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
Cramers Method
Cramers method is a mathematical method which is highly used to calculate the solutions of the system of linear equations with the help of the determinants of the matrix . This method is one of the most easiest way to find the solution of the system of linear equations.
Let us consider we have a system of linear equations as below represented in the augmented matrix form.
| 1 | 5 | 2 | 4 |
| 5 | 3 | 1 | 8 |
| 6 | 7 | 9 | 2 |
In Cramers Method we need to calculate the determinant of the coefficient matrix then we replace the constants column which is last column of the augmented matrix with the all columns of the coefficient matrix one by one and we calculate the determinants as below.
D =
= -141
| 1 | 5 | 2 |
| 5 | 3 | 1 |
| 6 | 7 | 9 |
D0 =
= -170
| 4 | 5 | 2 |
| 8 | 3 | 1 |
| 2 | 7 | 9 |
D1 =
= -162
| 1 | 4 | 2 |
| 5 | 8 | 1 |
| 6 | 2 | 9 |
D2 =
= 208
| 1 | 5 | 4 |
| 5 | 3 | 8 |
| 6 | 7 | 2 |
Now we have all the information that is required to find the values of the variables. we can find the value of each variables as below:
= 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 |
