Math Problem Statement

An augmented matrix for a linear system is given below. {(1,0,0,9),(0,0,1,-4)} Identify all of the free variables. Note that more variables may be listed than are actually in the system! Check all that apply from x1, x2, x3, x4, x5

Solution

The given augmented matrix is:

1 & 0 & 0 & | & 9 \\ 0 & 0 & 1 & | & -4 \end{pmatrix}$$ This matrix corresponds to the following system of equations: 1. $$x_1 = 9$$ 2. $$x_3 = -4$$ From this, we can see: - $$x_1$$ and $$x_3$$ are **bound** variables, as they are explicitly solved in terms of constants. - The matrix provides no information about $$x_2$$, $$x_4$$, and $$x_5$$, so they are **free variables**. ### Conclusion: The free variables are $$x_2$$, $$x_4$$, and $$x_5$$. --- Do you have any questions or would you like more details? Here are some related questions to consider: 1. What is the rank of the matrix? 2. How can you identify free variables in general for any augmented matrix? 3. What is the meaning of a "free variable" in the context of linear systems? 4. How can you express the general solution for this system? 5. What is the significance of the number of free variables relative to the number of equations? **Tip**: When solving linear systems, always check for consistency of the system (i.e., whether the system has one solution, infinitely many solutions, or no solution).

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Augmented Matrices
Systems of Linear Equations

Formulas

Row-Reduced Echelon Form (RREF)
Matrix Representation of Systems

Theorems

Fundamental Theorem of Linear Systems
Rank-Nullity Theorem

Suitable Grade Level

Undergraduate or High School (Grades 11-12)

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