This article elaborates on how to solve systems of linear equations by applying a theorem called the Cramer’s rule. This rule is an alternative to the elimination and substitution techniques. At the first stage of applying the rule, we have to rewrite the given system of linear equations as a matrix equation. More specifically, suppose that we have a system of linear equations in k unknowns as follows.

It can be expressed as the matrix equation below.

It has the form of *AX* = *B*, where

Note: *A* is called * coefficient matrix*.

**Theorem** **(Cramer’s Rule)**

Let *AX* = *B* be a k×k system of linear equations with nonsingular coefficient matrix *A* = (*a*_{ij}). Let *A _{i}* be the matrix obtained from

*A*by replacing the

*i*th column of

*A*by the column vector B. Let

*D*= |

*A*| and

*D*= |A

_{i}*| for i = 1, 2, 3, …, k. Then the system has the unique solution . (Note:*

_{i}*D*is the determinant of

*A*and

*D*is the determinant of

_{i}*A*for i = 1, 2, …, k.)

_{i}

**Example 1**

Find the solution set for the system of linear equations below.

**Answer**

Let x = x_{1} and y = x_{2}. Then the system of linear equations can be expressed as *AX* = *B* where

To find *A _{1 }*, replace the first column of

*A*with

*B*. This gives . Similarly, to find

*A*, replace the second column of

_{2 }*A*with

*B*. This gives . Now, compute

*D*,

*D*, and

_{1 }*D*as follows.

_{2}Consequently, and . Thus, the solution set is {(2,1)}.

**Example 2**

Find the solution set for the system of linear equations below.

**Answer**

The corresponding matrix equation for the system is *AX* = *B* where , , and . By following the similar procedure as in Example 1, we then have:

Consequently, and . Thus, the solution set is {(3,-2)}.

**Example 3**

Find the solution set for the system of linear equations below.

**Answer**

Let x = x_{1}, y = x_{2}, and z = x_{3}. Then the system of linear equations can be expressed as *AX* = *B* where , , and .

By following the similar procedure as in Example 1, we then have:

Now, compute *D*, *D _{1 }*,

*D*and

_{2}*D*as follows.

_{3}Accordingly, , , and . So, the solution set is {(2,1,1)}.

In fact, there are some other ways to solve a system of linear equations, such as by using the inverse of the coefficient matrix and Gauss-Jordan elimination method. These alternatives will be discussed in some other articles on this website.