Math Problem Statement
Solution
Let's examine the system of equations from the image, which is:
4x + y + z &= 15 \\ 3x - 6y - 4z &= 22 \\ -5x + y + 2z &= -19 \end{aligned}$$ ### Step 1: Convert to Matrix Form We can express the system as an augmented matrix: $$\begin{pmatrix} 4 & 1 & 1 & | & 15 \\ 3 & -6 & -4 & | & 22 \\ -5 & 1 & 2 & | & -19 \end{pmatrix}$$ ### Step 2: Use Row Operations to Solve I will now use Gaussian elimination to reduce this matrix to row-echelon form, allowing us to see if the system has a unique solution, no solution, or infinitely many solutions. I'll calculate this and provide the answer. Let me proceed with the calculations.The system of equations has a **unique solution**, which is: $$x = 4, \quad y = -3, \quad z = 2$$ Thus, the correct answer to the question is **B. Exactly 1** solution. Would you like to see the detailed steps of the row operations used in Gaussian elimination, or any other clarification? Here are 5 related questions to expand your understanding: 1. How does Gaussian elimination work for solving systems of equations? 2. Can systems with no solutions be identified early in Gaussian elimination? 3. What does it mean when a system has infinitely many solutions? 4. How can we use matrix inversion to solve systems of linear equations? 5. What is the difference between row-echelon form and reduced row-echelon form? **Tip:** When solving systems of equations, always check the determinant of the coefficient matrix to verify whether a unique solution exists (non-zero determinant).Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Equations
Matrices
Row Operations
Formulas
Gaussian Elimination
Matrix Representation of Linear Systems
Row-Echelon Form
Theorems
Gaussian Elimination Theorem
Uniqueness of Solutions in Linear Systems
Properties of Row-Reduction
Suitable Grade Level
High School - Grades 11-12
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