Math Problem Statement

Given the augmented matrix below, solve the associated system of equations. For your variables, use x1, x2, x3, x4, x5, and x6. [[1, -6, -5, -6, 3, 0 ,-1],[0,0,0,0,1,-5,0],[0,0,0,0,0,1,-9]] the solution is (,,,,,)

Solution

We are given the following augmented matrix:

1 & -6 & -5 & -6 & 3 & 0 & -1 \\ 0 & 0 & 0 & 0 & 1 & -5 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & -9 \\ \end{array}\right]$$ The goal is to solve the associated system of linear equations. We will express the solution in terms of the variables $$x_1, x_2, x_3, x_4, x_5, x_6$$. ### Step 1: Convert the augmented matrix to a system of equations From the augmented matrix, we can write down the system of equations: 1. $$x_1 - 6x_2 - 5x_3 - 6x_4 + 3x_5 = -1$$ 2. $$x_5 - 5x_6 = 0$$ 3. $$x_6 = 9$$ ### Step 2: Solve the system From equation (3), we already know: $$x_6 = 9$$ Substitute $$x_6 = 9$$ into equation (2): $$x_5 - 5(9) = 0 \implies x_5 = 45$$ Now substitute $$x_5 = 45$$ and $$x_6 = 9$$ into equation (1): $$x_1 - 6x_2 - 5x_3 - 6x_4 + 3(45) = -1$$ Simplify: $$x_1 - 6x_2 - 5x_3 - 6x_4 + 135 = -1$$ $$x_1 - 6x_2 - 5x_3 - 6x_4 = -136$$ ### Step 3: General solution We can express the solution in terms of the free variables $$x_2, x_3, x_4$$, since we have one equation with 4 unknowns. Let $$x_2 = t_1$$, $$x_3 = t_2$$, and $$x_4 = t_3$$, where $$t_1, t_2, t_3$$ are parameters. Thus, we get: $$x_1 = -136 + 6t_1 + 5t_2 + 6t_3$$ ### Final solution: $$(x_1, x_2, x_3, x_4, x_5, x_6) = (-136 + 6t_1 + 5t_2 + 6t_3, t_1, t_2, t_3, 45, 9)$$ This is the parametric solution to the system of equations, where $$t_1, t_2, t_3$$ are free parameters. Would you like further details or clarifications? Here are five related questions for further exploration: 1. What are free variables in a system of linear equations? 2. How does Gaussian elimination help solve systems of equations? 3. What does it mean for a system to have infinitely many solutions? 4. How do augmented matrices represent systems of equations? 5. How can we verify a solution to a system of linear equations? **Tip**: When solving systems of equations with more unknowns than equations, it's common to express solutions in terms of free variables.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations
Matrix Representation

Formulas

General solution of a linear system
Matrix equation representation
Gaussian Elimination

Theorems

Gaussian Elimination
Parametric form of solutions

Suitable Grade Level

College level or advanced high school (Grades 11-12)

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