# Java Program to Solve Linear Equations

## How to solve a Linear Equation?

Let’s suppose we have to solve this:

```x + y + z = 6

2y + 5z = -4

2x + 5y - z = 27```

Now we can create matrix from this:

```1       1       1               6

0       2       5     and      -4

2       5       -1              27
```

Now we find Inverse of this matrix to get the solution

Now let’s write the program.

## Java Program to Solve Linear Equations:

```import java.util.Scanner;

public class Solve_Linear_Equation
{
public static void main(String args[])
{
char []var = {'x', 'y', 'z', 'w'};
System.out.println("Enter the number of variables in the equations: ");
Scanner input = new Scanner(System.in);
int n = input.nextInt();
System.out.println("Enter the coefficients of each variable for each equations");
System.out.println("ax + by + cz + ... = d");
double [][]mat = new double[n][n];
double [][]constants = new double[n][1];
//input
for(int i=0; i<n; i++)
{
for(int j=0; j<n; j++)
{
mat[i][j] = input.nextDouble();
}
constants[i][0] = input.nextDouble();
}
//Matrix representation
for(int i=0; i<n; i++)
{
for(int j=0; j<n; j++)
{
System.out.print(" "+mat[i][j]);
}
System.out.print("  "+ var[i]);
System.out.print("  =  "+ constants[i][0]);
System.out.println();
}
//inverse of matrix mat[][]
double inverted_mat[][] = invert(mat);
System.out.println("The inverse is: ");
for (int i=0; i<n; ++i)
{
for (int j=0; j<n; ++j)
{
System.out.print(inverted_mat[i][j]+"  ");
}
System.out.println();
}
//Multiplication of mat inverse and constants
double result[][] = new double[n][1];
for (int i = 0; i < n; i++)
{
for (int j = 0; j < 1; j++)
{
for (int k = 0; k < n; k++)
{
result[i][j] = result[i][j] + inverted_mat[i][k] * constants[k][j];
}
}
}
System.out.println("The product is:");
for(int i=0; i<n; i++)
{
System.out.println(result[i][0] + " ");
}
input.close();

}

public static double[][] invert(double a[][])
{
int n = a.length;
double x[][] = new double[n][n];
double b[][] = new double[n][n];
int index[] = new int[n];
for (int i=0; i<n; ++i)
b[i][i] = 1;

// Transform the matrix into an upper triangle
gaussian(a, index);

// Update the matrix b[i][j] with the ratios stored
for (int i=0; i<n-1; ++i)
for (int j=i+1; j<n; ++j)
for (int k=0; k<n; ++k)
b[index[j]][k]
-= a[index[j]][i]*b[index[i]][k];

// Perform backward substitutions
for (int i=0; i<n; ++i)
{
x[n-1][i] = b[index[n-1]][i]/a[index[n-1]][n-1];
for (int j=n-2; j>=0; --j)
{
x[j][i] = b[index[j]][i];
for (int k=j+1; k<n; ++k)
{
x[j][i] -= a[index[j]][k]*x[k][i];
}
x[j][i] /= a[index[j]][j];
}
}
return x;
}

// Method to carry out the partial-pivoting Gaussian
// elimination.  Here index[] stores pivoting order.

public static void gaussian(double a[][], int index[])
{
int n = index.length;
double c[] = new double[n];

// Initialize the index
for (int i=0; i<n; ++i)
index[i] = i;

// Find the rescaling factors, one from each row
for (int i=0; i<n; ++i)
{
double c1 = 0;
for (int j=0; j<n; ++j)
{
double c0 = Math.abs(a[i][j]);
if (c0 > c1) c1 = c0;
}
c[i] = c1;
}

// Search the pivoting element from each column
int k = 0;
for (int j=0; j<n-1; ++j)
{
double pi1 = 0;
for (int i=j; i<n; ++i)
{
double pi0 = Math.abs(a[index[i]][j]);
pi0 /= c[index[i]];
if (pi0 > pi1)
{
pi1 = pi0;
k = i;
}
}

// Interchange rows according to the pivoting order
int itmp = index[j];
index[j] = index[k];
index[k] = itmp;
for (int i=j+1; i<n; ++i)
{
double pj = a[index[i]][j]/a[index[j]][j];

// Record pivoting ratios below the diagonal
a[index[i]][j] = pj;

// Modify other elements accordingly
for (int l=j+1; l<n; ++l)
a[index[i]][l] -= pj*a[index[j]][l];
}
}
}
```

## OUTPUT:

```# javac Solve_Linear_Equation.java
# java Solve_Linear_Equation
Enter the number of variables in the equations:
2
Enter the coefficients of each variable for each equations
ax + by + cz + ... = d
1 2 3
3 2 1

1.0 2.0  x  =  3.0
3.0 2.0  y  =  1.0

The inverse is:
-0.49999999999999994  0.5
0.7499999999999999  -0.24999999999999997

The product is:
-0.9999999999999998
1.9999999999999996```