 Hello and welcome to the session. In this session we will discuss about the pair of linear equations in two variables and the algebraic method of solving these pair of linear equations. To find the solution of a pair of linear equations in two variables, we use algebraic method and we have several algebraic methods to solve these equations. First method that we shall discuss here is the substitution method. Let's consider a pair of linear equations in two variables. Now we shall solve this pair of linear equations in two variables using substitution methods step by step. Our first step would be we take one equation out of these two linear equations like we consider this equation. Now in the next step we express one variable of the above equation that we have just taken in terms of the other variable. Hence we get this step what we do is we substitute this value of the variable in the equation of the given pair of linear equations that is this. On doing this we get this equation which on further solving would give us the value of the other variable. In the next step we substitute this value of the variable that we have just found in the equation formed in step two that is this equation. This would give us the value of the other variable. Hence we have solved the given pair of linear equation and this is the solution of the given pair of linear equations. So this was one method. Now the next method that we shall discuss is the elimination method which is also used to solve the pair of linear equations. This method is basically more convenient than the substitution method that we have just discussed above. Our basic idea of this method is to eliminate one variable and get an equation in one variable only. Consider the same pair of linear equations that we have taken in the substitution method. We shall solve this pair of linear equations now with elimination method. In the first step of this method we multiply both equations tuitable non-zero constants to make coefficients of some variable numerically equal. Like in this case we multiply the first equation by the coefficient of the x of the second equation. And we multiply the second equation by the coefficient of x of the first equation. So that coefficients of the variable x would be numerically equal. So on doing this we get this pair of linear equations. We add or subtract one equation from the other so that one variable gets eliminated. In case we will subtract both these equations so as to eliminate the variable x. As we get this equation in one variable in the next step we solve this equation in one variable to get the value of the variable. On doing this we get the value of the variable as this. Now in the last step we will substitute this value of the variable that we have just found in either of the original equations. Just to get the value of the other variable. We substitute this value in the first equation which will give us the value of the other variable and that comes out to be this. So this is the solution of the given pair of linear equations on solving by elimination method. Next method that we have is the cross multiplication method. This method basically works for any pair of linear equations in two variables of the form a1x plus b1y plus c1 equal to 0 and a2x plus b2y plus c2 equal to 0. This diagram that we have drawn will help us solve these pair of linear equations using cross multiplication method. Note that the arrows between the two numbers indicate that they are to be multiplied and the second product is to be subtracted from the first. Let's consider the pair of linear equations that we have taken in the above two methods. Now this could also be written in the form. We will draw a similar diagram for this pair of linear equations. We know that the arrows between the numbers means that they are to be multiplied so we have from this we can find out the value of the variables x and y. So we get these values. You must have observed that for a given pair of linear equations in two variables the solution comes out to be same in all the three methods. So to solve a given pair of linear equations in two variables we can use any of the three methods. Now in general if we are given a pair of linear equations a1x plus b1y plus c1 equal to 0 and a2x plus b2y plus c2 equal to 0 then three situations can arise. First if a1 upon a2 is not equal to b1 upon b2 then this pair of linear equations is said to have a unique solution and this system is said to be consistent. Also if we have a1 upon a2 is equal to b1 upon b2 is not equal to c1 upon c2 then this pair of linear equations will have no solution and we say that the system is inconsistent. Then the third situation that we would have is if a1 upon a2 is equal to b1 upon b2 is equal to c1 upon c2 then in this case the pair of linear equations would have infinitely many solutions and thus the system would be dependent or consistent. Next we discuss about the equations reducible to a pair of linear equations in two variables. Like sometimes we are given pair of equations which are not linear so we can reduce them to linear form by making some suitable substitutions. Suppose we have a pair of equations as a1 upon x plus b1 upon y is equal to c1 and a2 upon x plus b2 upon y is equal to c2. As you can see this pair of equations is not in a linear form so to reduce this to a linear form what we do is we make some suitable substitutions like we take 1 upon x as p and 1 upon y as q. So we get a1p plus b1q is equal to c1 and a2p plus b2q is equal to c2. Now this pair of equations is in the linear form and this can be solved by any of the three methods that we have discussed above. Let's consider a pair of equations which is not in a linear form. We shall make some suitable substitutions to reduce this in a linear form. We take 1 upon x minus 1 as a and 1 upon y minus 2 as b so as to get this pair of equations which is in a linear form and can be solved by any of the three methods we have discussed above. This completes the session. Hope you have understood the concept of pair of linear equations in two variables and algebraic method to solve these equations. Also the three situations that we can come across when we are given a pair of linear equations in two variables and how to reduce a given pair of equations to a linear form.