Math Problem Statement

Solve

Solution

The problem presented is a system of linear equations that you need to solve. The system is given by:

x_1 + x_2 + 2x_3 = 2 \\ 3x_1 + 2x_2 - 3x_3 = 1 \end{cases}$$ The equation can be represented in matrix form as: $$\begin{bmatrix} 1 & 1 & 2 \\ 3 & 2 & -3 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 2 \\ 1 \end{bmatrix}$$ Now, let's solve this system. Since the system has two equations and three unknowns, there may be infinitely many solutions, depending on the rank of the coefficient matrix and the augmented matrix. ### Step 1: Express the system in augmented matrix form $$\begin{bmatrix} 1 & 1 & 2 & \vert & 2 \\ 3 & 2 & -3 & \vert & 1 \end{bmatrix}$$ ### Step 2: Use Gaussian elimination to solve the system. We'll perform row operations to reduce the matrix to row echelon form. 1. **Row 2** = **Row 2** - 3 × **Row 1**: $$\begin{bmatrix} 1 & 1 & 2 & \vert & 2 \\ 0 & -1 & -9 & \vert & -5 \end{bmatrix}$$ 2. **Row 2** = -1 × **Row 2**: $$\begin{bmatrix} 1 & 1 & 2 & \vert & 2 \\ 0 & 1 & 9 & \vert & 5 \end{bmatrix}$$ ### Step 3: Back-substitution to find solutions. From the second row: $$x_2 + 9x_3 = 5 \quad \text{(Equation 2)}$$ So, $$x_2 = 5 - 9x_3$$ Substituting this into the first equation: $$x_1 + (5 - 9x_3) + 2x_3 = 2$$ Simplify: $$x_1 - 7x_3 = -3$$ So, $$x_1 = 7x_3 - 3$$ ### General Solution: $$x_1 = 7x_3 - 3, \quad x_2 = 5 - 9x_3, \quad x_3 = x_3$$ Let $$x_3 = t$$ (where $$t$$ is a parameter), then the general solution can be written as: $$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 7t - 3 \\ 5 - 9t \\ t \end{bmatrix}$$ This represents the infinite number of solutions based on the value of $$t$$. Would you like more details or have any questions? Here are some related questions you might find helpful: 1. What is the significance of the rank of a matrix in solving systems of equations? 2. How do you interpret the parameter $$t$$ in the solution of this system? 3. What are the conditions for a system of linear equations to have a unique solution, no solution, or infinitely many solutions? 4. How does the Gaussian elimination method work for solving larger systems of equations? 5. Can you explain the concept of row echelon form and how it helps in solving linear systems? **Tip:** When dealing

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Math Problem Analysis

Mathematical Concepts

Linear equations
Matrix operations
Gaussian elimination

Formulas

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Theorems

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Suitable Grade Level

Advanced High School and Above

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