduced form, assume that it is equivalent to an augmented matrix corresponding to a system of linear equations.

1 0 5

0 1 −7 0 0 0 Find the solution(s) to the system, if it exists. State the solution as a point, using the parameter(s) s and t if needed.

Solution

The given augmented matrix is in row-reduced form:

1 & 0 & 5 \\ 0 & 1 & -7 \\ 0 & 0 & 0 \end{pmatrix}$$ We can write the corresponding system of linear equations as: $$x_1 + 5 = 0$$ $$x_2 - 7 = 0$$ From the first equation, we solve for $$x_1$$: $$x_1 = -5$$ From the second equation, we solve for $$x_2$$: $$x_2 = 7$$ Since there is no third row, this system is consistent, and the solution is: $$(x_1, x_2) = (-5, 7)$$ Thus, the solution to the system is the point $$(-5, 7)$$. Would you like more details on how to interpret this matrix or how to solve other types of systems? ### Related questions: 1. How do we interpret a row of all zeros in a row-reduced augmented matrix? 2. What happens when there is a row of the form $$0 \ 0 \ 0 = 1$$ in the augmented matrix? 3. How can we solve a system of linear equations when there are free variables (i.e., infinite solutions)? 4. What is the significance of the pivot positions in a row-reduced augmented matrix? 5. How do you convert a system of linear equations into an augmented matrix and reduce it? ### Tip: Always look for free variables (those not corresponding to pivot columns) to identify infinite solutions in systems of equations.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Row-Reduced Echelon Form
Systems of Linear Equations

Formulas

Row-Reduced Form Interpretation
Solving Systems of Equations

Theorems

Row Reduction Theorem
Existence and Uniqueness of Solutions for Linear Systems

Suitable Grade Level

Grades 9-12

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