Math Problem Statement

Find all solutions to the system of linear equations. (If there are an infinite number of solutions use s1 as your parameter. If there is no solution, enter NO SOLUTION.) x1 − x2 + 6x3 = 0 −2x1 + x2 − x3 = −1 3x1 − 2x2 + 7x3 = 1 (x1, x2, x3) =

Solution

We are given the following system of linear equations:

1. & \quad x_1 - x_2 + 6x_3 = 0 \\ 2. & \quad -2x_1 + x_2 - x_3 = -1 \\ 3. & \quad 3x_1 - 2x_2 + 7x_3 = 1 \end{aligned}$$ We can solve this system using Gaussian elimination or matrix methods. I'll solve it step by step. First, we'll write the system as an augmented matrix: $$\begin{pmatrix} 1 & -1 & 6 & 0 \\ -2 & 1 & -1 & -1 \\ 3 & -2 & 7 & 1 \end{pmatrix}$$ Now, we'll apply Gaussian elimination to solve for $$x_1$$, $$x_2$$, and $$x_3$$. Let me perform the necessary steps.It seems that the matrix has produced invalid values during the Gaussian elimination process, likely due to a division by zero or a degenerate case. This could suggest that the system might have either no solution or an infinite number of solutions. Let me carefully analyze the system and detect if it falls into one of these cases.The determinant of the coefficient matrix is extremely close to zero, indicating that the system is either dependent (leading to infinite solutions) or inconsistent (leading to no solution). Let's now investigate further to confirm whether this system has no solution or infinite solutions.The row-reduced echelon form (RREF) of the augmented matrix is: $$\begin{pmatrix} 1 & 0 & -5 & 1 \\ 0 & 1 & -11 & 1 \\ 0 & 0 & 0 & 0 \end{pmatrix}$$ This shows that the third row is all zeros, indicating that there are infinitely many solutions. We can express the solution in terms of a free parameter $$x_3 = s_1$$, and solve for $$x_1$$ and $$x_2$$ accordingly: $$\begin{aligned} x_1 &= 1 + 5s_1 \\ x_2 &= 1 + 11s_1 \\ x_3 &= s_1 \end{aligned}$$ Thus, the general solution is: $$(x_1, x_2, x_3) = (1 + 5s_1, 1 + 11s_1, s_1)$$ Would you like more details or have any questions? Here are 5 related questions: 1. How can we interpret the geometric meaning of this solution? 2. What does it mean when a system has infinite solutions in terms of the rank of the matrix? 3. Can we analyze the consistency of systems with dependent equations? 4. How would the solution change if a constraint was added to $$x_3$$? 5. How does Gaussian elimination help simplify complex systems? **Tip**: Systems with infinite solutions usually arise when the equations are linearly dependent. This means they describe the same geometric plane or line in space.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations
Gaussian Elimination
Matrix Operations

Formulas

Row-reduced echelon form (RREF)
Parametric solutions in linear systems

Theorems

Fundamental theorem of linear algebra
Gaussian elimination method

Suitable Grade Level

Grades 11-12, College

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