Math Problem Statement

. Test for consistency and solve the system of equations βˆ’2π‘₯ + 𝑦 + 𝑧 = 3; π‘₯ βˆ’ 2𝑦 + 𝑧 = 2; π‘₯ + 𝑦 βˆ’ 2𝑧 = βˆ’5.8

Solution

To test for consistency and solve the system of equations:

-2x + y + z &= 3 \\ x - 2y + z &= 2 \\ x + y - 2z &= -5.8 \end{aligned}$$ ### **Step 1: Convert the system into augmented matrix form** $$\begin{bmatrix} -2 & 1 & 1 & |3 \\ 1 & -2 & 1 & |2 \\ 1 & 1 & -2 & |-5.8 \end{bmatrix}$$ ### **Step 2: Convert to Row Echelon Form (Gaussian Elimination)** #### **Swap Row 1 and Row 2 (to get a leading 1 in the first column)** $$\begin{bmatrix} 1 & -2 & 1 & |2 \\ -2 & 1 & 1 & |3 \\ 1 & 1 & -2 & |-5.8 \end{bmatrix}$$ #### **Eliminate first column below the leading 1** - Add **2 Γ— (Row 1)** to **Row 2** - Subtract **Row 1** from **Row 3** $$\begin{bmatrix} 1 & -2 & 1 & |2 \\ 0 & -3 & 3 & |7 \\ 0 & 3 & -3 & |-7.8 \end{bmatrix}$$ #### **Eliminate second column below leading 1** - Add **Row 2** to **Row 3** $$\begin{bmatrix} 1 & -2 & 1 & |2 \\ 0 & -3 & 3 & |7 \\ 0 & 0 & 0 & |-0.8 \end{bmatrix}$$ ### **Step 3: Check for Consistency** The third row translates to: $$0x + 0y + 0z = -0.8$$ which is a **contradiction**. Since this is a false statement, the system is **inconsistent** and has **no solution**. --- ### **Conclusion** The given system of equations is **inconsistent** and has **no solution**. --- Do you want a different method to verify or any clarification? Here are 5 related questions: 1. What is the difference between consistent and inconsistent systems? 2. How do you use Gaussian elimination to solve systems of equations? 3. How can row operations help in determining the rank of a matrix? 4. What is the role of augmented matrices in solving linear systems? 5. How can we interpret inconsistent systems in real-world applications? **Tip:** If a system is inconsistent, that means the equations represent planes that do not intersect at a common point.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Equations
Gaussian Elimination
Matrix Row Operations

Formulas

Augmented Matrix Representation
Row Echelon Form
Gaussian Elimination Steps

Theorems

Consistency of a System of Equations
Row Reduction Theorem

Suitable Grade Level

Grades 10-12