 So, this question is about consistency of linear equations and the question says for what value of k the system of equations x plus 2y equals 5 and 3x plus ky plus 15 equals 0 has 1 a unique solution and 2 no solution. So, before attempting this question let us first do a quick recap of what was consistency of linear equations. So, if there are two linear equations let us say a1x plus b1y plus c1 equals 0 this is equation number 1 and let us say there are another equation is a2x plus b2y plus c2 equals 0 this is equation number 2. So, if these are the two equations then we know that for consistency what is consistency consistency consistency of linear equations. What is consistency of linear equation that means they have solution there if you solve them you will get solution. So, one case number one is unique solution unique solution that the it means that the equation has only one pair of x and y that is called unique solution and for that condition is a1 by a2 should not be equal to b1 by b2 correct. And then in unique solution itself let us say this is category a in consistency there are two cases one is unique and then second one is infinitely infinitely many solutions many solutions in this case what is the condition a1 by a2 must be equal to b1 by b2 must be equal to c1 by c2 this is for consistency and for inconsistency that is no solution no solution will be when a1 by a2 is equal to b1 by b2 but not equal to c1 by c2. Now, have to be careful that in this format all the variables and constants are on the left hand side nothing is on the right hand side except 0 ok. So now let us solve the given problem so the question is x plus 2y equals 5 so let me rewrite the question these are the two equations given x plus 2y equals 5 equals 5 and hence it is not in standard form so I am writing it as x plus 2y minus 5 equals 0 let it be equation 1 and second is 3x plus ky ky plus 15 equals 0 so this is equation number 2 so we have to find out the value of k for unique solution so let us solve first case first case is for unique solution so unique solution I know what do we know a1 by a2 must not be equal to b1 by b2 ok so let us say what is a1 by a2 here 1 a1 is 1 if you see this is 1 1 by 3 must not be equal to 2 by k so if you cross multiply so cross multiply multiplication in equation is same like equation so hence k is k must not be equal to 3 into 2 which is 6 so k must not be equal to 6 so k can be k can have any value except 6 for the system of equation to be to be having unique solution ok so this is part number 1 how about part number 2 question number 2 they are saying no solutions for no solution what should what should be no solution for no solution what is the criteria it should be a1 by a2 should be equal to b1 by b2 and should not be equal to c1 by c2 so let us find out a1 by a2 and all so if you see a what is a1 so 1 by 3 must be equal to 2 by k and shouldn't be equal to minus 5 by 15 so obviously it is not equal to minus 5 by 15 because 1 by 3 anyways is not equal to not equal to 5 by 15 minus 5 by 15 right so hence from these two equations this this this equation I can get 1 by 3 must be equal to 2 by k is it it so 2 by k so that means k must be equal to 6 right and k must be equal to 6 and 2 yeah that will be all that will be all why because 1 by 3 is anyways not equal to minus 5 by 15 okay so k if k is 6 then this particular equation this particular system of equation will have no solution so answer is k is equal to 6