 Hello everyone, myself AS Falmari. In this video, we are going to discuss how to solve the system of simultaneous linear equations. At the end of this session, students will be able to solve the system of simultaneous linear equations. The system of simultaneous linear equations is categorized into two parts. The first one is non-homogeneous linear equations and second one is homogeneous linear equations. We will discuss one by one. First, let us consider non-homogeneous linear equations. The most general form of the non-homogeneous linear equations is a11 x1 plus a12 x2 plus o1 plus a1n xn equal to b1, a21 x1 plus a22 x2 plus o1 a2n xn equal to b2. Continuously and the last equation will be a m1 x1 plus a m2 x2 plus o1 plus a mn xn equal to bm. Such a type of a system of linear equations is called as the non-homogeneous system of linear equations. Now, here are in all n unknowns are present that are x1, x2, so on up to xn and in all m equations are present and all these a11, a12, so on up to a mn and these b1, b2, bm are all constants. In this topic, we are going to solve such kind of a system. Here we will use the concept of a matrix. We try to write these system of linear equations in terms of matrix form. Let us see how we are going to write. First of all, you have to collect the coefficients of the variables x1 to xn from each of these equations. From the first equation, when I collect all these constants from the left hand side, I will get them as a11, a12, so on up to a1n. From the second equation, the coefficients of the variables are a21, a22, so on up to a2n and continue up to the last equation. From the last equation, we get a m1, a m2, a mn. We will insert one square bracket to this. So, this is now looking like a matrix. Now, we will create one column matrix of the unknowns like this x1, x2, so on up to xn and finally, we will equate this product of matrices to one column matrix, which is the matrix of constants of the right hand sides of this system b1, b2 and so on up to bm. Now, if we take the multiplication of these two matrices on the left hand side and when we equate it to the corresponding position, we can see that we will get one by one all these equations. So, this is what the equivalent matrix form of this system of non-homogeneous linear equations. This first matrix is called as the coefficient matrix. The second column matrix is called as the matrix of unknowns as the unknowns are involved and this last one called as the constant matrix as the right hand side constants are involved here. Now, definition of augmented matrix. The matrix of the form in a square bracket a, b, that is the matrix formed by the coefficient matrix and the constant matrix is called the augmented matrix. That is for the above general system, what will be the augmented matrix? First, we have to write what the coefficient matrix. Now, this is what the coefficient matrix, now we have to join the column matrix. We can join that column matrix by drawing vertical dotted lines here and that will be b1, b2, bm. This is what the augmented matrix of above system is. Solution, the set of n numbers s1, s2, so on up to sn such that if we put x1 equal to s1, x2 equal to s2, x3 equal to s3 and so on, xn equal to sn. In the left hand side of the above system 1, we get right hand side constants is called the solution set. What we are saying? The solution set means this is what the given system of linear equations. Now, on this left hand side, we are going to replace x1 by s1, x2 by s2 and so on xn by sn. Then, if it results the right hand side constants b1, b2, bm, at that time we can say that these set of n numbers form a solution for the given system of non-homogeneous linear equations. That is substituting x1 as s1, x2 as s2, xn as sn. In each equation of the left hand side of the system, what we get? We will get these new equations. If equation first reduces to b1, second reduces to b2 and last reduces to bm, then we can say that these set of n numbers forms a solution set for the given system of non-homogeneous linear equations. Consistent, inconsistent. If the system has a solution, then we can say that the given system is consistent. Otherwise, we can say that the given system is inconsistent. That is, if the given system has a solution, we say that consistent. If it does not has any kind of solution, we say that the given system is inconsistent. Note, suppose the given system of equation contains n unknowns in n equations. If the determinant of coefficient matrix A is not equal to 0, then the system is always consistent. And if the determinant of coefficient matrix A is equal to 0, then the system is always inconsistent. Condition of consistency. Let the given system of m equations in n unknowns we are having. If the rank of the augmented matrix AB is exactly equal to the rank of the matrix A, then we can say that the given system is consistent. If the rank of the augmented matrix AB is not equal to rank of the matrix A, then we can say that the system is inconsistent. These two conditions are used to check the consistency of the system. Working rule. Step number one will be writing the given system of m linear equations in n unknowns in matrix form as A into x equal to b, where A is coefficient matrix, x is the matrix of unknowns and b is the matrix of constants. Step number two, consider the augmented matrix AB and apply elementary row transformations on coefficient matrix A and on constant matrix B till you get an echelon form. You have to apply the row transformations on the augmented matrix AB until you get the echelon form of that augmented matrix AB. Once you get the augmented matrix AB in echelon form, then let us find out the rank of the matrix A and the rank of the augmented matrix AB. So in this case, two cases are arising. The case number one, what it will be? Rank of the matrix A and rank of augmented matrix AB we have obtained either they are equal and not equal. In case number one, we consider that rank of the matrix A not equal to rank of the augmented matrix AB. According to our condition of consistency, the given system is inconsistent. It means it has no solution. Case number two, rank of the matrix A is exactly equal to rank of the augmented matrix AB. According to the condition of consistency, the given system is consistent. That is the given system has a solution. Now there are two types of solutions for that purpose we are saying here. Case number two has following two sub cases. Let us denote r as the rank of A and n will be the number of unknowns. The sub case number one, if rank of the matrix A is exactly equal to the number of unknowns, then the given system of non-homogeneous linear equations has the unique solution. That is x1, x2, xn has the unique numbers. Sub case two, rank of the matrix A is not equal to number of unknowns. In this case, the given system of non-homogeneous linear equations has infinitely many solutions. This infinite solution is obtained by choosing n minus r unknowns out of n unknowns as parameters and expressing remaining unknowns in terms of these parameters. So this is what the working rule for finding the solution of non-homogeneous linear equations.