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How Cramer's rule is used to solve a system of equations


Define Cramer's rule

Define Cramer's rule: How Cramer's rule is

used to solve a system of equations?

There are different ways to solve a single problem in math. This is why it is so complicated and easy at the same time. One can solve a problem by the method of one’s liking.

For example, there are many ways to write an equation i.e. slope-intercept form, standard form, point-slope form, etc.

Similarly, equations are also solved using different methods. One of such method is Cramer’s rule and it is the most favored one. 

What is Cramer’s rule?

Cramer’s rule is used to solve a system of complex mathematical equations. It is named after a Genevan mathematician, Gabriel Cramer

In Cramer’s rule, we find the value of an unknown variable, let’s say z, by replacing the z column of the matrix and finding its determinant. After that, dividing that with the value of the original matrix’s determinant.   


It is an easy way but it can be inefficient and tiresome when the system consists of more than three equations. Moreover, it is only efficient if the system has a unique solution.

If the determinant for the matrix of the linear equation system is zero, you will need to find some other method to solve for the value of variables. This is because Cramer’s rule is not applicable in such cases. 

How to solve equations using Cramer’s rule?

Even though there are some drawbacks of Cramer’s rule but where it can be applied, it is the best choice. This method has made computing the value of variables very easy.


          One thing should be noted that Cramer’s rule only works on square matrices. That is, the number of unknown variables is equal to the number of equations.

Solving a system of 2 linear equations: 

It is easy to solve a system consisting of two equations, for the obvious reason of fewer calculations.

Let’s consider two general equations:


                     a1x + b1y = d1                   … Equation 1

                     a2x + b2y = d2                   … Equation 2

To solve it for the values of x and y, we need to convert them into matrix form.



                        a1      b1              d1


                        a2      b2              d2

                              ↑            ↑

                        X-Column          Constant column


The 2 by 2 matrix above is the coefficient matrix A and the 1 by 2 matrix is the constant matrix. The formula for the value of a variable is:

                 =     |Dx| / |D|  , |Dy| / |D|

By using this formula, we can find the value of the x and y variables. The letter “D” represents the determinant of matrix A.  

To find the other determinants (|Dx| or |Dy|), we have to place the constant matrix at the place of the column of the variable whose value we want to find. 

In the general example above, if we want to find the value of the variable X, we will replace the X-column with the constant matrix. Like this:


                               d1       b1

                               d2       b2


It is now a matrix X and its determinant is |Dx|. Hopefully, this all makes sense. Now, let’s solve an example for better understanding.


Find the value of the X and Y variables for these equations.

              6x  +  6y  =  10

              1x  +  2y  =   3


Step 1: Form the matrix.

              6        6          10


              1        2           3

                  A          Constant 

Step 2: Find the X and Y matrix.


          10      6

           3       2


            6      10       

          1       3

Step 3: Find the determinants of all three matrices i.e (|D|, |Dx|, and |Dy|).

Determinant of matrix A:

   = (6)(2) - (1)(6)

   = 12 - 6

   = 6

Determinant of matrix X:

   = (10)(2) - (3)(6)

   = 20 - 18

   = 2

Determinant of matrix Y:

   = (6)(3) - (1)(10)

   = 18 - 10

   = 8

Step 4: Use Cramer’s rule formula to calculate the value of X and Y variables.

For X:

          |Dx | / |D| = 2 / 6 = 1 / 3

For Y:

         |Dy | / |D| = 8 / 6 = 4 / 3

So, the answer is 

         X = 1 / 3    And   Y = 4 / 3

Solving a system of 3 Linear equation:

To find the values of variables in a system of three equations is very similar to finding the values in a 2 equation system.

The only difference when there are 3 equations is that you have to solve for an extra variable. Let’s see the general example.

                    a1x       b1y      c1z  = d1

                    a2x       b2y      c2z  = d2

                    a3x       b3y      c3z  = d3


On conversion into matrices, it will look like this:

              a1      b1     c1            d1


              a2      b2     c2    =      d2   

              a3      b3     c3            d3

This time we will have to find the Z variable as well. For this purpose, a determinant |Dz| will be computed. The rest of the process is similar to the system of the 2 equations.

Alternate method: through calculator:

Even though calculating the values of variables is easy through Cramer’s rule, you cannot deny that it’s time taking.  And let’s not forget the possibilities of minor errors which cause a major impact.

This is why it is suggested to use the online Cramer’s rule calculator. With an efficient tool, the process to solve even complex equations will be easy and quick. 

You only have to enter the problem (question or equation) and the tool will solve it accordingly. The only thing you have to keep in mind is the input process. 

You should add appropriately to avoid misinterpretation by a calculator. 

To wrap up:

Cramer’s rule uses determinants to solve systems of equations. It is suggested for 2 and 3-equation systems. You can also pick a calculator from a vast collection of tools available over the internet.

Also read: Essay on how to overcome the fear of maths

Also read: Importance of mathematics in daily life essay

Also Read: 10 Importance Of Mathematics In Daily Life 



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