Solving Linear Equations Using Reduced Row Echelon Form

Math Problem Statement

Use the reduced row echelon form above to solve the system

Solution

The image provides a linear system of equations along with its reduced row echelon form (RREF). The system to solve is:

-x + y + 2z &= 2, \\ x - 2y - z &= -6, \\ -x + y + 2z &= 2. \end{aligned}$$ The RREF of the augmented matrix for this system is: $$\begin{bmatrix} 1 & 0 & -3 & 2 \\ 0 & 1 & -1 & 4 \\ 0 & 0 & 0 & 0 \end{bmatrix}.$$ ### Step 1: Interpret the RREF The RREF gives us the following system of equations: $$\begin{aligned} x - 3z &= 2, \\ y - z &= 4. \end{aligned}$$ ### Step 2: Solve for $$x$$ and $$y$$ - From $$x - 3z = 2$$, we solve for $$x$$: $$x = 3z + 2.$$ - From $$y - z = 4$$, we solve for $$y$$: $$y = z + 4.$$ ### Step 3: Parametrize the Solution Since $$z$$ is a free variable, the general solution can be written in terms of $$z$$: $$\begin{aligned} x &= 3z + 2, \\ y &= z + 4, \\ z &= z. \end{aligned}$$ Thus, the parametric solution for the system is: $$\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 3z + 2 \\ z + 4 \\ z \end{bmatrix}.$$ ### Summary of Solution: The solution is parametrized by $$z$$, and the general form is: $$x = 3z + 2, \quad y = z + 4, \quad z = z.$$ Let me know if you need any further clarification! --- Here are some related questions you might find useful: 1. How do you compute the RREF of a matrix? 2. What are free variables, and why are they important in systems of equations? 3. How do you interpret the results from an inconsistent system? 4. Can every system of linear equations be solved using RREF? 5. How do you use matrix operations to solve a system of equations? **Tip:** When solving systems of equations, always check if the system is consistent by looking at the last row of the RREF matrix!

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
Matrix Operations
Parameterization

Formulas

x = 3z + 2
y = z + 4
z = z

Theorems

Row Reduction
System Solution Using RREF

Suitable Grade Level

College

Related Recommendation

General Solutions of a Linear System using Augmented Matrix and RREF

Solving Systems of Equations Using Reduced Row Echelon Form (RREF)

Solve a System of Equations from a Matrix in Reduced Echelon Form

Solving Linear Systems Using Gaussian Elimination and RREF

Solving System of Linear Equations Using RREF