Math Problem Statement

Solve for x and y the following system of Linear Equations using Determinant of a matrix:

Solution

To solve the given system of linear equations using determinants (Cramer's Rule), we first set up the system in matrix form.

The given system is:

x = 3y + 1 \\ 2x + 4y = 12 \end{cases}$$ Rearranging the first equation, we get: $$x - 3y = 1$$ Thus, the system can be rewritten as: $$\begin{cases} x - 3y = 1 \\ 2x + 4y = 12 \end{cases}$$ Now, we represent this system in matrix form as $$AX = B$$, where $$A = \begin{pmatrix} 1 & -3 \\ 2 & 4 \end{pmatrix}, \quad X = \begin{pmatrix} x \\ y \end{pmatrix}, \quad B = \begin{pmatrix} 1 \\ 12 \end{pmatrix}$$ To use Cramer's Rule, we first calculate the determinant of matrix $$A$$, denoted as $$\text{det}(A)$$. $$\text{det}(A) = \begin{vmatrix} 1 & -3 \\ 2 & 4 \end{vmatrix} = (1)(4) - (-3)(2) = 4 + 6 = 10$$ Since $$\text{det}(A) \neq 0$$, the system has a unique solution. We proceed to find $$x$$ and $$y$$ using Cramer's Rule. 1. **For $$x$$**: Replace the first column of $$A$$ with $$B$$ and calculate the determinant of the new matrix $$A_x$$. $$A_x = \begin{pmatrix} 1 & -3 \\ 12 & 4 \end{pmatrix}$$ $$\text{det}(A_x) = \begin{vmatrix} 1 & -3 \\ 12 & 4 \end{vmatrix} = (1)(4) - (-3)(12) = 4 + 36 = 40$$ $$x = \frac{\text{det}(A_x)}{\text{det}(A)} = \frac{40}{10} = 4$$ 2. **For $$y$$**: Replace the second column of $$A$$ with $$B$$ and calculate the determinant of the new matrix $$A_y$$. $$A_y = \begin{pmatrix} 1 & 1 \\ 2 & 12 \end{pmatrix}$$ $$\text{det}(A_y) = \begin{vmatrix} 1 & 1 \\ 2 & 12 \end{vmatrix} = (1)(12) - (1)(2) = 12 - 2 = 10$$ $$y = \frac{\text{det}(A_y)}{\text{det}(A)} = \frac{10}{10} = 1$$ Thus, the solution to the system is $$x = 4$$ and $$y = 1$$. Would you like further details or have any questions? --- **Relative Questions:** 1. What is Cramer's Rule? 2. How do you calculate the determinant of a 2x2 matrix? 3. Can Cramer's Rule be used for systems where $$\text{det}(A) = 0$$? 4. How does the matrix method compare to substitution and elimination methods? 5. What is the significance of the determinant in linear algebra? 6. How can we verify the solution to a system of equations? 7. What are some real-life applications of solving systems of linear equations? 8. How would the method change if the system had three variables? **Tip:** Remember that Cramer's Rule is only applicable when the determinant of the coefficient matrix is non-zero, ensuring a unique solution.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Determinants
Cramer's Rule

Formulas

Determinant of a 2x2 matrix

Theorems

Cramer's Rule

Suitable Grade Level

Grades 10-12

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