Math Problem Statement
3x 2y þ z ¼ 2 x 3y þ 2z ¼ 1 2x þ y 3z ¼ 5
Solution
The system of linear equations you provided is:
1. \quad 3x - 2y + z &= 2 \\ 2. \quad x - 3y + 2z &= 1 \\ 3. \quad 2x + y - 3z &= -5 \end{aligned}$$ To solve this system of equations, we can use various methods, such as substitution, elimination, or matrix methods (Gaussian elimination or Cramer's rule). I'll proceed with Gaussian elimination for clarity. ### Step 1: Write the augmented matrix We rewrite the system in matrix form $$[A|b]$$, where $$A$$ is the coefficient matrix and $$b$$ is the vector of constants on the right-hand side: $$\begin{bmatrix} 3 & -2 & 1 & | & 2 \\ 1 & -3 & 2 & | & 1 \\ 2 & 1 & -3 & | & -5 \end{bmatrix}$$ ### Step 2: Perform Gaussian elimination 1. Make the pivot in the first row, first column a 1 by swapping row 1 with row 2: $$\begin{bmatrix} 1 & -3 & 2 & | & 1 \\ 3 & -2 & 1 & | & 2 \\ 2 & 1 & -3 & | & -5 \end{bmatrix}$$ 2. Eliminate the first element in rows 2 and 3 by subtracting multiples of row 1: - $$\text{Row 2} = \text{Row 2} - 3 \times \text{Row 1}$$ - $$\text{Row 3} = \text{Row 3} - 2 \times \text{Row 1}$$ $$\begin{bmatrix} 1 & -3 & 2 & | & 1 \\ 0 & 7 & -5 & | & -1 \\ 0 & 7 & -7 & | & -7 \end{bmatrix}$$ 3. Eliminate the second element in row 3 by subtracting row 2 from row 3: $$\begin{bmatrix} 1 & -3 & 2 & | & 1 \\ 0 & 7 & -5 & | & -1 \\ 0 & 0 & -2 & | & -6 \end{bmatrix}$$ ### Step 3: Back-substitution 1. From the third equation: $$-2z = -6 \implies z = 3$$ 2. Substitute $$z = 3$$ into the second equation: $$7y - 5z = -1 \implies 7y - 5(3) = -1 \implies 7y - 15 = -1 \implies y = 2$$ 3. Substitute $$y = 2$$ and $$z = 3$$ into the first equation: $$x - 3(2) + 2(3) = 1 \implies x - 6 + 6 = 1 \implies x = 1$$ ### Final Solution: The solution to the system is: $$x = 1, \quad y = 2, \quad z = 3$$ Would you like any more details or explanations on the steps? Here are some related questions for further practice: 1. Solve the system using substitution instead of Gaussian elimination. 2. Use matrix inversion to solve the system. 3. Apply Cramer's Rule to solve the system. 4. Formulate a different system of equations with three variables and solve it. 5. Use a determinant to determine if the system has a unique solution. **Tip:** Always check your solution by substituting the values back into the original equations to verify correctness.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Equations
Gaussian Elimination
Back-Substitution
Formulas
Gaussian elimination steps
Back-substitution
Matrix representation of linear systems
Theorems
Gaussian Elimination
Suitable Grade Level
Grades 10-12
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