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The document outlines the steps to solve a system of equations using Cramer’s Rule. It presents a specific system of equations, converts it into matrix form, calculates the determinants of the matrices, and ultimately finds the values of x and y. The final solution is x = 4/3 and y = 3.

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25 views1 page

14

The document outlines the steps to solve a system of equations using Cramer’s Rule. It presents a specific system of equations, converts it into matrix form, calculates the determinants of the matrices, and ultimately finds the values of x and y. The final solution is x = 4/3 and y = 3.

Uploaded by

usamaraza0786123
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as TXT, PDF, TXT or read online on Scribd
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To solve the system of equations using Cramer’s Rule, we follow these steps.

Given system of equations:


1. 3x + 2y = 10
2. -6x + 4y = 4

Step 1: Write the system in matrix form A \cdot X = B

The system is:


\begin{bmatrix} 3 & 2 \\ -6 & 4 \end{bmatrix} \cdot \begin{bmatrix} x \\ y \
end{bmatrix} = \begin{bmatrix} 10 \\ 4 \end{bmatrix}

Where:
• A = \begin{bmatrix} 3 & 2 \\ -6 & 4 \end{bmatrix}
• X = \begin{bmatrix} x \\ y \end{bmatrix}
• B = \begin{bmatrix} 10 \\ 4 \end{bmatrix}

Step 2: Find the determinant of matrix A, denoted as \text{det}(A)

\text{det}(A) = (3)(4) - (-6)(2) = 12 + 12 = 24

Step 3: Find matrices A_x and A_y by replacing columns of A with the column matrix
B
• For A_x, replace the first column of A with B:
A_x = \begin{bmatrix} 10 & 2 \\ 4 & 4 \end{bmatrix}
\text{det}(A_x) = (10)(4) - (4)(2) = 40 - 8 = 32
• For A_y, replace the second column of A with B:
A_y = \begin{bmatrix} 3 & 10 \\ -6 & 4 \end{bmatrix}
\text{det}(A_y) = (3)(4) - (-6)(10) = 12 + 60 = 72

Step 4: Solve for x and y using Cramer’s Rule

x = \frac{\text{det}(A_x)}{\text{det}(A)} = \frac{32}{24} = \frac{4}{3}


y = \frac{\text{det}(A_y)}{\text{det}(A)} = \frac{72}{24} = 3

Final Solution:

x = \frac{4}{3}, \quad y = 3

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