Math Problem Statement

c) { x - y - z = 0 x + 2y - 5z = 2 3x - 2y - 4z = 1 }

Solution

To solve this system of equations, we can use methods such as substitution, elimination, or matrix operations. Here, I'll use matrix operations to find the solution.

The system of equations is:

x - y - z = 0 \\ x + 2y - 5z = 2 \\ 3x - 2y - 4z = 1 \end{cases} $$ We can represent this system in matrix form $$ Ax = b $$, where $$ A $$ is the coefficient matrix, $$ x $$ is the vector of variables, and $$ b $$ is the constant vector: $$ A = \begin{pmatrix} 1 & -1 & -1 \\ 1 & 2 & -5 \\ 3 & -2 & -4 \end{pmatrix}, \quad x = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad b = \begin{pmatrix} 0 \\ 2 \\ 1 \end{pmatrix} $$ To solve for $$ x $$, we can use the inverse of matrix $$ A $$: $$ x = A^{-1}b $$ First, we need to find the inverse of $$ A $$. Let's proceed with the calculations: 1. **Find the determinant of $$ A $$:** $$ \text{det}(A) = 1 \cdot (2 \cdot (-4) - (-5) \cdot (-2)) - (-1) \cdot (1 \cdot (-4) - (-5) \cdot 3) + (-1) \cdot (1 \cdot (-2) - 2 \cdot 3) $$ $$ \text{det}(A) = 1 \cdot (-8 - 10) - (-1) \cdot (-4 - 15) + (-1) \cdot (-2 - 6) $$ $$ \text{det}(A) = 1 \cdot (-18) + 1 \cdot (-19) + 1 \cdot (-8) $$ $$ \text{det}(A) = -18 - 19 - 8 = -45 $$ Since the determinant is not zero, the inverse exists. 2. **Find the inverse of $$ A $$, $$ A^{-1} $$:** $$ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) $$ Where \(\text{adj}(A)\) is the adjugate of $$ A $$. Calculate the minors and cofactors to find \(\text{adj}(A)\): $$ \text{adj}(A) = \begin{pmatrix} -14 & -11 & -4 \\ -7 & -11 & -5 \\ -4 & -9 & 3 \end{pmatrix} $$ $$ A^{-1} = \frac{1}{-45} \cdot \begin{pmatrix} -14 & -11 & -4 \\ -7 & -11 & -5 \\ -4 & -9 & 3 \end{pmatrix} $$ $$ A^{-1} = \begin{pmatrix} \frac{14}{45} & \frac{11}{45} & \frac{4}{45} \\ \frac{7}{45} & \frac{11}{45} & \frac{5}{45} \\ \frac{4}{45} & \frac{9}{45} & -\frac{3}{45} \end{pmatrix} $$ 3. **Multiply $$ A^{-1} $$ by $$ b $$:** $$ x = A^{-1}b = \begin{pmatrix} \frac{14}{45} & \frac{11}{45} & \frac{4}{45} \\ \frac{7}{45} & \frac{11}{45} & \frac{5}{45} \\ \frac{4}{45} & \frac{9}{45} & -\frac{3}{45} \end{pmatrix} \begin{pmatrix} 0 \\ 2 \\ 1 \end{pmatrix} $$ $$ x = \begin{pmatrix} \frac{14 \cdot 0 + 11 \cdot 2 + 4 \cdot 1}{45} \\ \frac{7 \cdot 0 + 11 \cdot 2 + 5 \cdot 1}{45} \\ \frac{4 \cdot 0 + 9 \cdot 2 - 3 \cdot 1}{45} \end{pmatrix} = \begin{pmatrix} \frac{22 + 4}{45} \\ \frac{22 + 5}{45} \\ \frac{18 - 3}{45} \end{pmatrix} = \begin{pmatrix} \frac{26}{45} \\ \frac{27}{45} \\ \frac{15}{45} \end{pmatrix} = \begin{pmatrix} \frac{26}{45} \\ \frac{27}{45} \\ \frac{1}{3} \end{pmatrix} $$ So, the solution is: $$ x = \frac{26}{45}, \quad y = \frac{27}{45}, \quad z = \frac{1}{3} $$ Would you like further details or explanations on any step? Here are some other questions you might find helpful: 1. How can I solve a system of equations using substitution? 2. What are the properties of determinants? 3. How can I use row reduction to solve a system of linear equations? 4. What is the significance of the adjugate matrix in finding the inverse? 5. How can I verify the solution of a system of equations? **Tip:** When dealing with systems of linear equations, always check the determinant of the coefficient matrix to determine if the system has a unique solution.

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Math Problem Analysis

Mathematical Concepts

Systems of Linear Equations
Matrix Operations
Determinants
Inverse Matrices

Formulas

Matrix representation of linear systems
Inverse matrix formula

Theorems

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Suitable Grade Level

High School