# System of Linear Equations in three variables using Cramer’s Rule

Cramer’s rule: In linear algebra, Cramer’s rule is an explicit formula for the solution of a system of linear equations with as many equations as unknown variables. It expresses the solution in terms of the determinants of the coefficient matrix and of matrices obtained from it by replacing one column by the column vector of the right-hand-sides of the equations. Cramer’s rule is computationally inefficient for systems of more than two or three equations.
Suppose we have to solve these equations:
a1x + b1y + c1z = d1
a2x + b2y + c2z = d2
a3x + b3y + c3z = d3
Following the Cramer’s Rule, first find the determinant values of all four matrices. [Tex]D_1 = \begin{vmatrix} d_1 & b_1 & c_1\\ d_2 & b_2 & c_2\\ d_3 & b_3 & c_3\\ \end{vmatrix}   [/Tex] [Tex]D_3 = \begin{vmatrix} a_1 & b_1 & d_1\\ a_2 & b_2 & d_2\\ a_3 & b_3 & d_3\\ \end{vmatrix}   [/Tex]

There are 2 cases:
Case I : When D ≠ 0 In this case we have,
x = D1/D
y = D2/D
z = D3/D
Hence unique value of x, y, z will be obtained.
Case II : When D = 0
(a)  When at least one of D1, D2 and D3 is non zero: Then no solution is possible and hence system of equations will be inconsistent.
(b)  When D = 0 and D1 = D2 = D3 = 0: Then the system of equations will be consistent and it will have infinitely many solutions.

Example

Consider the following system of linear equations.
[2x – y + 3z = 9], [x + y + z = 6], [x – y + z = 2] [Tex]D_1 = \begin{vmatrix} 9 & -1 & 3\\ 6 & 1 & 1\\ 2 & -1 & 1\\ \end{vmatrix}   [/Tex] [Tex]D_3 = \begin{vmatrix} 2 & -1 & 9\\ 1 & 1 & 6\\ 1 & -1 & 2\\ \end{vmatrix}   [/Tex]

[x = D1/D = 1], [y = D2/D = 2], [z = D3/D = 3]

Below is the implementation.

## C++

 // CPP program to calculate solutions of linear // equations using cramer's rule #include  using namespace std;   // This functions finds the determinant of Matrix double determinantOfMatrix(double mat) {     double ans;     ans = mat * (mat * mat - mat * mat)           - mat * (mat * mat - mat * mat)           + mat * (mat * mat - mat * mat);     return ans; }   // This function finds the solution of system of // linear equations using cramer's rule void findSolution(double coeff) {     // Matrix d using coeff as given in cramer's rule     double d = {         { coeff, coeff, coeff },         { coeff, coeff, coeff },         { coeff, coeff, coeff },     };     // Matrix d1 using coeff as given in cramer's rule     double d1 = {         { coeff, coeff, coeff },         { coeff, coeff, coeff },         { coeff, coeff, coeff },     };     // Matrix d2 using coeff as given in cramer's rule     double d2 = {         { coeff, coeff, coeff },         { coeff, coeff, coeff },         { coeff, coeff, coeff },     };     // Matrix d3 using coeff as given in cramer's rule     double d3 = {         { coeff, coeff, coeff },         { coeff, coeff, coeff },         { coeff, coeff, coeff },     };       // Calculating Determinant of Matrices d, d1, d2, d3     double D = determinantOfMatrix(d);     double D1 = determinantOfMatrix(d1);     double D2 = determinantOfMatrix(d2);     double D3 = determinantOfMatrix(d3);     printf("D is : %lf \n", D);     printf("D1 is : %lf \n", D1);     printf("D2 is : %lf \n", D2);     printf("D3 is : %lf \n", D3);       // Case 1     if (D != 0) {         // Coeff have a unique solution. Apply Cramer's Rule         double x = D1 / D;         double y = D2 / D;         double z = D3 / D; // calculating z using cramer's rule         printf("Value of x is : %lf\n", x);         printf("Value of y is : %lf\n", y);         printf("Value of z is : %lf\n", z);     }     // Case 2     else {         if (D1 == 0 && D2 == 0 && D3 == 0)             printf("Infinite solutions\n");         else if (D1 != 0 || D2 != 0 || D3 != 0)             printf("No solutions\n");     } }   // Driver Code int main() {       // storing coefficients of linear equations in coeff matrix     double coeff = {         { 2, -1, 3, 9 },         { 1, 1, 1, 6 },         { 1, -1, 1, 2 },     };       findSolution(coeff);     return 0; }

## Java

 // Java program to calculate solutions of linear // equations using cramer's rule class GFG {   // This functions finds the determinant of Matrix static double determinantOfMatrix(double mat[][]) {     double ans;     ans = mat * (mat * mat - mat * mat)         - mat * (mat * mat - mat * mat)         + mat * (mat * mat - mat * mat);     return ans; }   // This function finds the solution of system of // linear equations using cramer's rule static void findSolution(double coeff[][]) {     // Matrix d using coeff as given in cramer's rule     double d[][] = {         { coeff, coeff, coeff },         { coeff, coeff, coeff },         { coeff, coeff, coeff },     };           // Matrix d1 using coeff as given in cramer's rule     double d1[][] = {         { coeff, coeff, coeff },         { coeff, coeff, coeff },         { coeff, coeff, coeff },     };           // Matrix d2 using coeff as given in cramer's rule     double d2[][] = {         { coeff, coeff, coeff },         { coeff, coeff, coeff },         { coeff, coeff, coeff },     };           // Matrix d3 using coeff as given in cramer's rule     double d3[][] = {         { coeff, coeff, coeff },         { coeff, coeff, coeff },         { coeff, coeff, coeff },     };       // Calculating Determinant of Matrices d, d1, d2, d3     double D = determinantOfMatrix(d);     double D1 = determinantOfMatrix(d1);     double D2 = determinantOfMatrix(d2);     double D3 = determinantOfMatrix(d3);     System.out.printf("D is : %.6f \n", D);     System.out.printf("D1 is : %.6f \n", D1);     System.out.printf("D2 is : %.6f \n", D2);     System.out.printf("D3 is : %.6f \n", D3);       // Case 1     if (D != 0)      {         // Coeff have a unique solution. Apply Cramer's Rule         double x = D1 / D;         double y = D2 / D;         double z = D3 / D; // calculating z using cramer's rule         System.out.printf("Value of x is : %.6f\n", x);         System.out.printf("Value of y is : %.6f\n", y);         System.out.printf("Value of z is : %.6f\n", z);     }           // Case 2     else     {         if (D1 == 0 && D2 == 0 && D3 == 0)             System.out.printf("Infinite solutions\n");         else if (D1 != 0 || D2 != 0 || D3 != 0)             System.out.printf("No solutions\n");     } }   // Driver Code public static void main(String[] args) {     // storing coefficients of linear     // equations in coeff matrix     double coeff[][] = {{ 2, -1, 3, 9 },                         { 1, 1, 1, 6 },                         { 1, -1, 1, 2 }};     findSolution(coeff);     } }    // This code is contributed by PrinciRaj1992

## Python3

 # Python3 program to calculate  # solutions of linear equations  # using cramer's rule   # This functions finds the  # determinant of Matrix def determinantOfMatrix(mat):       ans = (mat * (mat * mat -                         mat * mat) -            mat * (mat * mat -                         mat * mat) +            mat * (mat * mat -                         mat * mat))     return ans   # This function finds the solution of system of # linear equations using cramer's rule def findSolution(coeff):       # Matrix d using coeff as given in      # cramer's rule     d = [[coeff, coeff, coeff],          [coeff, coeff, coeff],          [coeff, coeff, coeff]]           # Matrix d1 using coeff as given in      # cramer's rule     d1 = [[coeff, coeff, coeff],           [coeff, coeff, coeff],           [coeff, coeff, coeff]]           # Matrix d2 using coeff as given in      # cramer's rule     d2 = [[coeff, coeff, coeff],           [coeff, coeff, coeff],           [coeff, coeff, coeff]]           # Matrix d3 using coeff as given in      # cramer's rule     d3 = [[coeff, coeff, coeff],           [coeff, coeff, coeff],           [coeff, coeff, coeff]]       # Calculating Determinant of Matrices      # d, d1, d2, d3     D = determinantOfMatrix(d)     D1 = determinantOfMatrix(d1)     D2 = determinantOfMatrix(d2)     D3 = determinantOfMatrix(d3)           print("D is : ", D)     print("D1 is : ", D1)     print("D2 is : ", D2)     print("D3 is : ", D3)       # Case 1     if (D != 0):                 # Coeff have a unique solution.          # Apply Cramer's Rule         x = D1 / D         y = D2 / D                   # calculating z using cramer's rule         z = D3 / D                     print("Value of x is : ", x)         print("Value of y is : ", y)         print("Value of z is : ", z)       # Case 2     else:         if (D1 == 0 and D2 == 0 and             D3 == 0):             print("Infinite solutions")         elif (D1 != 0 or D2 != 0 or               D3 != 0):             print("No solutions")   # Driver Code if __name__ == "__main__":       # storing coefficients of linear      # equations in coeff matrix     coeff = [[2, -1, 3, 9],              [1, 1, 1, 6],              [1, -1, 1, 2]]       findSolution(coeff)   # This code is contributed by Chitranayal

## C#

 // C# program to calculate solutions of linear // equations using cramer's rule using System;   class GFG {   // This functions finds the determinant of Matrix static double determinantOfMatrix(double [,]mat) {     double ans;     ans = mat[0,0] * (mat[1,1] * mat[2,2] - mat[2,1] * mat[1,2])         - mat[0,1] * (mat[1,0] * mat[2,2] - mat[1,2] * mat[2,0])         + mat[0,2] * (mat[1,0] * mat[2,1] - mat[1,1] * mat[2,0]);     return ans; }   // This function finds the solution of system of // linear equations using cramer's rule static void findSolution(double [,]coeff) {     // Matrix d using coeff as given in cramer's rule     double [,]d = {         { coeff[0,0], coeff[0,1], coeff[0,2] },         { coeff[1,0], coeff[1,1], coeff[1,2] },         { coeff[2,0], coeff[2,1], coeff[2,2] },     };           // Matrix d1 using coeff as given in cramer's rule     double [,]d1 = {         { coeff[0,3], coeff[0,1], coeff[0,2] },         { coeff[1,3], coeff[1,1], coeff[1,2] },         { coeff[2,3], coeff[2,1], coeff[2,2] },     };           // Matrix d2 using coeff as given in cramer's rule     double [,]d2 = {         { coeff[0,0], coeff[0,3], coeff[0,2] },         { coeff[1,0], coeff[1,3], coeff[1,2] },         { coeff[2,0], coeff[2,3], coeff[2,2] },     };           // Matrix d3 using coeff as given in cramer's rule     double [,]d3 = {         { coeff[0,0], coeff[0,1], coeff[0,3] },         { coeff[1,0], coeff[1,1], coeff[1,3] },         { coeff[2,0], coeff[2,1], coeff[2,3] },     };       // Calculating Determinant of Matrices d, d1, d2, d3     double D = determinantOfMatrix(d);     double D1 = determinantOfMatrix(d1);     double D2 = determinantOfMatrix(d2);     double D3 = determinantOfMatrix(d3);     Console.Write("D is : {0:F6} \n", D);     Console.Write("D1 is : {0:F6} \n", D1);     Console.Write("D2 is : {0:F6} \n", D2);     Console.Write("D3 is : {0:F6} \n", D3);       // Case 1     if (D != 0)      {         // Coeff have a unique solution. Apply Cramer's Rule         double x = D1 / D;         double y = D2 / D;         double z = D3 / D; // calculating z using cramer's rule         Console.Write("Value of x is : {0:F6}\n", x);         Console.Write("Value of y is : {0:F6}\n", y);         Console.Write("Value of z is : {0:F6}\n", z);     }           // Case 2     else     {         if (D1 == 0 && D2 == 0 && D3 == 0)             Console.Write("Infinite solutions\n");         else if (D1 != 0 || D2 != 0 || D3 != 0)             Console.Write("No solutions\n");     } }   // Driver Code public static void Main() {     // storing coefficients of linear     // equations in coeff matrix     double [,]coeff = {{ 2, -1, 3, 9 },                         { 1, 1, 1, 6 },                         { 1, -1, 1, 2 }};     findSolution(coeff);     } }    // This code is contributed by 29AjayKumar

Output:
D is : -2.000000
D1 is : -2.000000
D2 is : -4.000000
D3 is : -6.000000
Value of x is : 1.000000
Value of y is : 2.000000
Value of z is : 3.000000


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