Example of Solving a System of Linear Equations by Cramer's Rule.
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It is necessary to solve the system of linear equations using Cramer's rule.
3  x_{1}  +  3  x_{2}  =  9  
x_{1}  +  2  x_{2}  =  9 
Let's write the Cramer's rule:
x_{1} = det A_{1} / det A
x_{2} = det A_{2} / det A
It is impossible to divide by zero. Therefore, if the determinant of A is zero, then it is impossible to use Cramer's rule.
Let's calculate the determinant A. more info
The determinant A consists of the coefficients of the left side of the system.
3  x_{1}  +  3  x_{2}  =  9  
x_{1}  +  2  x_{2}  =  9 
det A =  3  3  
1  2 
= 3 * 2  3 * 1 = 6  3 = 3
The determinant A is not zero. It is possible to use the Cramer's rule.
Let's calculate the determinant A_{1}. more info
It is necessary to change column 1 in determinant A to the column of the right side of the system.
System  det A  det A_{1}  



det A_{1} =  9  3  
9  2 
= 9 * 2  3 * 9 = 18  27 = 9
Let's calculate the determinant A_{2}. more info
It is necessary to change column 2 in determinant A to the column of the right side of the system.
System  det A  det A_{2}  



det A_{2} =  3  9  
1  9 
= 3 * 9  9 * 1 = 27  9 = 18
Result:
x_{1} = det A_{1} / det A = 9/3 = 3
x_{2} = det A_{2} / det A = 18/3 = 6