 Hi, I'm Zor. Welcome to Unisor Education. I would like to start a new topic in algebra called systems of equations. System of equations. This is just an introductory lecture. There will be others related more to how to solve the system of equations. So this is just general talk about what are the systems of equations. I do recommend you immediately after you listen to this lecture to go back to Unisor.com and read again the notes for the lecture. I think it will be very helpful. Alright, let me first remind about what actual equations are for. Equations are certain numerical or algebraic rather conditions which we established about certain unknown variables. So if you have some condition about unknown variable which we call x, let's say 1.5 times x equals to 6. Now what does it mean? It means basically that there is some unknown variable which if multiplied by 1.5 gives 6. Now the next question is what is this unknown variable? Well, we have already learned that using certain transformations with the equation we can come to a conclusion x is equal to 4. This is also kind of equation but it's an explicit equation which actually shows the value of the x. So equation is a condition which we impose on certain unknown and which allow us to find the value of this unknown variable. Okay, that's clear. Now let's just do it in a little bit more formal representation. This condition which we impose on our equation can be expressed as an algebraic function f of x. And I will put it equal to 0 because I have to put some equation. Whatever the algebraic expression which contains x is, I can always reduce it to this particular form. Something like if 2x plus 6 is equal to 57, I can always say 2x minus 51 equals to 0. So it's easy to put it equal to 0 to give certain standard notation. So we are considering equations of this particular format that some function being applied against the variable will give the variable of 0. Now it's a condition and if we are able to convert this condition into something like this, that means that we have found a solution. So if we are saying that x is equal to a is a solution, it means that if a is substituted into this function, then I will have an identity, a numerical identity 0 on the left and 0 on the right. Alright, so now talking about systems of equation, the next step is, well, what if I have more than one variable? x is just one variable, what if I have two, three variables, n number of variables, whatever. That's actually how we are approaching the systems of equation. So what's important is that we have certain number, n whatever, of unknown variables and certain number of conditions these variables must satisfy. If we have these conditions satisfied with certain values of the variables, these values are called solutions. So let me just give you an example which is not necessarily like algebraic expression of our condition. Here is the one, for instance, x plus y is equal to 2 and another condition is x and y are integers greater than 0. This is also a condition. So I have one condition expressed as an algebraic expression and another condition is basically certain words which describe certain restrictions on our variables. Now, is it enough to solve the equation? Well, let's just think about it. If x and y are integer and positive, it means they are at least one or greater. So if both are equal to one, both are minimal, that actually satisfies the first algebraic equation. If any one of these is greater than one, then we will not have the first condition satisfied, which means this particular number of conditions, two conditions actually, also allow you to find the value of variables. Can I say that this is a systems of equation? No, because one is an equation and another is just some kind of words which specify the condition. So if all the conditions are specified in this algebraic form, then we can talk about systems of equation. So all conditions must be expressed as an equation. Now let's talk about certain examples first. Now, before I just gave you one simple example of two variables and two conditions which allow us to find the solutions. Now, these algebraic expressions, which we are talking about as participating in the systems of equation, can be absolutely different. Now, let me give you one particular example. x plus y is equal to 3, 2x plus y is equal to 4, and 3x plus 2y equals to 7. Now, is this a system of equations? Well, yes, it's a system of three equations with two unknowns. However, let's just take a look at one particular property. Oh, by the way, it doesn't look like something is equal to zero. It's easier for me to do it in this way, but obviously I can put it x plus y minus 3 is equal to zero plus 2x plus y minus 4 is equal to zero, and same thing. So it doesn't really matter whether I put constant on the right or on the left. Now, what's kind of peculiar about this particular system of equations? Well, look at the third equation. If you will add one and two conditions together, x plus 2x is 3x, y plus y is 2y, 3 plus 4 is 7. So what does it mean? If I found a solution to this system of equations, so if I found the value of x and the value of y, which satisfies these two, this will be automatically satisfied. So this condition is basically, it doesn't bring any new restriction onto our variable. It does not help us to find the unknowns. It does not bring any new information into whatever our knowledge about our unknown actually is. These two, they are independent, and we will talk about what is independent actually. But you feel that these two are completely unrelated to each other. And this one is dependent on the previous two, which means it's really extraneous. Usually, system of equations, whenever it's like presented to be solved, usually it does not contain anything like that. However, I mean, just as a trick, you might always expect something. But just, let's just be very, very clear. If you have one of the equations derived in some way from the others, then it doesn't really bring you any new information, and it can be completely discarded from the system without any laws of the villages to solve the system. Alright, this is just an example, for instance. So let's now talk about a general form of the system of equations. As I was saying, system of equations is a set of conditions expressed in an algebraic form. Now, what does it mean in an algebraic form? It means there is some kind of function, f, that's called function number one, which depends on a certain number of unknowns. And the condition is like this. Now, we have another function, which is also dependent on the same n unknowns, etc. And you have function by the number m of the same set of unknowns. And all together, we are saying that all these m functions represent a system of m equations with n unknowns. This is general definition of the system of equations. I'm not talking about linear equation, quadratic equation. This is all completely outside of the scope of this introductory lecture. We're talking about equations in general. There is an equal sign, and that's why it's equation. Now, I put zero on the right just for a definitiveness. And some function on the left, algebraic function. Well, which means we can basically know what kind of manipulations to do with x1, x2, etc. We can add, subtract, multiply, use exponential function, logarithmic function, whatever functions we use and their composition, that's what it means, algebraic function. So, all these functions used in this particular equation on the left. And all together, it gives you a system of n equations with n unknowns. Now, just to be on a semi-precise note, I would like to say that in general, if you would like to find out the values of n variables, most likely you will need n equations. So, the correspondence between the number of equations and the number of variables should actually exist. Why is a different question? I mean, it's not necessarily that if you have n equations with n variables, then the system will have one and only one solution. It might not have it at all, etc. But, however, generally speaking, you should expect for n variables, which are known to you to have n conditions, because every condition might allow you to find one particular extra variable. So, you can just reduce the number of variables from n down to 1. And then, number of equations can be used for this purpose. And that's how you can solve the whole system of equations. Again, not necessarily true. There are systems of n equations with n variables which have no solutions, one single solution, k solutions, or infinite number of solutions. Everything happened. But in general, you should really expect that the number of equations and the number of variables correspond to each other. Okay. So, this is just a general definition of the system of equations. Now, the solution is if you find some set of values which being substituted into each one of these results in corresponding number of identities, all n should be numerical identities. If it is possible, then these a1, etc. an represent a solution. It might not exist at all. It might be only one unique set of values, or it may be two or three or k sets of values. It can be infinite number of values, but all of them are solutions. All right. Now, it's time to give you a few examples. So, I would like to give some examples of the systems which have no solutions, one solution, two solutions, whatever. That probably would help. All right. If we are talking about system of equations, by the way, I didn't really say it before, let me say it now. Question is what kind of values are unknown can take? So, basically, if you wish, where exactly we are looking for solutions? We can look among integer numbers, we can look among positive numbers, we can look among all real numbers, we can look among complex numbers, etc. So, whenever the system of equations is specified, it must also be specified what exactly is the set of values where we are looking for solutions. Now, if nothing is said about this, traditionally it means we are looking for real number solutions. Everything else, like if we are looking for complex solutions or we are looking for integer solutions or whatever else, is specified. It mustn't be specified. Because, again, traditionally, by default, if you wish, we are always looking for real number solutions. Okay. Now, I will assume exactly this. I will assume that we are looking for real number solutions unless specifically set otherwise. So, let me give an example of a few systems which have certain number of solutions. Okay. One, two. Okay. This is the system of two equations with two unknown variables, x and y. So, my question is what are the solutions of the system? The answer is there is none. Now, why there is no solutions? And, again, we are talking about real number solutions. Well, look at this. x square plus y square. x and y are real number. Positive, negative doesn't really matter. But x square and y square are always positive or zero, right? Now, can it be positive? No, because the sum will not be equal to zero. If the sum is equal to zero, it means each one of them is equal to zero. So, x is equal to zero and y is equal to zero. We basically get the value of various of our unknown variables, x and y, right from the first equation. As you see in this particular case, it's sufficient to have one equation with two variables, but it's such a specific type of equation that we can get the values without any other conditions. But now I have the second condition, that their product must be equal to one. Well, obviously, these are zero and zero, the product will not be equal to one. So, the second condition will never be satisfied if the first is satisfied, which means the system has no solutions, all right? No solutions. Okay, next. Next, I have an example with one solution. x plus y plus z is equal to 6. 2x plus y plus z is equal to 7. And 3x plus y plus 5z is equal to 20. All right. Now, in this particular case, this is an example of a system which has one and only one solution. Now, how can I find out this particular solution? Well, in this case, it's really easy. Let's do it this way, since if x and y and z are solutions, which means all these three equations are actually identities. Now, we can subtract from the second one, we can subtract the first one, and what do we see? y and z would nullify each other, so x and x, if I subtract from this, I subtract this, I will have only one x here, and 7 minus 6 would be 1. So immediately, I see that the x is equal to 1. The necessity of these two equations actually results in one and only one value for the x. Now, if I have that, I can always transfer my system of three equations into a system of two equations, namely, I will take this one, and instead of x, I will use 1, which means y plus z is equal to 5. y plus z, this is 1, this is 6. So y plus z is equal to 5. And from this, the third one, 3x is 3. So y plus 5z is equal to 321, so it's 17, right? So now, if I will subtract this one from this, I will get 4z is equal to 12, and z is equal to 3. Now, if z is equal to 3, then y is equal to 2. So I have basically solved, and we're not talking about how I solved it, it doesn't really matter, but I came to a conclusion that from these, necessary to have these three as values. Now you can obviously check and everything will work out. So this is just an example of a system of three equations, linear, by the way, because everything is in the first degree of power, which has only one solution. Now, next is, I want multiple solutions. x squared plus y is equal to 3, and x squared minus y is equal to minus y. Alright, this is the system of two equations with two variables. Again, solving this system very easy. I will add them up, and I will have what? 2x squared is equal to 2, 3 and minus 1, y and minus 1, minus y will nullify each other, from which x squared is equal to 1, from which x is equal to either 1, or minus 1, right? To get x squared is equal to equal to 1. Now, knowing two different values of x, I obviously have two different values of y, or maybe the same one, but it will be two different pairs. In both cases, x squared is equal to 1, so 1 plus something is equal to 3, that something must be equal to 2, right? And the same thing would be here. If this is 1 minus y is equal to minus 1, so y is equal to 2, the same thing. So I have two different pairs, two different solutions, one pair and another pair. So this is the case when one particular system of two equations with two variables has two different solutions, one solution and another solution. Okay, and finally, I wanted to give you an example when I have infinite number of solutions. Infinite number of solutions. x plus y plus z is equal to 6, 2x plus y plus z is equal to 7. What can we say about this? This is the system of two equations with three different unknowns. Let's just do it this way. If I will assign a new variable, y plus z is equal to t, then I will have x plus t is equal to 6 and 2x plus t is equal to 7. Subtracting from this, this, I will get x is equal to 1 and t is equal to 5. But t is y plus z, which means any combination of y and z, which sounds at 5, basically would satisfy this particular equation. So I have an infinite number of solutions. 1, then if z is equal to 0, for instance, y is equal to 5. So it's 5, 0. 1, for x it's still 1. Then if z is equal to 1 and y is equal to 4, 4, 1. This is also a solution. 1, 3, 2. And now I'm talking about the integer solution. So obviously there are all kinds of real number solutions. So anything, whatever z is, if y is equal to 5 minus z and z is anything, that's a solution. So all these triplets, when the x is equal to 1, z is anything, and y is equal to 5 minus z, all of these would satisfy this system of two equations with three unknowns as a solution. So this is basically an example of a system which has infinite number of solutions. And by the way, notice that this is a system with two equations with three unknowns. So it's again typical, again not necessarily, but it's typical that if number of equations is less than the number of unknowns, you might expect something like infinite number of solutions. Again, not necessarily because as I was saying, something like x squared plus y squared is equal to 0, this particular one equation with two unknowns has only one solution, x is equal to 0 and y is equal to 0. That's it. However, in most of the cases, you can expect that if number of equations is less than the number of unknowns, you might expect a lot of different solutions, maybe infinite. And now I will talk about something which I did touch before, a system of equations when one equation is derived from others. Let's just consider one of them. x plus y plus z minus 6 is equal to 0, 2x plus y plus z minus 7 equals to 0 and 3x plus 2y plus 2z minus 13 is equal to 0. Alright, we did actually mention this system of equations before. The only thing was I put 6 and 7 and 13 on the right. It doesn't really matter. So anyway, what's peculiar about this particular system of equations depends on the third one can be derived by the previous two by adding them together. You see, x plus x, 3x, y plus y, 2y, z plus z, 2, z minus 6 minus 7 minus 13. Excuse me. And as I was saying before, this is a case when the third equation doesn't really bring us any new information. So in theory, if anything satisfies the first two, it will definitely satisfy the third equation as well. So as if I don't have it, basically that's what it is. So this case is called a derived equation, derived from the previous ones in this particular case. And the system becomes basically under-determined. So you have less conditions than the number of variables and it's so much less that it results in the number of variables to have indefinite number of solutions, infinite number of solutions. All right. Now, what's very important about derived equations is, well, let's try to formalize it somehow. So let's go back into the original general form of the system of equations. Now, general form, as I was saying before, was some function of N variables is equal to zero, then another function, etc., and then function number N from the same. Now, what does it mean that the system is really containing certain equations derived from others? Well, derived how derived? I mean, that's not easy to define, right? Okay, here is something which I consider to be a decent definition. Let me just, for definitiveness, talk about derivation of the last one, f nth function, or equation actually, from the previous one. What does it mean that f nth of this, the nth equation, is derived from the previous m-1 equations? It means the following, that the function itself can be derived from the previous functions using some derivation function, f, which is supposed to be known, f1, 2, m-1. Where these are x1, x2, xn, x1, x2, xn, x1, x2, xn, etc. So, if my mth function can be expressed as algebraic function f, known function, from the other m-1 functions, that actually means that it's derived, and it means that basically it doesn't bring us any new information. The only thing which is really important in this case is that if x1, etc., xn are solutions, then this is equal to 0. Each of these is also equal to 0, which means that I should have the equation 0 is equal to f of 0, etc., 0, as a necessary condition for this derivation. So, what does it mean that the mth function is derived from the previous one? The mth's equation is derived from the previous one. It means that the function itself, fm, can be expressed as a function of functions, in this case, and there is this condition which function f must actually satisfy. Well, let's just go back to our original example and figure out what exactly... I don't need this anymore... what exactly I mean in this case. If I have these three equations as before, let me do it the way how I did it before, everything on the left, and 0 on the right. Okay. Now, as I was saying before, the third equation is a sum of these two. What does it mean? It means that our function f of two variables, only two, should be equal to u plus v. Where u is this, so f of x plus y plus z minus 6, and the second, which is 2x plus y plus z minus 7, is equal to sum of these, which is x plus 2x3, xy plus y2y, z plus z2z, minus minus 6, minus 13, and indeed we get this particular solution, this particular equation, this function. So the functions are related using this particular form of our derivation function. So f of uv is equal to u plus v is a derivation function. And the way how it's derived, basically I've just described. Now, let's check the second condition, this one. Well, indeed, if I will put 0 and 0 instead of u and v, I will have 0 plus 0, which is 0. So function really satisfies this particular condition. Now, what if this particular condition, f of 0, 0, 0 equals to 0, is not satisfied? Well, let me give you another example. Let's just assume that instead of these two, these three equations, rather, I have something slightly different. Instead of 13, I will have, let's say, minus fiction. Now, can I find a function which will connect these two to this one? Well, yes, indeed. How about this? x plus y plus z minus 6 as the first argument, then, as the second argument, I will have this function, I will have u plus v minus 37, which is 2 plus x plus 2x is 3x, y plus y is 2y, z plus z is 2z, minus 6 minus 7 is minus 13, and minus 37, and this is minus 50. So, I have found my function. This is my function, which is a derivation function which allows me to derive the third equation from the first two. However, how about this condition? Not at all. If I put 0 and 0, I will get 0, 0, 0 plus 0 minus 37. So, this condition is not satisfied, which means that derivation, I derive the left parts of these equations, but I cannot derive the right parts. 0 will not be derived in the same way as the unknown. So, basically, what it means is that the third is not really properly derived from the previous two. Basically, that's it for introduction. I just wanted to talk about certain terminology, about general concepts of what real systems of equations are, and I didn't really touch any specific types of the systems, like systems of equations of the first degree, linear system of equations, second degree or polynomial system of equations, logarithmic, whatever. This is the subject of subsequent lectures. But as a general introduction, I do recommend you to go again back into the Unizor.com notes for this particular lecture. Read it again. It's always helpful. Thanks very much.