Math Problem Statement
solve the system of equations:2x + 3y - z = 7, x - 2y + 4z = -2, 3x + y + z = 10
Solution
To solve the system of equations:
we can use methods such as substitution, elimination, or matrix operations (like Gaussian elimination). Here, I'll proceed using matrix methods for clarity.
Step 1: Set up the augmented matrix
The system can be represented as an augmented matrix:
2 & 3 & -1 & | & 7 \\ 1 & -2 & 4 & | & -2 \\ 3 & 1 & 1 & | & 10 \\ \end{bmatrix}$$ ### Step 2: Apply Gaussian elimination 1. **Make the first element of the first row equal to 1** (by dividing the first row by 2): $$\begin{bmatrix} 1 & \frac{3}{2} & -\frac{1}{2} & | & \frac{7}{2} \\ 1 & -2 & 4 & | & -2 \\ 3 & 1 & 1 & | & 10 \\ \end{bmatrix}$$ 2. **Eliminate $$x$$ from the second and third rows** by subtracting appropriate multiples of the first row: - Row 2: $$R_2 - R_1$$ - Row 3: $$R_3 - 3 \times R_1$$ After performing these operations, the matrix becomes: $$\begin{bmatrix} 1 & \frac{3}{2} & -\frac{1}{2} & | & \frac{7}{2} \\ 0 & -\frac{7}{2} & \frac{9}{2} & | & -\frac{11}{2} \\ 0 & -\frac{7}{2} & \frac{5}{2} & | & \frac{1}{2} \\ \end{bmatrix}$$ 3. **Make the second pivot a 1** by dividing the second row by $$-\frac{7}{2}$$: $$\begin{bmatrix} 1 & \frac{3}{2} & -\frac{1}{2} & | & \frac{7}{2} \\ 0 & 1 & -\frac{9}{7} & | & \frac{11}{7} \\ 0 & -\frac{7}{2} & \frac{5}{2} & | & \frac{1}{2} \\ \end{bmatrix}$$ 4. **Eliminate $$y$$ from the first and third rows** using the updated second row: - Row 1: $$R_1 - \frac{3}{2} \times R_2$$ - Row 3: $$R_3 + \frac{7}{2} \times R_2$$ After these operations, the matrix becomes: $$\begin{bmatrix} 1 & 0 & \frac{5}{7} & | & \frac{1}{7} \\ 0 & 1 & -\frac{9}{7} & | & \frac{11}{7} \\ 0 & 0 & 4 & | & 4 \\ \end{bmatrix}$$ 5. **Solve for $$z$$** in the third row: $$z = 1$$. 6. Substitute $$z = 1$$ into the second row to find $$y$$: $$y - \frac{9}{7} \cdot 1 = \frac{11}{7} \Rightarrow y = 2$$ 7. Substitute $$y = 2$$ and $$z = 1$$ into the first row to find $$x$$: $$x + \frac{5}{7} \cdot 1 = \frac{1}{7} \Rightarrow x = -1$$ ### Solution The solution to the system is: $$x = -1, \quad y = 2, \quad z = 1$$ Would you like further details on any of the steps? ### Related Questions 1. How can Gaussian elimination be applied to larger systems of equations? 2. What are the advantages of using matrices to solve systems of equations? 3. Can systems of equations always be solved using Gaussian elimination? 4. What are other methods to solve systems of linear equations? 5. How can augmented matrices represent real-world problems? **Tip:** Practicing row operations on matrices can greatly improve your efficiency in solving systems of linear equations!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Systems of Linear Equations
Matrix Operations
Formulas
Gaussian elimination steps
Matrix row operations for elimination
Theorems
Gaussian elimination theorem
Suitable Grade Level
Grades 10-12
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