Math Problem Statement

Use Gauss-Jordan row reduction to solve the given system of equations. (If there is no solution, enter NO SOLUTION. If the system is dependent, express your answer using the parameters x and/or y.)

xy =  0 3x − y  =  9 x  −  y  =  −1(xy) =

Solution

We can solve the given system of equations using the Gauss-Jordan row reduction method. The system of equations is:

x + y &= 0 \quad (1) \\ 3x - y &= 9 \quad (2) \\ x - y &= -1 \quad (3) \end{aligned}$$ We first convert the system of equations into an augmented matrix: $$\left[ \begin{array}{ccc|c} 1 & 1 & 0 & 0 \\ 3 & -1 & 0 & 9 \\ 1 & -1 & 0 & -1 \end{array} \right]$$ Now, we will perform row reduction to get this matrix into reduced row-echelon form (RREF). ### Step 1: Row operations - Leave $$R_1$$ unchanged. - Replace $$R_2$$ with $$R_2 - 3R_1$$. - Replace $$R_3$$ with $$R_3 - R_1$$. This gives us the new matrix: $$\left[ \begin{array}{ccc|c} 1 & 1 & 0 & 0 \\ 0 & -4 & 0 & 9 \\ 0 & -2 & 0 & -1 \end{array} \right]$$ ### Step 2: Eliminate below the pivot - Divide $$R_2$$ by $$-4$$ to make the pivot $$1$$. - Replace $$R_3$$ with $$R_3 - \left(\frac{1}{2}\right)R_2$$. The matrix becomes: $$\left[ \begin{array}{ccc|c} 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & -\frac{9}{4} \\ 0 & 0 & 0 & \frac{7}{2} \end{array} \right]$$ ### Step 3: Check consistency The last row $$\left[ 0 \ 0 \ 0 \ \frac{7}{2} \right]$$ implies the equation $$0 = \frac{7}{2}$$, which is a contradiction. ### Conclusion: Since this leads to an inconsistency, the system has **no solution**. --- Would you like more details or have any questions? Here are five related questions to explore further: 1. How can we interpret inconsistent systems in terms of geometry? 2. What is the difference between row-echelon form and reduced row-echelon form? 3. Can Gaussian elimination always determine if a system is consistent or inconsistent? 4. How does Gauss-Jordan elimination compare with other methods for solving systems of equations? 5. What happens when a system has infinitely many solutions? **Tip**: Always check for inconsistencies early in the row-reduction process to avoid unnecessary steps.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Equations
Gauss-Jordan Elimination

Formulas

Gauss-Jordan Row Reduction
Matrix Representation of Linear Systems

Theorems

Gauss-Jordan Elimination Theorem
Consistency of Linear Systems

Suitable Grade Level

Grades 10-12