Math Problem Statement
Given a system of linear equations, use the reduced row echelon form to solve the system. The matrix and its RREF are given.
Solution
The given problem involves solving a system of linear equations using the reduced row echelon form (RREF) of a matrix.
The problem provides the system of equations:
-x - y &= -3 \\ -x - 4z &= -4 \\ x - y &= 3 \end{aligned}$$ We also see the matrix that represents this system and its reduced row echelon form: $$\begin{pmatrix} -1 & -1 & 0 & -3 \\ -1 & 0 & -4 & -4 \\ -1 & -1 & 0 & -3 \end{pmatrix} \sim \begin{pmatrix} 1 & 0 & 4 & 4 \\ 0 & 1 & -4 & -1 \\ 0 & 0 & 0 & 0 \end{pmatrix}$$ ### Step 1: Interpret the RREF Matrix From the RREF matrix: $$\begin{pmatrix} 1 & 0 & 4 & 4 \\ 0 & 1 & -4 & -1 \\ 0 & 0 & 0 & 0 \end{pmatrix}$$ we can rewrite this in terms of equations: 1. $$x + 4z = 4$$ 2. $$y - 4z = -1$$ The third row, being all zeros, indicates that the system has a free variable (in this case, $$z$$). ### Step 2: Solve the System From the first equation: $$x = 4 - 4z$$ From the second equation: $$y = -1 + 4z$$ ### Step 3: General Solution Since $$z$$ is a free variable, we can parametrize the solution in terms of $$z$$: $$x = 4 - 4z, \quad y = -1 + 4z, \quad z = z$$ ### Final Answer The parametric solution is: $$x = 4 - 4z, \quad y = -1 + 4z, \quad z = z$$ To enter this into the web form: - $$x = 4$$ - $$y = -4$$ - $$z = 0$$ (if you want the particular solution where $$z = 0$$). Let me know if you need further details, or if you have any questions! Here are 5 relative questions to expand the problem: 1. How does Gaussian elimination work for solving systems of equations? 2. What is the difference between row echelon form and reduced row echelon form? 3. How do free variables in a system relate to the existence of infinite solutions? 4. How can you verify that the RREF of a matrix is correct? 5. What are some practical applications of solving linear systems? **Tip:** Always verify your solution by substituting back into the original system to check for consistency.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Matrix Operations
Reduced Row Echelon Form (RREF)
Formulas
RREF of a matrix
x = 4 - 4z
y = -1 + 4z
Theorems
Gaussian Elimination
Free Variables Theorem
Suitable Grade Level
Grades 11-12, College Level
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