Math Problem Statement
Find the general solution of the system whose augmented matrix is given below:
1 -7 0 -1 0 -9
0 1 0 0 3 7
0 0 1 0 8 9
0 0 0 0 0 0
Solution
Let's solve the system of linear equations from the augmented matrix provided:
The augmented matrix is:
1 & -7 & 0 & -1 & 0 & -9 \\ 0 & 1 & 0 & 0 & 3 & 7 \\ 0 & 0 & 1 & 0 & 8 & 9 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix}$$ This represents the following system of equations: 1. $$x_1 - 7x_2 - x_4 = -9$$ 2. $$x_2 + 3x_5 = 7$$ 3. $$x_3 + 8x_5 = 9$$ We need to solve this system for $$x_1, x_2, x_3, x_4, x_5$$. ### Step 1: Identify the pivot variables From the matrix, we can identify that: - $$x_1$$, $$x_2$$, and $$x_3$$ are **basic** variables (they have leading 1's in their columns). - $$x_4$$ and $$x_5$$ are **free** variables (they do not have leading 1's). ### Step 2: Express basic variables in terms of free variables - From equation (1): $$x_1 = 7x_2 + x_4 - 9$$ - From equation (2): $$x_2 = 7 - 3x_5$$ - From equation (3): $$x_3 = 9 - 8x_5$$ ### Step 3: General solution - $$x_1 = 7(7 - 3x_5) + x_4 - 9 = 49 - 21x_5 + x_4 - 9 = 40 - 21x_5 + x_4$$ - $$x_2 = 7 - 3x_5$$ - $$x_3 = 9 - 8x_5$$ - $$x_4$$ and $$x_5$$ are free. ### Conclusion The correct solution corresponds to **option A**, where: - $$x_1 = 40 - 21x_5 + x_4$$ - $$x_2 = 7 - 3x_5$$ - $$x_3 = 9 - 8x_5$$ - $$x_4$$ is free - $$x_5$$ is free Would you like further details on any of the steps? --- Here are 5 related questions for you to consider: 1. How do you determine free and basic variables from a matrix? 2. What does it mean for a system to be inconsistent? 3. Can you always express solutions in terms of free variables? 4. What happens if all variables are free? 5. How does row-reduction help in solving a system of linear equations? **Tip**: When solving augmented matrix problems, always check if the last row corresponds to a valid equation. If it results in something like $$0 = 1$$, the system is inconsistent.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Augmented Matrices
Systems of Linear Equations
Formulas
Row reduction to echelon form
Expressing basic variables in terms of free variables
Theorems
Gaussian Elimination
Free and Basic Variables
Suitable Grade Level
Undergraduate or Advanced High School