# Service for Solving Linear Programming Problems

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## Example №2. Solving of a 3x3 System of Linear Equations by the Cramer's Rule

This solution was made using the calculator presented on the site.
It is necessary to solve the system of linear equations using Cramer's rule. x1 + 2 x2 - 2 x3 = -8 - 3 x1 - 3 x2 + 3 x3 = 18 - x1 - 2 x2 + 3 x3 = 5
Let's write the Cramer's rule:
x1 = det A1 / det A
x2 = det A2 / det A
x3 = det A3 / 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.
The determinant A consists of the coefficients of the left side of the system. x1 + 2 x2 - 2 x3 = -8 - 3 x1 - 3 x2 + 3 x3 = 18 - x1 - 2 x2 + 3 x3 = 5
 det A = 1 2 -2 = -3 -3 3 -1 -2 3
 1 2 -2 -3 -3 3 -1 + 1 -2 + 2 3 + ( -2)
This elementary transformation does not change the value of the determinant.
 = 1 2 -2 = -3 -3 3 0 0 1
 1 2 -2 -3 -3 3 0 0 1
Row number 3
Column number 1
Element Row 3 and column 1
have been deleted
( -1) 3 + 1 * 0 *
 2 -2 -3 3
 1 2 -2 -3 -3 3 0 0 1
Row number 3
Column number 2
Element Row 3 and column 2
have been deleted
( -1) 3 + 2 * 0 *
 1 -2 -3 3
 1 2 -2 -3 -3 3 0 0 1
Row number 3
Column number 3
Element Row 3 and column 3
have been deleted
( -1) 3 + 3 * 1 *
 1 2 -3 -3
Products are summed. If the element is zero than product is zero too.
 = ( -1) 3 + 3 * 1 * 1 2 = -3 -3
 = 1 2 = -3 -3
= 1 * ( -3) - 2 * ( -3) =
= -3 + 6 =
= 3
The determinant A is not zero. It is possible to use the Cramer's rule.
It is necessary to change column 1 in determinant A to the column of the right side of the system.
System det A det A1 x1 + 2 x2 - 2 x3 = -8 - 3 x1 - 3 x2 + 3 x3 = 18 - x1 - 2 x2 + 3 x3 = 5
 1 2 -2 -3 -3 3 -1 -2 3
 -8 2 -2 18 -3 3 5 -2 3
 det A1 = -8 2 -2 = 18 -3 3 5 -2 3
 -8 2 -2 + 2 18 -3 3 + ( -3) 5 -2 3 + ( -2)
This elementary transformation does not change the value of the determinant.
 = -8 2 0 = 18 -3 0 5 -2 1
 -8 2 0 18 -3 0 5 -2 1
Row number 1
Column number 3
Element Row 1 and column 3
have been deleted
( -1) 1 + 3 * 0 *
 18 -3 5 -2
 -8 2 0 18 -3 0 5 -2 1
Row number 2
Column number 3
Element Row 2 and column 3
have been deleted
( -1) 2 + 3 * 0 *
 -8 2 5 -2
 -8 2 0 18 -3 0 5 -2 1
Row number 3
Column number 3
Element Row 3 and column 3
have been deleted
( -1) 3 + 3 * 1 *
 -8 2 18 -3
Products are summed. If the element is zero than product is zero too.
 = ( -1) 3 + 3 * 1 * -8 2 = 18 -3
 = -8 2 = 18 -3
= -8 * ( -3) - 2 * 18 =
= 24 - 36 =
= -12
It is necessary to change column 2 in determinant A to the column of the right side of the system.
System det A det A2 x1 + 2 x2 - 2 x3 = -8 - 3 x1 - 3 x2 + 3 x3 = 18 - x1 - 2 x2 + 3 x3 = 5
 1 2 -2 -3 -3 3 -1 -2 3
 1 -8 -2 -3 18 3 -1 5 3
 det A2 = 1 -8 -2 = -3 18 3 -1 5 3
 1 -8 -2 + 1 -3 18 3 + ( -3) -1 5 3 + ( -1)
This elementary transformation does not change the value of the determinant.
 = 1 -8 -1 = -3 18 0 -1 5 2
The elements of row 1 multiplied by 2 are added to the corresponding elements of row 3.   more info
 1 -8 -1 -3 18 0 -1 + 1 * 2 5 + ( -8) * 2 2 + ( -1) * 2
This elementary transformation does not change the value of the determinant.
 = 1 -8 -1 = -3 18 0 1 -11 0
 1 -8 -1 -3 18 0 1 -11 0
Row number 1
Column number 3
Element Row 1 and column 3
have been deleted
( -1) 1 + 3 * -1 *
 -3 18 1 -11
 1 -8 -1 -3 18 0 1 -11 0
Row number 2
Column number 3
Element Row 2 and column 3
have been deleted
( -1) 2 + 3 * 0 *
 1 -8 1 -11
 1 -8 -1 -3 18 0 1 -11 0
Row number 3
Column number 3
Element Row 3 and column 3
have been deleted
( -1) 3 + 3 * 0 *
 1 -8 -3 18
Products are summed. If the element is zero than product is zero too.
 = ( -1) 1 + 3 * ( -1) * -3 18 = 1 -11
 = - -3 18 = 1 -11
= - ( -3 * ( -11) - 18 * 1 ) =
= - ( 33 - 18 ) =
= -15
It is necessary to change column 3 in determinant A to the column of the right side of the system.
System det A det A3 x1 + 2 x2 - 2 x3 = -8 - 3 x1 - 3 x2 + 3 x3 = 18 - x1 - 2 x2 + 3 x3 = 5
 1 2 -2 -3 -3 3 -1 -2 3
 1 2 -8 -3 -3 18 -1 -2 5
 det A3 = 1 2 -8 = -3 -3 18 -1 -2 5
 1 2 -8 -3 -3 18 -1 + 1 -2 + 2 5 + ( -8)
This elementary transformation does not change the value of the determinant.
 = 1 2 -8 = -3 -3 18 0 0 -3
 1 2 -8 -3 -3 18 0 0 -3
Row number 3
Column number 1
Element Row 3 and column 1
have been deleted
( -1) 3 + 1 * 0 *
 2 -8 -3 18
 1 2 -8 -3 -3 18 0 0 -3
Row number 3
Column number 2
Element Row 3 and column 2
have been deleted
( -1) 3 + 2 * 0 *
 1 -8 -3 18
 1 2 -8 -3 -3 18 0 0 -3
Row number 3
Column number 3
Element Row 3 and column 3
have been deleted
( -1) 3 + 3 * -3 *
 1 2 -3 -3
Products are summed. If the element is zero than product is zero too.
 = ( -1) 3 + 3 * ( -3) * 1 2 = -3 -3
 = - 3 * 1 2 = -3 -3
= - 3 * ( 1 * ( -3) - 2 * ( -3) ) =
= - 3 * ( -3 + 6 ) =
= -9
Result:
x1 = det A1 / det A = -12/3 = -4
x2 = det A2 / det A = -15/3 = -5
x3 = det A3 / det A = -9/3 = -3