Math Problem Statement

solve

Find the row-reduced echelon form of the following system and hence its solution set: 2x_1 - x_2 - x_3 = 6, x_1 + x_2 + 2x_3 = 1, x_1 - 2x_2 = 2

Solution

Let's solve the given system of equations by converting it into row-reduced echelon form (RREF).

Given system:

2x_1 - x_2 - x_3 &= 6 \\ x_1 + x_2 + 2x_3 &= 1 \\ x_1 - 2x_2 &= 2 \end{aligned}$$ ### Step 1: Write the Augmented Matrix $$\begin{bmatrix} 2 & -1 & -1 & | 6 \\ 1 & 1 & 2 & | 1 \\ 1 & -2 & 0 & | 2 \end{bmatrix}$$ ### Step 2: Convert to Row Echelon Form #### Swap Row 1 and Row 2 (to get a leading 1 in the first column) $$\begin{bmatrix} 1 & 1 & 2 & | 1 \\ 2 & -1 & -1 & | 6 \\ 1 & -2 & 0 & | 2 \end{bmatrix}$$ #### Subtract 2 × (Row 1) from Row 2 and subtract Row 1 from Row 3: $$\begin{bmatrix} 1 & 1 & 2 & | 1 \\ 0 & -3 & -5 & | 4 \\ 0 & -3 & -2 & | 1 \end{bmatrix}$$ #### Add Row 2 to Row 3: $$\begin{bmatrix} 1 & 1 & 2 & | 1 \\ 0 & -3 & -5 & | 4 \\ 0 & 0 & -7 & | 5 \end{bmatrix}$$ ### Step 3: Convert to Row-Reduced Echelon Form #### Divide Row 3 by -7: $$\begin{bmatrix} 1 & 1 & 2 & | 1 \\ 0 & -3 & -5 & | 4 \\ 0 & 0 & 1 & | -\frac{5}{7} \end{bmatrix}$$ #### Eliminate the third column in Rows 1 and 2: - Row 1 → Row 1 - 2 × (Row 3) - Row 2 → Row 2 + 5 × (Row 3) $$\begin{bmatrix} 1 & 1 & 0 & | \frac{17}{7} \\ 0 & -3 & 0 & | \frac{23}{7} \\ 0 & 0 & 1 & | -\frac{5}{7} \end{bmatrix}$$ #### Divide Row 2 by -3: $$\begin{bmatrix} 1 & 1 & 0 & | \frac{17}{7} \\ 0 & 1 & 0 & | -\frac{23}{21} \\ 0 & 0 & 1 & | -\frac{5}{7} \end{bmatrix}$$ #### Eliminate second column from Row 1: $$\begin{bmatrix} 1 & 0 & 0 & | \frac{88}{21} \\ 0 & 1 & 0 & | -\frac{23}{21} \\ 0 & 0 & 1 & | -\frac{5}{7} \end{bmatrix}$$ ### Step 4: Solution From the final matrix, we obtain: $$x_1 = \frac{88}{21}, \quad x_2 = -\frac{23}{21}, \quad x_3 = -\frac{5}{7}$$ Thus, the solution set is: $$\left( \frac{88}{21}, -\frac{23}{21}, -\frac{5}{7} \right)$$ --- Would you like a step-by-step breakdown of any specific part? ### Related Questions: 1. How do you determine if a system of equations has a unique solution? 2. What is the difference between row echelon form and reduced row echelon form? 3. How do elementary row operations affect the solution of a system? 4. Can a system of equations have infinitely many solutions in RREF? 5. How does the rank of a matrix relate to the number of solutions of a system? ### Tip: Always check for inconsistencies (like a row of the form $$[0 \quad 0 \quad 0 \quad | b]$$ with $$b \neq 0$$) to determine if the system is inconsistent.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Linear Algebra
Row-Reduced Echelon Form
Systems of Linear Equations

Formulas

Row-Reduced Echelon Form
Elementary Row Operations

Theorems

Gauss-Jordan Elimination

Suitable Grade Level

Grades 10-12, College

Related Recommendation

Solving Systems of Linear Equations Using Row-Reduced Echelon Form

Solve a 3x3 System of Equations Using Row Operations and Matrices

Solve System of Linear Equations Using Matrix Reduction to RREF

General Solutions of a Linear System using Augmented Matrix and RREF

Solving a System of Linear Equations Using Reduced Row Echelon Form