Mastering Cramer's Rule: A Simple Way to Solve System of Linear Equations

In the world of mathematics, solving linear equations is an essential skill for any student, teacher, or professional. Whether you're a high school student or a college professor, understanding how to solve systems of linear equations can be a daunting task. However, with cramer's rule , you can eliminate the complexity and arrive at the solution with ease.

Cramer's rule is a method for solving systems of linear equations with two or more variables. It is named after Gabriel Cramer, a Swiss mathematician who developed the rule in the 18th century. The rule is based on the concept of determinants, which are mathematical objects used to describe the properties of square matrices.

Using Cramer's rule is relatively simple. First, you need to write your system of linear equations in the form of:

Ax + By = C

Dx + Ey = F

where A, B, C, D, E, and F are constants, and x and y are the variables. Next, you need to evaluate the determinant of the coefficient matrix, which is:

| A B | | x | | C |

| D E | = | y | | F |

The determinant is calculated by multiplying the entries in each row and column, then adding the results. The resulting value is denoted by the letter Delta (Δ).

Once you have calculated the determinant, you can use it to solve for each variable. The formula to solve for the first variable (x) is:

x = Δ(AF - BD) / Δ

Similarly, the formula to solve for the second variable (y) is:

y = Δ(CE - AD) / Δ

Let's consider a simple example to illustrate how to apply Cramer's rule. Suppose we have the following system of linear equations:

2x + 3y = 10

x - 2y = -3

First, we need to write the coefficient matrix:

| 2 3 | | x | | 10 |

| 1 -2 | = | y | | -3 |

The determinant of the coefficient matrix is:

Δ = 2(-2) - 1(3) = -4 - 3 = -7

Now, we can use the formulas to solve for each variable:

x = Δ(10(-2) - 3(-3)) / Δ = -14 / -7 = 2

y = Δ((2)(-3) - 1(10)) / Δ = -6 / -7 = 6/7

Therefore, the solution to the system of linear equations is x = 2 and y = 6/7.

Cramer's rule is a powerful tool for solving systems of linear equations. By understanding how to apply the rule, you can eliminate the complexity and arrive at the solution with ease. Whether you're a student or a professional, Cramer's rule is an essential skill to master in your mathematical toolkit.