﻿ Cramer's Rule 2x2 Step by Step

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 x1 + 3 x2 = 9 x1 + 2 x2 = 9
Let's write the Cramer's rule:
x1 = det A1 / det A
x2 = det A2 / 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. 3 x1 + 3 x2 = 9 x1 + 2 x2 = 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.
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 3 x1 + 3 x2 = 9 x1 + 2 x2 = 9
 3 3 1 2
 9 3 9 2
 det A1 = 9 3 9 2
= 9 * 2 - 3 * 9 = 18 - 27 = -9
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 3 x1 + 3 x2 = 9 x1 + 2 x2 = 9
 3 3 1 2
 3 9 1 9
 det A2 = 3 9 1 9
= 3 * 9 - 9 * 1 = 27 - 9 = 18
Result:
x1 = det A1 / det A = -9/3 = -3
x2 = det A2 / det A = 18/3 = 6