Math Problem Statement
Find all the solutions of Ax = 0, where A is row equivalent to the given matrix. Identify the free variables and write the solution in parametric vector form.
Solution
We are tasked with finding the solutions to the homogeneous system , where is the matrix given by:
1 & 5 & -9 & 6 \\ 0 & 1 & -1 & 6 \end{pmatrix}$$ ### Step 1: Analyze the Row-Echelon Form of the Matrix The matrix $$A$$ is already in row echelon form, so we can move to solving the system by identifying the pivot and free variables. - **Pivot variables**: These are the variables corresponding to the leading entries (non-zero entries) in each row. - From the first row: $$x_1$$ is a pivot variable. - From the second row: $$x_2$$ is a pivot variable. - **Free variables**: The variables corresponding to the columns without pivots. - $$x_3$$ and $$x_4$$ are free variables. ### Step 2: Solve the System Using Back-Substitution From the second row equation: $$x_2 - x_3 + 6x_4 = 0 \quad \Rightarrow \quad x_2 = x_3 - 6x_4$$ From the first row equation: $$x_1 + 5x_2 - 9x_3 + 6x_4 = 0$$ Substituting $$x_2 = x_3 - 6x_4$$: $$x_1 + 5(x_3 - 6x_4) - 9x_3 + 6x_4 = 0$$ Simplify: $$x_1 + 5x_3 - 30x_4 - 9x_3 + 6x_4 = 0 \quad \Rightarrow \quad x_1 - 4x_3 - 24x_4 = 0$$ Thus: $$x_1 = 4x_3 + 24x_4$$ ### Step 3: Write the Parametric Vector Form of the Solution Let the free variables $$x_3$$ and $$x_4$$ be parameters: $$x_1 = 4x_3 + 24x_4, \quad x_2 = x_3 - 6x_4, \quad x_3 = x_3, \quad x_4 = x_4$$ The solution can be written as: $$\mathbf{x} = x_3 \begin{pmatrix} 4 \\ 1 \\ 1 \\ 0 \end{pmatrix} + x_4 \begin{pmatrix} 24 \\ -6 \\ 0 \\ 1 \end{pmatrix}$$ This represents the general solution in parametric vector form. ### Summary of Solution: - **Free variables**: $$x_3$$, $$x_4$$ - **Parametric solution**: $$\mathbf{x} = x_3 \begin{pmatrix} 4 \\ 1 \\ 1 \\ 0 \end{pmatrix} + x_4 \begin{pmatrix} 24 \\ -6 \\ 0 \\ 1 \end{pmatrix}$$ Would you like further details or have any specific questions? ### Related questions: 1. How do you interpret free variables in a system of equations? 2. What are the steps for row-reducing a matrix? 3. How can you identify if a system has infinitely many solutions? 4. What is the significance of the parametric vector form in solving systems of equations? 5. How does the rank of a matrix affect the number of free variables in a system? ### Tip: Always identify pivot and free variables after putting the matrix into row echelon form to quickly determine the solution structure of the system.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Row Echelon Form
Homogeneous Systems
Parametric Vector Form
Formulas
Solution to Ax = 0 in row echelon form
Parametric vector form
Theorems
Free and pivot variables in a system of linear equations
Suitable Grade Level
College Level - Linear Algebra