System Of Linear Equations Solver
System Of Linear Equations Solver is a free online calculator which help you to find out the solutions of linear equations if they exist. This calculator is very useful in finding the multi variable linear equations which can take multiple hours to solve by hand. Go ahead and try this calculator to solve your linear equations problems.
System Of Linear Equations are the set of two or more than two linear equations which contains the same variables i.e. x,y,z,.....,k. Linear Equations are those equations in which the variables are raised to the power of one.The path of the linear equations are straight. Equations in which variables are raised to any other powers rather then one are non-linear equations. The path of the non-linear functions are curve. The general form of 2-variable system of linear equations is given by:Similarly, the general form of 3-variable system of linear equations looks is :
In the above examples x,y,z are the variables and are the coefficients.These coefficients are used to compute the value of the variables. The value of these variables is called the solutions of the linear equations.
We can solve system of linear equations by using various methods.In this calculator we offer only three methods of calculations.
1. Row Echelon Method or Gauss–Jordan elimination
2. Cramers Method
3. Matrix Inversion Method
1. Row Echelon Method or Gauss–Jordan elimination
2. Cramers Method
3. Matrix Inversion Method
Matrix Inversion Method
Matrix Inversion Methodis a mathematical technique in which Inverse of a matrix is used to calculate the solution of the linear system of equations.
It works on the principle that if we multiply a matrix with its inverse we get the identity matrix.
............... (i)
We have a system of linear equations in the form of
Multiplying both sides with the inverse of matrix A, we get:
............ (ii)
Using (i), we can write (ii) as:
It works on the principle that if we multiply a matrix with its inverse we get the identity matrix.
............... (i)
We have a system of linear equations in the form of
Multiplying both sides with the inverse of matrix A, we get:
............ (ii)
Using (i), we can write (ii) as:
Let us suppose we have a following system of linear equation written in the form of :
| 1 | 5 | 2 |
| 5 | 3 | 1 |
| 6 | 7 | 9 |
| X0 |
| X1 |
| X2 |
| 4 |
| 8 |
| 2 |
Inorder to find the value of X we should find the inverse of matrix A
=
x
=
And We find the inverse of matrix A to be :
| -20/141 | 31/141 | 1/141 |
| 13/47 | 1/47 | -3/47 |
| -17/141 | -23/141 | 22/141 |
After that, Writing the matrix in the form of
| X0 |
| X1 |
| X2 |
| -20/141 | 31/141 | 1/141 |
| 13/47 | 1/47 | -3/47 |
| -17/141 | -23/141 | 22/141 |
| 4 |
| 8 |
| 2 |
Now,
| X0 |
| X1 |
| X2 |
| 170/141 |
| 54/47 |
| -208/141 |
Types of Solutions in System of Linear Equations.
1. Unique Solution
If the determinant of the coefficient matrix is not equal to zero then we will have unique solution. We can also say that thenumber of variables is equal to the number of equations and no rows in the augmented matrix is dependent to each other or redundant or filled with zero The rank of the coefficient matrix is equal to the rank of the augmented matrix.
For Example:
For Example:
| 1 | 0 | 3 | 4 |
| 6 | 3 | 1 | 5 |
| 3 | 4 | 5 | 6 |
2. No Solution
If the determinant of the coefficient matrix is equal to zero then we have no solution. We can also determine this by comparing the rank of augmented matrix and coefficient matrix. If the rank of the augmented matrix is greater than the rank of the coefficient matrix then we have no solution.
For Example:
For Example:
| 1 | 0 | 3 | 4 |
| 6 | 3 | 1 | 5 |
| 0 | 0 | 0 | 6 |
3. Infinite Solutions
If the determinant of the coefficient matrix is equal to zero and and at the same time if the rank of the augmented matrix is equal to the rank of the coefficient matrix then we have infinite many solutions. We can also determine this if all the elements in a row of augmented matrix are zero or if the rows are duplicates or dependent to each other.
For Example:
For Example:
Augmented matrix with a row with all zeros
| 1 | 0 | 3 | 4 |
| 6 | 3 | 1 | 5 |
| 0 | 0 | 0 | 0 |
Augmented matrix with two rows dependent to each other. Here, row2 = 2*row3
| 1 | 0 | 3 | 4 |
| 4 | 8 | 6 | 12 |
| 2 | 4 | 3 | 6 |
