https://file.aiquickdraw.com/mathbot/file/1722331759666b60512tz.png

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

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:

\[
\begin{cases}
x = 3y + 1 \\
2x + 4y = 12
\end{cases}
\]

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.

Learn how to solve a system of linear equations using determinants with Cramer's Rule. Detailed step-by-step solution provided for the equations x = 3y + 1 and 2x + 4y = 12. Suitable for Grades 10-12 students studying linear algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation