- Form the coefficient matrix and calculate its determinant \(D = 4\).
- Create matrices for \(x\) and \(y\) by substituting the constants and calculate \(D_x = -40\) and \(D_y = 36\).
- Apply Cramer's Rule to find \(x = \frac{D_x}{D} = -10\) and \(y = \frac{D_y}{D} = 9\).
- The solution to the system is \(\boxed{x = -10, y = 9}\).
Explanation
- Problem Analysis and Cramer's Rule Setup
The system of equations is given by:
\[ \begin{cases} x + 2y = 8 \ -3x - 2y = 12 \end{cases} \]
To solve this system using Cramer's rule, we first represent the system in the matrix form \(AX = B\), where \(A\) is the coefficient matrix, \(X\) is the variable vector, and \(B\) is the constant vector:
\[ \begin{pmatrix} 1 & 2 \ -3 & -2 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 8 \ 12 \end{pmatrix} \]
Cramer's rule states that if the determinant of the coefficient matrix \(A\) (denoted by \(D\)) is non-zero, the variables \(x\) and \(y\) are given by \(x = \frac{D_x}{D}\) and \(y = \frac{D_y}{D}\), where \(D_x\) and \(D_y\) are the determinants of the matrices obtained by replacing the first and second columns of \(A\) with the constant vector \(B\), respectively.
- Calculating Determinants
First, we calculate the determinant of the coefficient matrix \(A\):
\[ D = \det\begin{pmatrix} 1 & 2 \ -3 & -2 \end{pmatrix} = (1)(-2) - (2)(-3) = -2 + 6 = 4. \]
Next, we calculate \(D_x\) by replacing the first column of \(A\) with the constant vector \(\begin{pmatrix} 8 \ 12 \end{pmatrix}\):
\[ D_x = \det\begin{pmatrix} 8 & 2 \ 12 & -2 \end{pmatrix} = (8)(-2) - (2)(12) = -16 - 24 = -40. \]
Then, we calculate \(D_y\) by replacing the second column of \(A\) with the constant vector \(\begin{pmatrix} 8 \ 12 \end{pmatrix}\):
\[ D_y = \det\begin{pmatrix} 1 & 8 \ -3 & 12 \end{pmatrix} = (1)(12) - (8)(-3) = 12 + 24 = 36. \]
Since \(D = 4
eq 0\), the system has a unique solution.
- Calculating the Variables and Final Conclusion
Using the values found, we solve for \(x\) and \(y\):
\[ x = \frac{D_x}{D} = \frac{-40}{4} = -10 \]
\[ y = \frac{D_y}{D} = \frac{36}{4} = 9 \]
Thus, the solution to the system is \(x = -10\) and \(y = 9\). Substituting these back into the original equations: \(-10 + 2(9) = -10 + 18 = 8\) and \(-3(-10) - 2(9) = 30 - 18 = 12\), which confirms the solution is correct.
Examples
Imagine you are managing a logistics warehouse and need to determine the optimal number of two types of delivery trucks to use given two specific constraints, such as total load capacity and total time available. By setting up the constraints as a system of linear equations, you can solve for the exact number of each truck type required using methods like Cramer's Rule. This mathematical approach helps in streamlining operations and making data-driven decisions for efficient resource allocation in real-world scenarios.