Solving Simultaneous Equations Using Cramer's Rule: A Comprehensive Guide

Cramer's Rule is a powerful method for solving systems of linear equations. It utilizes determinants to find the values of the variables in the equation system. This article will walk you through the process of solving the system of equations:

1. The Given Equations

The system of equations is:

2. Expressing the Equations in Matrix Form

First, express the equations in matrix form:

A  
| 3  -1 |
|  1   4 |
B  
| 7 |
|11 |

3. Calculating the Determinant of the Coefficient Matrix (D)

The coefficient matrix (A) is:

A  | 3  -1 |
    |  1   4 |

The determinant (D) is calculated as:

D |3 -1| (3 * 4) - (-1 * 1) 12 1 13

4. Calculating Determinants of the Matrices (D_x) and (D_y)

4.1 Determinant (D_x)

To find (D_x), replace the first column of (A) with the constants from the right side:

A_x  
| 7  -1 |
|11   4 |

The determinant (D_x) is calculated as:

D_x |7 -1| (7 * 4) - (-1 * 11) 28 11 39

4.2 Determinant (D_y)

To find (D_y), replace the second column of (A) with the constants from the right side:

A_y  
| 3   7 |
| 1  11 |

The determinant (D_y) is calculated as:

D_y |3 7| (3 * 11) - (7 * 1) 33 - 7 26

5. Calculating (x) and (y)

Using Cramer's Rule:

x D_x / D 39 / 13 3

y D_y / D 26 / 13 2

6. Conclusion

The solution to the system of equations is:

x 3, y 2

Alternative Methods: Simultaneous Method and Graphical Method

While Cramer's Rule is a powerful and efficient method, there are other ways to solve simultaneous equations:

6.1 Simultaneous Method

One can also solve the given equations by the simultaneous method. Multiply the first equation by 4, and then subtract the second equation to eliminate one variable:

1. 3x - y 7

Multiply by 4:

2. 12x - 4y 28

Add it to the second equation:

13x 39, so x 3

Place x 3 back into the first equation to find y:

9 - y 7, so y 2

6.2 Graphical Method

The graphical method involves plotting both lines and finding the point of intersection.

6.3 Matrix Method

Using matrix methods, such as the inverse matrix method, can also be effective.

Conclusion

Cramer's Rule is a straightforward and efficient method for solving systems of linear equations, especially when dealing with small systems. It is particularly useful in theoretical and applied mathematics, engineering, and other fields requiring the solution of linear equations.