 जो़ of the same, this is best for me भी ईदास its ok वापे आहडी शामा भुगर कि लगे तो तीशनी दोर दो अँर यह दोवेंग जो तीशना है ट्र जितसें यो ड़े अग्जेक्तिव is to get problem solved उसंदा आद लनेशीन चेर वैं थिज़ा। इसे वेडिया में, वो गडोत of recap on algebra, the linear equations come into this. you have what exactly is in 10th for you so in algebra we have polynomials you know polynomials okay we have also linear equations linear equations into variables hello then linear equations is two variables then we have quadratic equation what else do we have polynomial linear equations into variables quadratic equation we have sequence in series what else then we have number system which is studied under number k and we have geometry and in geometry there is a very important topic we are going to discuss and that is sign where you will discuss properties of similarity in all the things triangle then we have circles you have studied little bit of circles last time but this time around it will be more this is geometry then we have something called coordinate geometry coordinate geometry we will talk about points and section formula and all the things coordinate geometry then what else do we have what is trigonometry trigonometry so this is trigonometry trigonometry thanks for the trigonometry then we have something called dimension area then volumes right and then in dimension itself there is one other chapter called area related to circles so we talk about lots of circles then what else then we have stacks and prompts probability and statistics and one thing I missed in geometry is construction any guess I missed this is what you will be saying sorry so this is what you are going to study throughout the year perfect that's it so you will be studying this okay yes if you are willing to and I need to go with me then we will go a little bit more deeper and this one is a little bit more so yeah we will be going in much deeper details of this and then maybe some of it of 11 very important I heard you are going to say coordinate geometry psychology what are you talking about on this idea probability and then there will be a chance of taking commerce Veda what are you doing in 11th grade commerce what is this anyways so let's start with linear equations in two variables that's what is happening in these two yeah okay so first thing is we will give you a just small brief of what is a we will start with something called polynomial until that is clear it becomes difficult to understand what's polynomial right you know polynomial in my career studied polynomials right yes or no so let's start with very basic thing before polynomial also what is term term in algebra what was term so yes so term is nothing but you know a constant constant multiplied with a variable variable and there is a power onto the variable and there could be multiple variables multiple variables also variable this is all okay example could be 3x inspired example could be root 2 x to the power minus 1 which is nothing but root 3 upon 56 by 7 x to the power of 5 y all our terms how is that root 3 by x to the power x to the power minus 1 1 but isn't it root 2 oh my bad so root 2 becomes root 3 thanks for pointing out okay this is term now what is an expression you know what is an expression in algebra yeah so combination of terms right so your combination of terms and example so let's say 3x plus 4y so if you see there is a plus sign yes we are combining with plus or minus right 4y minus 25x square y this is another expression these are all expressions right how many terms here number of terms such expressions are called binomials hence this is called a binomial number of terms here is 2 so this is also binomial so example of a trinomial will be so let's say number of terms number of terms is 3 it's called trinomial okay so example of trinomial is 2x square minus 3y plus 4z these are all trinomial and then anything above we can generalize them as polynomials poly means many so many terms are there then it is called a polynomial clear right now sorry we will come to you know there is one particular catch about polynomial and this is what I will tell you so not all expressions are polynomial guys not all expressions are polynomials why I will tell you so expressions could be any combination right any combination of terms but for polynomial is defined like this a polynomial you guys are not writing polynomial what is polynomial so polynomial is defined as p within brackets x that means this is polynomial invert variable x and it is given as a0 plus a1x plus a2x square plus a3xq plus dot dot plus anx right where if you see all the powers where what is it a0 and a1 a2 till an all are are real numbers all are real numbers right a0 a1 a2 a3 all are real numbers yeah tell me a0 this is our index to differentiate this is a variable a0 a1 a2 it is not multiplication subscript a0 a1 a2 a3 an right and if you see 1 2 3 4 all are non negative integers right so hence all n n is the non negative what is non negative integer what is non negative integer as well as 0 so something which is not negative non negative integer so 0 1 2 3 4 all whole numbers are called non negative so non positive yes okay this is you know so now tell me whether this polynomial is in one variable you can have polynomials in multiple variables as well for example let's say p x y I can have an example 1 plus 2 x y plus 3 x y square plus 4 y plus 5 x plus 9 x square y square this is a polynomial how many variables how many variables but we are not going to devote much time on polynomials of yeah two or multiple variables but then we need to just know okay now another important thing is degree of a polynomial degree of a polynomial yes the power basically which power I am going to talk about the power of the variable power of the variable but the highest power of the variable right so in this case let's say this is an increasing order so what is the degree of this polynomial is it so 1 2 3 4 maximum is n so what is the degree n what about this one yes what is the degree of this no in such cases you just check it's 4 why because find out the highest sum of powers of all the variables so what is our sum of powers here 2 yeah yeah yeah 4 so the highest is the degree is 4 yes tell me I am going to say 4 y is equal to degree is equal to come again sorry you said something yeah yeah but 1 4 y is equal to 4 y where say degree of polynomial like the term 4 y 1 only one variable y no variables not constants you have to consider only variables this is degree of polynomial you understood how to find out degree of polynomial so now basis the degree of polynomial we have classified polynomials before that tell me I am giving you write down these expressions and tell me which one are polynomial which one is not 1 is x plus y 2 x 3 x to the power 5 by 2 4 root x plus 5 y plus 7 then 5 25 under root x square y 7 6 5 x by y plus 7 y by x plus z which one is polynomial which one is not which one is polynomial so Gopika says x plus y is polynomial all of you agree yeah yes next is x polynomial yes or no polynomial polynomial no so polynomial don't go by the yes to as far as English is concerned poly has to be many so but then everything is considered in the family of polynomial even polynomial will be falling under the family so whether it is polynomial by the definition of polynomial does it satisfy that definition that because what is the coefficient here 1 is it real yes what is the power of variable it is a non-negative integer so that means it is a right what was this no why it will be negative 2 by 5 negative 2 by 5 why since it is not a rational number no it is a rational number 5 by 2 is 2.5 rational number sir why is it not it is polynomial how many if you say it is polynomial does it satisfy that how sir n is not negative n is a real number n is a real number n is an integer sir you didn't write that it is written very much here non-negative now tell me is it a polynomial why because it is why it is not a polynomial because here it is not an integer is it an integer so power must be non-negative integer what is this 5 by 2 integer not integer why it is a rational number what about this no why half x power is half not an integer so ruled out yes or no it is a polynomial because this is real and this is very much polynomial what about this why power of y is negative 1 okay so this is correct okay fair enough now now if we have now let us classify polynomials in terms of their degrees so sir if you have something like 25 to the power 5 by 2 x will that be a polynomial you said okay the question is you learn the definition let's go by the definition 25 to the power 5 by 2 polynomial yes or no yes why 39 to the power 5 by 2 x polynomial yes or no no it is not it is this is real this has to be just real number coefficient should be real power should be non-negative integer sir how do you solve that who is asking you to solve remove this see when I am just studying something mathematics is not always okay come to the conclusion come to the solution simply if I know atleast as far as knowledge goes this is polynomial forget about it I have to use logarithms or calculate or whatever that's the second issue who is asking you to solve and find the value no one okay perfect this is the polynomial now I am talking about how to classify polynomials in terms of its degrees okay so let us say let us say I have polynomial Px this is a usual notation guys this is how P for polynomial brackets x means the polynomial is in which variable x so let us say it is 5 is it a polynomial yeah is it a polynomial why not yet there is it is is it real number and it is non-negative integer so every constant also I understood so this is what is the degree here very good sir would it be 5 to the power 1 for degree you have to just consider the variable okay what if Px is equal to 3 plus 2x what is the degree very good now degree this is a degree 1 what is Pxy is equal to 2x minus 3y plus 4 degree degree degree is yeah there is a confusion still can why to I told you each term check all the terms and find the highest power term if there are 2 terms club there exponents together you will get the degree of that okay what is this if Px again is let us say 2x squared plus 3x plus 4 what is the degree no one is writing anything now Px Px plus 4x minus 2 by 3x plus 5 by 2 what is the degree now degree 1 is called linear degree 2 is called quadratic linear polynomial quadratic polynomial and this one is cubic and there are other names as well by quadratic we take this that x y right we are not interested in the nomenclature we are interested in understanding their behavior is it okay fair enough so now you are understanding what our focus of this chapter is going to be so this is what we will be studying in our we will come to the will you know graduate to linear equations right now we are only talking about polynomials of degree 1 they are called linear polynomial now you can sense what is going to be the next thing so linear polynomial transitions to linear equation when when you equate the linear polynomial to something it becomes a linear equation correct so and then in other other chapter when you equate this quadratic polynomial to something it will become quadratic getting it now so let us now what is the difference between these two both are linear one is with one variable another little bit so now you are again zeroing into what we are going to study this one first you already studied in ninth grade did you not study in the ninth grade linear equation in one variable now we are going to we have not yet started equation we are still in the realm of polynomials we will go and transition into equation sooner but then this is what we are going to deal with is that okay now why is the linear equation called a linear equation by the way so because it is linear and it is an equation why so the degree of the equation is 1 so why it is called linear why it is not called another linear equation so linear as in a line as in a line very much very important linear as in line that means there is some line connected to those okay now linear equation in two variables you are not going to linear equation in one variable linear equation in one variable was sorry linear polynomial first of all so first of all I am trying to say what is it I am trying to say what is linear polynomial in two variables you saw that and then I am saying I am equating it to something else okay then it becomes a linear equation so hence now let us consider all linear polynomial okay PXY PXY yeah or let us say yeah before linear equation in two variables let us say we have PX PX equals let us say 2X plus 1 this is a linear polynomial with how many variables 1 let us say for the time being we are considering it to be 1 PX equals Y okay Y I will tell you so the idea is let us plot let us do something I don't know what I am doing but let us do something I am going to randomly put X values and corresponding values of PX I am going to find out and PX is nothing but Y so you know this Cartesian coordinate system you know or not Cartesian coordinate system you have already studied why it is called Cartesian coordinate system as I said like Cart dog would be nothing there was a guy called he was a French philosopher and mathematician so he is the guy who introduced this concept into mathematics so basically you what it is if in a plane or for that matter in a space in a three-dimensional space also if you have two perpendicular X and Y axis like that then any point on that plane here and if it is three-dimensional then point in space can be uniquely defined its position can be uniquely defined how so you draw perpendicular yes or no so this coordinates are X and Y ordered pair right why it is ordered pair because if we flip change the order of the numbers the position changes right so let us say X and Y so let us say X equal this is let us say 4 and this is Y and this one this distance is also 4 so what is the coordinate over there 4,4 you know X coordinate, Y coordinate apsica, ordinate and all that stuff okay this one is apsica this one is ordinate okay so X coordinate is the distance from the Y axis is called apsica and Y coordinate is also called ordinate is nothing but the points distance from X axis clear clear to everyone no problem no so now I am interested in this is one point but now I am interested in point plotting some more so how where do I get these points from so let us say I am putting some random values to this X let us say X equals to and I am making a table so this is X and corresponding Y so let us take X equals to 0 what is Y if X is 0 Y is what is Y when X is 1 what is X when X is minus 1 Y is minus 1 correct let us try to plot these points 0 so let us say this is 1 2 3 4 5 1 2 3 4 5 so 0 comma 1 where will we this is 0 comma 1 correct did I plot it correctly now 1 comma 3 1 1 2 3 so that is 4 oh sorry so this is 1 then minus 1 comma minus 1 so this is minus 1 let us try some more let us say I have 0.5 then 2 2 3 so this is 0.5 okay let us take minus 0.5 okay 0 0 okay so minus 0.5 is here so this is the point do you see some thread guys this is a new line hence it is called linear okay if you if you just notice what is this Y equals to 2 X plus 1 right so I just got a linear equation okay now which is you can also say that if you have a polynomial with 2 variables X and Y you equate it to 0 you will get linear equation into variables alternatively if that itself becomes a linear equation Y is equal to 2 X plus 1 and whenever such kind of equation is there you will get a and hence it is called linear if it is quadratic then you will get a parabola then again a curve different type of cubic curve before another curve clear to everyone so why is it linear why if the linear equation is called linear because it lies in the straight line of the graph and it has 1 million linear equations graphical representation is a okay now if you see what are these points these points lie on this line or not let's try to again mention so this is 0 comma 1 then 1 comma 3 then minus 1 comma minus 1 then 0.5 what was it 0.5 was so this is 0.5 all these points see all these pairs represent a point on the plane now on the same plane these points are lying on which line all these points alternatively we can say all these points whatever the coordinates are if you deploy there the equation gets satisfied so hence guys we call these as roots to this solution all these are what are they called roots slash equation clear clear this is called okay so how many roots are there in this linear equation infinite because there are lots of infinite if any points are there so this is clear right now infinite infinite now problem is or let's say we are studying pair of linear equations is this now let us say I have another line let us say y equals let us say we have another line y equals minus 2x plus 1 another line so let's try and plot that so I will use here so y is equal to negative 2x plus 1 let me say this is x this is y what is x if x is 0 what is y 1 tell me gopika what is y when x is minus 1 ishani x is 1 y shreya x is 2 and veda x is minus 2 okay let us use another color to represent these points 0 comma 1 hey we have 0 comma 1 already then minus 1 comma 3 then 1 comma minus 1 1 comma minus 1 then 2 comma 3 2 comma minus 3 and then minus 2 comma 5 do you see a trend again line it will be state because my drawing is not that great okay minus 2 comma sorry this was 1 comma 2 comma minus 3 these 2 lines intersecting where right now this point is called 0 comma 1 6 both on that line which line y goes to x plus 1 and the moment I change did you notice the moment I change the value of 2 the coefficient of x the line itself got changed so it is now correct so this one is new line and these 2 are intersecting if there are 2 lines intersect or they will be parallel or they will superimpose each other 3 conditions here it is what condition it is intersecting what is the point 0 comma 1 now 0 comma 1 geometrically sits on both the lines correct and 0 comma 1 algebraically satisfies both the equations y goes to x plus 1 as well as y goes 0 comma 1 is the solution to the pair of linear equations you understood so hence we say so what is our inference or what is the conclusion conclusion is 0 comma 1 is the solution to the pair pair of linear equations linear equations linear equations we have a pair see if you had only one equation you had infinitely many solutions but the moment there are 2 linear equations now you are getting only one right and only one when they are intersecting if they are superimposing on each other that means if I had another line superimposing on each other then both the linear equations would have infinitely many points in common so infinitely many solutions agree so hence 0 1 is the solution to the pair of linear equations or the system of linear equations also it is called at times so pair of linear equations what pair y equals to 2x plus 1 and y equals to negative 2x you understood this is called solution of pair of linear equations it could be a case where let's say okay let us plot another equation which is let's say y equals 2x plus 3 another equation let us try to plot that so z 8 yeah so here let us say now x and y and what is this y equals to 2x plus 3 now tell me Gopika x is 0 y is a 5 no 3 x is minus 1 ishaani 1 vedha when x is 2 7 7 and it is minus 2 okay x is minus 2 it is minus 1 let us now plot the different color so 0 comma 3 where is 0 comma 3 0 1 2 3 next minus 1 comma 1 minus 1 comma 1 then 1 comma 5 1 5 then 2 comma 7 6 let's say 7 do you see a trend again but does this line cut the first line parallel so hence there are now look closely what are these two lines what is the difference and what is the similarity y equals to 2x only difference is so what is the inference if the coefficient of y coefficient of x in any pair of two equations are same then the lines are going to be parallel but make sure that these two we will talk about this this thing also but make sure the ratio of this actually how it works is the ratio of the coefficient of y must be equal to ratio of the coefficient of x is but must not be equal to the ratio of the constant terms we will talk about all these little later but we just have this idea here so what are the possibilities of two lines now either they will be intersecting or they will be parallel or they will be superimposing so how it superimposes for example if you have this equation 2y equals to 4x plus 2 if you solve this yeah we will come to consistency when it is parallel it is called inconsistent okay when it is not parallel it is consistent consistency means we will find solution but inconsistency means we will not find solution so hence only in case of parallel lines we will not find but then in school we learnt that parallel lines are inconsistent it happens consistent it happens inconsistency if they impose the infinite and consistent if they intersect then the consistency can have the unique solution if they are parallel they have consistent and don't have consistent so this is how okay now let us let us first understand next is you know the conditions of what you are talking about consistency and inconsistency let's talk about but before that let us now generalize this whatever we have learnt now what is consistency consistency of what pair of of the system of we call it system of pair it is more than 2 it will become the system of linear equation of linear equation now what how do we you know actually find out let's say what is the general expression for linear equation 2 variable system of so there are this is how now I am defining first of all what is a pair of system or pair of linear equation so what are there so let's say a1x plus y plus c1 equals 0 okay this is equation number 1 this is a general form of a linear equation in 2 variables where a1, b1, c1 are all real numbers you already know now second is a2x plus b2y plus c2 now what is it yes where where a1, a2, b1, b2 and c1, c2 are real numbers are real numbers are real numbers such that such that a1 square plus b1 square should not equal to 0 and a2 square plus b2 should not equal to 0 why am I giving you these constraints this is simply to say that both a1 and b1 cannot be 0 why why why this square because the only way to express this that it's like in your computers both a1 and b1 cannot be simultaneously 0 and a2 a1 cannot be 0 and c in this case a1 square plus b1 square should not be equal to 0 is a method or is a mechanism to say what am I conveying through this that both of them cannot be 0 simultaneously when I say that why look at this expression also let us talk about this a2 plus b2 is equal to 0 when is this possible when both are 0 is it possible when one is non 0 another is 0 and then sum is 0 a2 plus b2 is 0 only when a as well as b both are 0 now if I have to tell you that both a and b should not be 0 then I must tell you that yes or no because in any other case you will not be able to say that so for example you take a plus b equal to 0 is it possible for a plus b to be 0 but a and b to be non 0 yes it is possible when a is 1 b is minus 1 correct but there is no way see we have to basically communicate this fact we cannot be together 0 and the only way of expressing that is this in all other cases you will get clear so hence please understand this this only communicate don't get confused by this this only communicates that at the same time both a1 and b1 cannot be one of them can be 0 but not both clear with clear so this is the definition of let's say this is how our system of pair of linear equations are expressed now what I am talking about whether you will get solution or not how do I know whether this is parallel this is not parallel work so let us do some mathematics around it and this is what I am going to do now what did we understand about solution the line must intersect when the line intersects that means that particular x and y value will be in both of them both of these equations will be satisfied by that x and y so let us first express this first equation like this can I write this equation as b1y equals minus a1x minus c1 correct rearranging I took a1x and c1 on the right hand side can I write like that all of you agree with the agree both of you agree agree wherever you stop next can I not say y equals minus a1 by b1 x minus c1 by b1 yes no I agree disagree hello I just now divided the entire equation by b1 so I took this b1 on the right hand side what will happen I will have to divide the entire equation by b1 so this by b1 am I right similarly if you do this you will get y is equal to minus a2 by b2 x minus c2 by b2 am I right yes so both are intersecting that means the y coordinate of that intersection is there will be one y coordinate there will be one point whose y coordinate will be same for both so hence can I not equate this 3 and this 4 what do I get I am writing it here so or rather here itself so let us say these are same so I can say minus a1 by b1x minus c1 by b1 equals minus a2 by b2x minus c2 by b2 can I just cancel all the minus signs multiply the entire equation by minus 1 what will happen this will get cancelled so minus sign gets cancelled become positive now now let us rearrange the x terms together at the constant together so hence what will y get a1x b1 minus a2b2 x is equal to c2 by b2 minus c2 by b2 minus c1 by b2 am I right so would you tell me the difference if you cancel out the negative sign just for comfort I love positivity hence as many positive signs as possible here it is all the negative so everything is positive now why is it minus a2x so a1 by b1x now bring this on the left-hand side and bring this on the right-hand side correct now simplify what will it be so let me just do it here itself can you take the LCM so a1 b2 now can I take x common in the left-hand side so a1 b2 minus a2 b1 upon no upon b1 b2 times x is equal to c2 b1 minus c1 b2 upon b1 b2 go so what is x x equals to c2 b1 minus c1 b2 upon a1 b2 minus a2 b1 now guys this is the ratio all c2 b2 a2 b2 all that stuff all are real numbers yes or no now this is real number in numerator this is real number in denominator so can you take b1 minus b2 common how so listen what is the constraint on this ratio this denominator what is the problem with this this cannot be 0 if this becomes 0 x becomes unreal now it becomes 0 x becomes unreal that means there is a point unreal point where the lines are intersecting so that means if this becomes 0 lines become parallel correct if this becomes 0 then light will be there will be no point of intersection because this is not defined that means the lines are intersecting at infinity if this becomes 0 x becomes in other way if this becomes 0 x becomes something divided by 0 infinite it's not true as such but you understand like that you can understand right so that means the lines are intersecting at infinity so what do I get for parallel lines for parallel lines if two which two lines the two lines which I started with you guys so those two parallel lines what is the condition a1 b2 minus a2 b1 must be equal to correct what does that means that is if you rearrange it you will get a1 by a2 is equal to if the ratios a1 by a2 is equal to b1 by b2 same then then there will be parallel then there will be parallel but for intersecting intersecting lines for that a1 by b2 minus a2 b1 must not be equal to a1 by a2 and for you will get overlapping lines where this becomes 0 by 0 when top four also becomes 0 it is overlapping so hence in that condition what will happen so now coincident lines 0 by 0 for many ways is not defined so hence but this is this condition is c2 b1 minus c1 b2 is equal to 0 that means b1 by b2 must be equal to right c2 right so hence combining these two so for overlapping they are parallel anyways and this condition is also true so combining these two what will happen a2 is equal to b1 by b2 is equal to c1 by c2 condition is a1 by a2 b1 by b2 so what are the three conditions if a1 by a2 is not equal to b1 by b2 system is consistent inconsistent that means there will be one if they are intersecting there will be one definition not more than one but one for sure if they are overlapping there are infinitely either they will have infinitely many solutions or they will have only one solution or they will have no solutions at all it cannot be a case that a pair of linear equation will have two solutions or three solutions only either it will have only one or infinite many or no solutions and then now you know the condition this is the condition for overlapping line parallel line so hence to add to this parallel line will be so to avoid this that parallel but not overlapping will be so the lines are parallel for parallel lines whether see overlapping lines are also parallel so this condition must be met but if they are not to be of their parallel but separated some distance this must not be true this must not be equal so this is for overlapping not overlapping but parallel and this is for irrelevant no I don't really care the moment this is there the moment you saw this it is bound to intersect the moment these two ratios are the moment these two ratios are not equal it is bound to now let us done done done I will pay you for it please write I will pay you for it are you guys feeling little cold here yeah so so let us come out of from the manual yes done done done now tell me I am now writing pair of linear equations so you will have to tell me which one are the questions would be something like this find the nature of the roots okay what do you mean by nature and consistency yeah nature of let us say the pair of whether consistent inconsistent any one solution many solutions are no solution now rapid fire I will write your answer 2x-3y equals 4 4x-5y plus 7 equals 0 there is one solution yes yes intersecting we are confident yes right now x plus y equals 1x equals minus y and rapid there is a solution there is a solution no solution no solution but it is not in this they are parallel lines yeah it is parallel okay y because the second line is symmetry x plus y equals did you get it so 1 upon 1 is equal to 1 upon so x plus y is equal to 1 x plus y is equal to 0 so does that mean 1 is equal to 0 hence hence there is no solution yes this is a linear equation yes or no yes sir this is a linear equation yes or no so since one it cannot be 0 that means they will never intersect that means you cannot equate these two there is no yes there is why because 2 by 4 is 1 by 2 minus 3 by 5 is so 1 by 2 is not equal to minus 3 by 5 the moment you saw that is those two ratios are not same it has to be consistent got it weather see what did I say for intersecting lines consistency what do you need to have a1 by a2 should not be equal to b1 by b2 check what is a1 2 what is a2 4 so 2 by 4 is half what is b1 minus 3 what is b2 5 so minus is 1 by 2 equal to minus 3 by 5 so got it next third 2x minus 9 equals 2y 2x 4x minus 18 equals 4y 3 yes there are infinite solutions infinite solutions why because x coefficient 2 by 4 2 by 4 actually I am doing it directly 2x minus 2y minus 9 equals 0 always reduce in this form a1 x plus b2 y plus a1 x plus b1 x plus a1 x plus b1 y plus c1 equals 0 and this is equal to 4x 4x minus 4y minus 18 equals 0 so a1 by a2 is 2 by 4 b1 by b2 they are coincident how many times next find the value of k if the pair of equation find the value of k if the pair of equations are parallel kx minus 4y plus 7 equals 0 2x minus 6 plus 6y equals minus 9 find the value of k find the value of k if these two lines are parallel find the value of k if these two lines are parallel find k if they are parallel others minus 4 by 3 minus 4 by 3 r where is r this is k you guys want to eat something shall I fill that off but then it will become hot you want to eat something are you mutton sticks I don't know fine tell me tell me brother no I'll eat sorry ok tell me k is equal to minus 4 by 3 there is something sharper than what sir nothing no blood there is no blood there is no blood in this body so you are running of water anyways yeah veda understood no see what did I ask what is the value of k if they are parallel what is the condition of parallel that a1 is equal to a1 by a2 is equal to b1 by b2 right so k by 2 must be equal to minus 4 by so k is minus 4 into 2 by 6 which is hello understood ok consistency inconsistency let's solve some bionic problems no valence guys don't spread valence spread ok let's give you solve find find the value of k so that the given system of equations has a unique solution equations are 2x minus 3y equals 1 2x minus 3y equals 1 and second is kx plus 5y equals 7 no no find the value of k so that this system of equation linear equations have a unique solution 2x minus 3y equals 1 kx plus 5y equals 7 equal to 10 by anything but right so you have right k is not equal to right so you have right k is not equal to weighty so you you just take k anything but that value you will always get 1solution if you change k you will get another set of linear equations no you have to find the value of k so that they have a unique solution case obviously so you have to say anything but k should not be equal to minus 2 so then it will become a parallel do you have to just write k not equal to that's it see k can be 1 can be 2 can be 5 can be 9 can be 1 million every time you get a new k you will get a 2 different set of equations 2 different lines see so one different line and one is already there if you change k what am I trying to say let us say these are the 2 lines so you have to get now there is only one value of k where it becomes parallel is it not the moment you change k what will happen this line will tilt correct and the moment it will intersect somewhere correct if you change k further change k further change k further keep changing k there will be infinite one line is fixed one line is already fixed another line was this which by changing the value of k you are changing the orientation so always it will intersect with this line but only for one case which is that case when they are parallel correct no so if you change k you will get a different set of equations every time there will be one solution to that set of equations correct any doubt even if it is there I will not be solving next whether you got it see what you need to do is what are the equations I will give you another solution on that also on the board so the question was can you write the equations please 2x-3y 2x-3y equal to 1 equal to 1 can I write that as this second second equation was kx-5y minus 5y is it plus 5y is equal to so minus 7 so this 2 equations to have so I am writing the three conditions unique solution unique solution say what is the condition for a unique solution a1 by a2 is not equal to b1 by b2 that's it you just need to check this and it's done unique ke baad no solution where gopika no solution a1 by b2 nishani now what was the question unique solution so what do I do 2yk must not be equal to minus 3 upon 5 so k must not be equal to 10 by minus 3 minus 10 by 3 k must not be equal to minus 10 by 3 you keep your word it doesn't matter it will intersect correct okay next for what value of k 2y plus 7 equals 0 x plus 2y plus 7 equals 0 for what value of k x plus 2y plus 7 equals 0 and 2x plus ky plus 14 2x plus ky plus 14 equals 0 infinitely many solutions or so for what value of k equation represent represent coincident lines 4 7 is to 14 sir 7 is to 14 4 is to 24 did I teach so well or you guys are really intelligent 8 is very good so humble good I appreciate yes did you understand good okay class is over 45 minutes left next see now she is okay next question write down hearts it's heart find the value of p yes find the value of p and q now do find the value of p and q for which the following system of equations has infinite number of solutions find the value of p and q the question is 2x plus 3 equations are 2x plus 3y equals 7 and p plus q x p plus qx then 2p minus qy 2p minus qy equals 21 p plus qx plus 2p minus qy equals 21 infinitely many solutions infinitely many solutions people will find qy equal to 1 for 1 minute 10 seconds extra yes sir that's all no unfair means yeah I got the same so I did it slightly I did it without writing much so you did it so you did it you have to calculate it I don't like this type I am doing it of course I am doing it of course we don't have no book maintenance we don't have no book maintenance what I did was I did it by myself I did it by myself so this is 1 by 3 so this has to be 2x6 this would be 9 exactly what I did so this would be all of them are equal to 1 by 3 1 by a2 is equal to b1 by m2 1 by c1 c1 by c2 is 1 by 3 so a1 by a2 and b1 by b2 what is the question you find out value of p can be q so that infinitely many solutions tell me the condition for infinitely many solutions a1 by a2 is equal to b1 by b2 is equal to 1 by 3 okay clear get it by heart okay now what is a1 in this case when a no x is never included x is the variable so coefficient 2 upon p plus q 2 by p plus q equals b1 what is b1 3 upon 2p minus this is the whole and this is nothing but 7 upon okay how do I solve them you equate these two both are same both are same so I can say 2 upon p plus q must be equal to 1 upon 3 7 by 21 is 1 upon 3 p plus q is equal to 6 I can say that I am in a class can I call you back I can call you back something later so it would be the same if we did 3 by 2p minus q is equal to you will have to use both you will not be able to solve this together in in 1 1 this is p plus q is equal to 6 equation number 1 you got p plus q is equal to 6 then if you equate these two what will you get you will get 3 upon 2p minus q is equal to 1 upon 3 again so this c if you see there are 3 2 equations what a is equal to b is equal to c that means a is also equal to c so use both conditions right 2p minus q is equal to 9 is it not how simple now actually we have not learnt how to solve linear equations but you can you can do tricks add both these equations you can just take p is equal to 6 p is equal to 6 whatever whichever way so this is equation so equation LHS LHS can be added RHS RHS can be added so hence add so p plus 2p how much and q minus q 0 6 plus 9 15 so what is p 5 so p is 5 if p is 5 put it back into here q is p is 5 q is 1 it is where why did you not say this the method where she first equated 2 by 3 plus q and 3 by 2 minus q yeah that is what I did but that will be complicated you can do that but that will be complicated but then we will have to get only one equation I mean like you get p is equal to 5 q and then you can just use it yeah so you can multiply the first one by 2 no solving equation we will see is it okay pair up next so I think now you are eligible to solve so hence let's go to did you learn graphics method of solution graphical method how to get on the graph solving equations by graph you did it in this year so now you plot the plot the lines both the lines and then where they are intersecting that is okay so we will talk about now we will come to now we learnt the conditions where we will get one solution no solution infinitely many solution now we will learn how to solve solving linear equations or solving a pair of linear equations in today solving a pair of linear equations what are the methods A is called graphical method okay graphical method there is a method to solve a pair of linear equations using graphs and then B is algebraic method in this method itself there are multiple in this method itself there are multiple such methods one is you know then there is something called elimination many times this they call it elimination by equating the coefficients on all that and then third is cross multiplication there are many more but we are not going into we will see if time permits in the course we will teach you there is something called Kramer's rule there is something called determinants method there are couple of more yes now so let's take this graphical method guys let's take this graphical method and let's do it using examples so let's say I have a x plus y equals 1 and x minus y equals 2 these are the two linear equations right and I have to solve it graphically okay so you remember these right okay graphical method and then algebraic methods are these okay so let us now solve these two or I will draw the graph say let us say this is my x and y coordinate okay now let us plot this x plus y equals to 1 see for linear equations if you want to plot a line you just need two points why because line is nothing but defined by two points only right now so this is what will you get as y 1 so hence the point is 0 comma is it did you understand this step guys I am saying put x equals to 0 no you can do that all but I am just telling you a quicker method put x equals to 0 y is 1 so the point is 0 comma 1 so where is that right now put y equals to 0 this is why so you got the line did you need to draw tables not to you next put x equals to 0 y is negative 2 so it is 1 somewhere here put y equals to 0 x is 2 this so point is point of intersection is somewhere 1.5 and 1.5 and no 1.5 and minus 1.5 and minus 1.5 and minus 0.5 see it works minus 0.5 1.5 and minus 0.5 minus 1 1.5 minus 0.5 is 1 no hence this point is 1.5 comma point of intersection solved now what is the problem with graphical method that takes too much time it takes a lot of time you need a graph but now using graphics calculator you can do a little quickly but let's say if you are doing it manually what is the biggest problem accuracy why because let's say here the best advantage was all the coefficients were integers what if it was something like this horribly looking root 7 x minus root of 251 y equals to root 13 37 how it is not a polynomial anyway it is not a polynomial it is an equation anyway next is let's say root of 21 by 6 x minus what is your birthday birthday 18 December 18 December so 1 8 1 2 what is your birthday so divide by 17 11 y equals what is your birthday 13 September all of you are second half fifth gen fifth gen so first half 0 5 0 1 what are you 7th always divide by 0 7 0 8 now solve graphically fun huh now you will really enjoy this okay even if you put x equals to 0 and try to find out why you will get nightmares so it becomes little difficult right if the coefficients are irrational then graphical method is not accurate so you should not be using logarithms you will have to use logarithms neither you will have to use calculators or you will have to use logs and again accuracy is the logarithms logarithms are magic minds we did it with this here we will teach you okay now same equation by substitution no that one same equation why it is too easy that one birthday equation so now we are talking about same equation x minus what about the equation by the way equation so now I am talking about substitution substitution as the name you have to represent or express one in terms of the other and then use that in the second equation not back into the first equation you will get 0 equals to 0 okay many people have this accurate out yeah so you for example what are the equations tell me x minus x plus y is equal to x minus y is equal to 2 right so let us let us say you express y in terms of x so y is equal to 1 minus x is it not now put this back into the same equation say you will get 1 is equal to 1 no point so hence you have to do what you have to use this so what will it be x and y so this y is that so 1 minus x so that means x minus 1 plus x equals 2 so 2x equals 1 point I do not know what point I will be understand now the moment I get x equals to 1 point 5 what do I do putting any one of these equations and find the other variable right so 1 minus 1 point 5 so y is okay this is called substitution substitution you guys are okay next next method is elimination but for elimination we will not use this simple looking use that I will fulfill your dream wait so now next set of equation is now I am talking about elimination how do we what is elimination you eliminate one of the variables how do I do it let us say we have 2x minus 3y equals 7 and 6x let us say 5x minus 2y equals 4 so how do I eliminate means I have to eliminate one of the variables choose any variable y eliminate y x you have to eliminate y but y sir because you multiply it with a less number of degrees so it is not for the category so anyway I will tell you the full method so let us say I want to eliminate x see if I want to eliminate x then I have to make sure make do something so that the coefficient of x here either are same or are having opposite sign that is it so hence what do I do I multiply this by 5 and multiply this by 2 so what will I get I will get 10x minus 15y equals 35 this one 10x minus 4y equals 8 and then we do subtraction if the sign is same then you subtract if the sign is opposite then you just simply add so hence subtract this will become plus this will become minus so this is 15 minus 15 plus 4y is minus 11y equals 8 35 minus 8 is 27 27 so y is negative 27 upon correct now how to find out x you substitute and find or eliminate again now this time eliminate y no this is much easier I am telling you this is much easier you multiply this by 2 and this by 3 so multiply this by 2 you will get you multiply this by 2 you will get 4x minus 6y equals 14 and this one 10x minus 6y you have to multiply by 3 15x minus 6y equals 12 now subtract again so you will get minus 11x equals 2 so x equals 2 is it easier now check if it is correct linear equations you will face a lot of you will make a lot of silly errors so be very very careful so let us check so let us put here in the first one 2x is so 2 into minus 2 into minus 2 by 11 minus 3 into minus 27 by 11 this must be equal to 7 let us check so it is nothing but minus 4 by 11 and this is 27 3 is minus 81 so plus 81 by 11 this is 77 by 11 which is check elimination done last cross multiplication also now let us solve this by cross multiplication cross multiplication is nothing but what we did for consistency you remember a1 b2 minus b2 a1 all that so let us I will give you that trick for solving let us say the same but before you cross multiplication you have to ensure a few things so now I am talking about cross multiplication cross multiplication first of all write these cross multiplication what is it it looks like a chain cross better so c by os n2 n2 nitrous oxide yes by any chance you inhale that anyways come back so what can I say about this equation first thing is represent the two equations in the standard form what is that so 2x minus 3y minus 7 equals okay 5x minus 2y minus only equals it is not that you can solve like this also but then follow one process whichever you are finding now what is a1 tell me a1 equals to a1 only a2 very good minus 3 go pick up v2 shreya c1 minus 7 anurag c2 so I cannot see this negative 4 okay now then what do you do you write x then you write y then you write 1 y then what do you do you start with v1 v2 like that so this is a shorthand now I am telling you the technique how we arrive at it what we were discussing in the consistency well apart where x equals to something by something remember that expression so we will try and prove that again but just for the sake of understanding the process let us learn the process and then get into okay so x is this write v1 by v2 then write c1 c2 then again write a1 a2 and then again write v1 v2 like that in cycles so vcav like that start with v go to c go to a go to v in that process it is cyclical right vcav now this first okay and this second remember like that top left to right bottom first then this this and what does it mean it means simply this you write x upon v1 v1 c2 minus v2 okay equals y c1 a2 minus c2 a1 equals just a minute upon a1 v2 minus this is the now how do I find out x now so what do you put up minus so this is the formula we will come this is just how to remember is how to remember because this is a complicated thing how will you remember you cannot really mug it up so just to have a mind map for that this is a trick and then what is the what is the final value then so x is equal to how much so x upon this is equal to what upon this isn't it so what will be x you will create these two what will you get x equals v1 c2 minus v2 c1 upon a1 v2 minus it will be what it will be substituted or it will be you need to solve for that now why you are doing cross multiplication go what go by this method only so y is equal to this equals to this y is equal to c1 a2 minus c2 a1 upon a1 v2 minus so for both we need to do cross multiplication yes for both you need to do cross multiplication let's check the previous one take care so let's now tell me so this is so much more complicated this is there it will not be for these kind of equations this is much easier if you didn't the value find out the in the calculator okay now tell me guys what is x v1 c2 tell me what is v1 minus 3 into c2 minus 4 minus v2 minus 2 minus 2 into c1 minus 7 divide by a1 into minus 2 then minus a2 5 into minus 3 4 12 minus minus minus is minus 14 minus minus minus is minus it is minus 4 plus 15 which is minus 2 upon did we get the same value last time so y denominator will remain the same what is y c denominator is same in both case what is y c1 minus 7 into minus 4 minus c2 minus 4 into by 11 so it is nothing but minus 35 plus 8 by 11 which is nothing but negative 27 by what is the same value but what is the third method so what is to be remembered this map x y1 v1 c1 a1 v2 c2 a2 then cross multiply this side first top down left to right down v1 c2 minus v2 c1 then c1 a2 minus c2 a1 then a1 v2 minus a2 how would you do this in another way in the textbook so can you derive it how will you do it if it is on this side this side no you told me this was a trick so how would the normal way go so basically this is the formula you have to maga this is the formula you have to maga but is it convenient to maga how do we arrive at it you remember by doing consistency we got x equals to this and we equated this to equal to 0 why then it will become unreal so that is how and then if you deploy same thing back to the equation you will get y value but the formula is complicated this is the general formula for any equation any pair of equation it will work and we did derive it isn't it sometime back when we were dealing with consistency we arrived at this point remember by that now but there we restricted ourselves only to x now if you put this x value in any of these equations you will get the y value it will be little complicated mathematics do you want me to do this I can solve that a pair of okay anyone is hungry now how do how much chocolate do you need anyone else you have only 3 pieces you will have to distribute please anyone wants to eat sanota no is that your chocolate it's okay anyone wants sanota I can share anyone wants say it oh yes you give half the chocolate there is more oh yes where is my can you see text I am standing here I am standing here I don't know for Priyanka's mother she likes she likes I love your she is working here on Monday oh yes it has to be both of them both of them looks a bit unclear otherwise yes it has never like that so I recently saw a picture of you like last year where I am like I had your card in my pocket look at it I had your card in your pocket every 2 seconds what is this I had a credit card in my pocket you have not seen I have come to this place how many times go find is that door over there it looks like you are holding a chemistry book chemistry oh my God what is that what is that why do you have this you can hold it and it will fall off because people who they drop their phones a lot they can just randomly especially when you are like watching isn't it when you are seeing them oh ok how you are so many times I would just like so stay away from phones sir sometimes it is required I need to check this I have heard these excuse a lot ok next sorry it is almost over you guys can't go back alone what late 8 o'clock 8 o'clock by cycle on H3 I have to go till 4 by afternoon 7 o'clock by 5 yes in the morning definitely I know in the morning definitely not that far in the morning definitely I know sir we always go alone but my cycle was punctured yeah so there was a one support a time here cycle but lost yeah yes sir I lost my key solved oh my God I am sorry I am a farmer I am not a custom whichever you feel like solve 2x plus 3y equals 7 2x plus 3y equals 7 and 6x plus 5y equals 11 and my dear friend 6x plus 5y equals 11 this is minus 1 by 4 x is 1 by 4 check check 1 by 7 yes sir all of you got it what is the answer minus 1 by 4 equal to x and y is equal to 10 by 12 10 by x in decimals is 0.25 x in decimals will be 0.25 correct minus 0.25 correct next cross multiplication method solve like that write down 3x minus 5y equals 20 so no write down 8x plus 5y is 9 8x plus 5y equals 9 and 3x plus 2y equals 4 8x plus 5y equals 9 and 3x plus 2y equals 4 sir we have to use cross multiplication 8x plus 5y equals 9 and 3x plus 2y equals 4 we have to have it we have to have it i will be asking done no looking up that means done otherwise you will be looking on your done irritating sir i know your expressions irritating shut up what are you trying to go pick as we bold done no sir which method are you using cross multiplication okay we use alphanage no this sunday you come down at 9 o clock here sunday on saturday we will do maths i got x is 2 x is minus 2 sorry i have been it done definitely there will be one question in boards using solve equations using cross multiplication then what will you do x is minus 2 and then you will get a unda what i got x is equal to x is because it is minus 20 x equals to minus 2 correct sir how is it minus 20 minus 2 minus 2 not minus 20 no i am minus 20 plus 18 are you using the formula then b1 into c2 it is minus 4 did you get y is equal to 5 did you get y is equal to 5 no wait i have asked for it sir is y is equal to 5 yes so every right should be negative 2 should be negative 6 which will make that that so next class we will be doing problem solving how to convert what problems into there is no restriction sir 20 word problems are easier eventually you will have to solve this what happened veda no worries take it out how is it minus 2 i got you are not taking you are not taking it to the other side sir can i please leave thanks for coming there is a dustbin outside sir you have to stop sir you have to stop sir i have sir sir this code formula come over here right right me sir gulmar sir gulmar sir we do not have class code because you three we would like you would like sir is it like no no why 9th graders 10th graders all are there 11th graders are there sir but typically i might not be able to come on Wednesdays what time i had your discussion with you yeah no worries we will see so like after june what what is the time like just after school sorry same time same time why class code is going on right we will have different time why sir x is minus 2 and y is 5 what is 5 brilliant very good so which one is wrong why sir why did this happen why is it like this but how do you do i don't think he wants to come specially specially but he actually classes on saturday or no saturday friday is there but saturday is a practice class only so sunday sunday so kathike i don't think he wants to come specially at this time because we were sad to play that day bye sir