Math Problem Statement

Selesaikan dengan determinan

Selesaikan dengan determinan

Solution

The image contains a system of linear equations that needs to be solved using the determinant method (Cramer's Rule). The system of equations is:

5x - 6y + 3z &= -9 \quad \text{(Equation 1)}\\ 2x - 3y + 2z &= -5 \quad \text{(Equation 2)}\\ 3x - 7y + 5z &= -16 \quad \text{(Equation 3)} \end{aligned}$$ ### Step 1: Write the system in matrix form This system can be written as: $$A \cdot \mathbf{x} = \mathbf{b}$$ where $$A$$ is the coefficient matrix, $$\mathbf{x}$$ is the vector of variables, and $$\mathbf{b}$$ is the constant vector: $$A = \begin{pmatrix} 5 & -6 & 3 \\ 2 & -3 & 2 \\ 3 & -7 & 5 \end{pmatrix}, \quad \mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} -9 \\ -5 \\ -16 \end{pmatrix}$$ ### Step 2: Compute the determinant of matrix $$A$$ The determinant of matrix $$A$$, denoted as $$\det(A)$$, is calculated as follows: $$\det(A) = \begin{vmatrix} 5 & -6 & 3 \\ 2 & -3 & 2 \\ 3 & -7 & 5 \end{vmatrix}$$ Let's calculate this determinant. ### Step 3: Calculate determinants for $$x$$, $$y$$, and $$z$$ - **Determinant $$\Delta_x$$**: Replace the first column of $$A$$ with $$\mathbf{b}$$: $$\Delta_x = \begin{vmatrix} -9 & -6 & 3 \\ -5 & -3 & 2 \\ -16 & -7 & 5 \end{vmatrix}$$ - **Determinant $$\Delta_y$$**: Replace the second column of $$A$$ with $$\mathbf{b}$$: $$\Delta_y = \begin{vmatrix} 5 & -9 & 3 \\ 2 & -5 & 2 \\ 3 & -16 & 5 \end{vmatrix}$$ - **Determinant $$\Delta_z$$**: Replace the third column of $$A$$ with $$\mathbf{b}$$: $$\Delta_z = \begin{vmatrix} 5 & -6 & -9 \\ 2 & -3 & -5 \\ 3 & -7 & -16 \end{vmatrix}$$ ### Step 4: Solve for $$x$$, $$y$$, and $$z$$ Using Cramer's Rule: $$x = \frac{\Delta_x}{\det(A)}, \quad y = \frac{\Delta_y}{\det(A)}, \quad z = \frac{\Delta_z}{\det(A)}$$ I'll calculate these determinants and the solutions for $$x$$, $$y$$, and $$z$$ now.The determinant of the coefficient matrix $$A$$ is $$\det(A) = 4.0$$. Using Cramer's Rule, the solutions for the variables are: $$\begin{aligned} x &= 3.0 \\ y &= 5.0 \\ z &= 2.0 \end{aligned}$$ Thus, the solution to the system of equations is: $$(x, y, z) = (3, 5, 2)$$ Would you like more details on the solution or have any other questions? Here are 5 related questions that might interest you: 1. How is Cramer's Rule derived from the determinant properties? 2. What are the conditions under which Cramer's Rule is applicable? 3. How do you solve a system of equations if the determinant of the coefficient matrix is zero? 4. Can you explain how the determinant of a matrix is calculated? 5. What are some other methods to solve systems of linear equations? **Tip:** When using determinants to solve a system of equations, always check the determinant of the coefficient matrix first. If it is zero, Cramer's Rule cannot be applied, and the system may be either dependent or inconsistent.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
System of Linear Equations
Determinants
Cramer's Rule

Formulas

Determinant of a 3x3 matrix
Cramer's Rule: x = Δx/Δ, y = Δy/Δ, z = Δz/Δ

Theorems

Cramer's Rule

Suitable Grade Level

Grades 10-12

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