 Okay, what else what else you got brother so you're halfway through this one What's the scoop with this one? This is again equation of a line and systems of equations is that? Yeah, go to it. Was that number five? Okay, so take a look at this Yeah, let me erase this So what we got so take a look at this now that we know what an equation of a line is right Okay So here's this here's this here's a line right Let's call this equation line one and let's say this is why one is equal to Mx plus B. Let's call this M1 and B1 and Y x1, right? It's just and subscripts in mathematics. It's like a last name is It's distinguishing this y from another y we might be talking about right So let's say we had this line that represents some type of relationship, right? If we draw another line, let's say here's another line All right, let's call this line two and Because this is line two the y and the x's and the m's and the B's they're different, right? Because this is the relationship as compared to that the only place they have the same x and m and Same x and y is here, right? So we're going to call this y2 is equal to m2 x2 Plus B2 does that make sense? what's important to us is To find out when does this line cross this line and They cross here, right? agree, okay This point it's a point it has an x and a y agree Now because this point exists both on this line and on this line This is the only location Where x1 and y1 are the same thing as x2 and y2 Right, they have the same values So this could be x1 and y1 or x2 and y2 Oops, y2, which means they're both x and y. They're both the same agreed. Are you okay with that so far? so How do you find out where these two lines cross? How do you find out what this point is? the way you find out you manually force Y1 to be equal to y2 because at this point Y1 is y2. Is it not? Okay, so if y1 is equal to y2 At this point then all we have to do to find out what they are What x and y are we're just going to set y1 equal to y2 So set y1 equal to y2. You okay with that? So what that means is we're going to set this because that's y1 equal to This which is y2 Does that make sense? So we're going to go m1 x1 plus b1 is equal to m2 x2 plus b2 Okay, so far Now if you know what this line is then what information do we know From this equation that explains this line What are the two things we need to know to know what an equation of a line is? The m and the b, right? If we know the m and the b for a line, we know how that line behaves Should we do an example? Okay, take a look at this. Let's do this in pink. Hopefully the pink comes out Okay. Yep. So take a look at this. What if I said graph the following line y is equal to negative 3x plus 5 Okay, how do we graph this line? We just want to graph it. We're not trying to solve for anything So if whenever you're trying to graph a line y is equal to mx plus b you go to the y-intercept and Then from that point you do what the m tells you to do So whenever you're trying to graph a line No, no, but yeah on the y-axis, right? So you go to the y-intercept To the y-axis and you go up five. Let's say this is one two three four five That's your y-intercept for this line and From here you do what the slope tells you to do. What is the slope telling you to do? So, what does that mean if the slope is negative three? What's what's what's the definition of a slope is rise over run, right? So you'll always have to think about it as a fraction, right? So if it's rise How do you write negative three as a fraction? How do you know negative three over negative three is one? had one right So any number can we put over one, right? So the slope m is negative three over one agree So to graph a line you go to the y-intercept, which is five, right? Here's five and then from the y-intercept You do the slope Negative three over one so you go down three rise is negative and I always use the negative with the rise up and down, right? one two three down and one over and The bottom will always be to the right if you use the negative just for the this so one over Let's assume that's one So this line is this You just graph that line Right, okay It's got for you For any line all you need to be able to graph a line is the m is The b and the m the y-intercept and the slope So if they've given you this equation that equation You're gonna know what the m is and you're gonna know what b is you're gonna know what m is and you're gonna know what b is So these numbers are just numbers these letters are just numbers. So in here, that's a number That's a number. That's a number. That's a number and at this point x1 is equal to x2 So they're also both the same x's So what you can do is just isolate the x Take a screen cap of this. We're gonna do that question that you just brought up on screen. Okay So we're gonna deal with the problem you have So keeping that visual in mind and by the way, there's one other thing I want to show you there is Three things that can happen when you're trying to solve a system of equations, which is what that question is asking you So there's basically three things that can happen when you're trying to solve a system of linear equations System of linear equations means you have two or more lines and you're trying to find out where they cross, right? One of them is if the lines cross each other, they'll be a solution another one is They may be parallel where there's no solution Or it may be the same system. It may be the same line and there's infinite number of solutions So when you're trying to solve this one, you're gonna get an x and a y. You will get an answer For this one, there is no solution And for this one, there's an infinite number of solutions Okay, this one has one solution solution No solutions, infinite number of solutions. Those are the three things that can happen When it comes to just talking about lines, right? So I'm going to erase this So let's write down the two equations that they gave you. What are they? 2y equals 8. That's your first equation. Let's put a 1 here. What's your second equation? x plus y is equal to 13 So whenever they give you these equations, you always have to keep in mind To graph a line, you want to put it in terms of y equals mx plus b So you have to rearrange these to have it as y equals mx plus b So how so basically it means you're going to get y by itself, right? So what do you need to do? Not minus. This is already minus so plus x. So you're going to grab the x and bring it over plus x agree So now we got 2y What's what's the what's the two doing to the y? It's multiplying. So it would be division and if you're doing bed mass Do addition, subtraction first and then multiplication and division, right? They have the same weight because you're solving now Right. So you're going the other way Sam dib, I guess they call it. I don't know Right So this becomes I'm going to rearrange this because I want it to be x first and then the b Plus eight now you still have to get y by itself So now you divide this by two and you got to divide each side by two, right? So i'm just going to divide each each one of these terms by two Okay, so this is x y plus 13. So our first equation really becomes y is equal to x over two is one One over two x plus eight divided by two is four. That's our first equation So you're rusty on your algebra Yes, that's what happens when you don't do mathematics for a couple of years you get rusty Right The second equation they gave you was x plus y equals 13. Is that correct? Yeah, so all we got to do to get y by itself Is move the x Over here minus x. So our second equation is really y is equal to And negative x means negative x I'm just going to rearrange it with x being the first because you want that to be there plus 13 This is our second equation right So graphically let's graph these so we get a visual of what's going on right The way we graph this is let's graph equation one first Equation one says the y intercept is four and the slope is one over two Equation two says y intercept is 13 and the slope is negative One because there's always a one in front if there's a negative number there, right So this becomes let's graph equation one y intercept is four one two three four And the slope is one over two slope is rise over run And they're both positive, right? So why Uh, the rise is one. So you go up one and then over two one two. So here's our first line Approximately, right? And our second line is the y intercept is 13 one two three four five six seven eight nine ten eleven twelve thirteen So here's our y intercept And the slope is one negative one over One rise rise is negative now over one. So one down one over Just over here So this was equation one and this is equation two and The sub the red side is always The negative one I applied for going down The right is positive left is negative Up is positive down is negative and I always and I always use the negative to go down Okay, so what we got right now is You can't go to down to the left, but you only have one negative number here, right? If the slope is negative one over one Or let's just call it here if the slope is negative one, right Let's assume our y intercept is still 13. This is 13, right The slope is negative one We need the slope to be a fraction rise over run. So what's negative one over one, right? Now You only have one negative here. You can use this negative to go either down or you can use the negative like this To go left Right up to you. Let's put this one on here first right now This one says negative one over one. So we're going to go down one That's the negative taken care of we used it up. We can't use it again, right? And then one here is positive. So we go right one right So that's there Let's graph it using this slope One over negative one. They're the same slope by the way, right? So if this if we're going to use one over negative one one is positive Right, so up is positive. So from here, we're going to go up one and then the one here is negative So we're going to left one Left one you end up at the same place. It's the same line Right, but for for me, I always use the negative for going down I don't use it for really going left very seldom very seldom Okay, it's just it's just convention for me because one reason I do that is whenever you have a number That's negative. Let's say negative five over four. You usually When you're trying to crunch numbers, you're usually writing negative four negative five over four You really never put the negative with the denominator. It makes calculations a little bit more difficult So the whole thing is you want to make things simple, right? And that's where practice and algebra comes up There's certain things you'll learn to make life easier as you grow older, right? Does that make sense Okay, so take a look at this so graphically The solution to this system of equations and this could represent something in real life and this could represent something in real life, right this could be I don't know anything, right? It could be hey if you travel for this long when do you cross a path over here if you Exercise this much. How much weight do you lose, right? Like if you're increasing your exercise so much, it could be if you eat less food, you lose more weight if you study more Well, hopefully your mark correlates with going up, right? So this is graphically We got the solution and the solution is here, right? This is the solution That x and that y is the answer, right? You're okay with this Okay. Now take a look at this This is visually appealing We see what's going on this line crosses this line here That's the solution to the system of equations, which means This x and y if we plug it in here, it'll make this equation true And this same x and y if we plug it in here will make this equation true This point x and y whatever that is Will work for this system and it'll work for this system Right if we go over here if we use the x and y here this x and y is only On this line. It's not on this line. So if we put this x and y whatever this might be Into this equation it doesn't work, right? It gives you a bogus answer Okay So for us visually we see what's going on, but what we really need to do is find exact values of things, right? So how do we find exact values of things? What we talked about At this point This y is the same as that y and that x is the same as that x So all we have to do to solve this equation is Set this y equal to this y because they're the same y at this point. They're the same y agreed So all we do we just set y1. I'm just going to call it y1 equal to y2 y1 equal to y2 Okay, that means I set this equal to this So all I'm going to do is going to go half x plus four is equal to negative x plus 13 You're okay with this So as soon as you do that You you're basically telling yourself Telling the system you want to be at this x coordinate at this x At this x We're not mixing the equations. We're setting them equal to each other We want to find out when they equal each other. So we're saying at this x at this point Both this y and this y are the same, right? And at this y This y whatever this y is Both this x and this x are the same. So basically at this point This equation is equal to this equation. They cross each other Right, so all we're doing we're saying, okay We have two equations. We have two unknowns the x and the y So what we need to do is we have to combine those two equations To end up with one equation with one unknown Right if we end up with one equation with one unknown we can solve that equation So how we're going to solve that equation? We're going to eliminate one of the variables or we're going to use the property that this y Is equal to this y By setting them equal to each other because they are equal to each other, right? So all we're going to do if you want to think about it, we're going to substitute This for this y. So we're subbing in because it's the same y So we set this guy equal to this guy. I hope that makes sense right You want me to give you an example here? What if I said y is equal to five? And w is equal to five right Can you set y and w equal to each other? Because they're the same So y is equal to w That's what we're doing here. We're saying y one Right does that make sense? Yeah, it's just it's just doing things mathematically Translating English into mathematics, right? We're we're saying they're both equal That just click for you. Yeah, just kick it down to simple Anyway, we'll do if something doesn't make sense. Let me know and we'll try to make a simplified version of it, right? But that's This is They're both y They're both exactly they're both equal to each other. So if that's the case Then we can set this equal to this and right now we notice that the only variable we have here is x So all we have to do just solve for x So the way we solve for x is I don't like fractions in my equations when I'm solving them So first thing I do I'm going to multiply this equation by two. Are you okay with that or do you want us to move things around first? Yeah, do you know why I'm I'm going to be doing that? Well, because one over two gets rid of that. Actually, let's not do that because that's a It simplifies things but for you, you don't need to know it right now. Let's do it this way When you're trying to solve for the x It means get the x by itself. So what you're trying to do is combine like terms, right? So you're going to grab this x bring it over You're okay with that What does it become? So what do you need to do you need to get rid of this x? So you're going to go plus x and you're going to go plus x here And then I'm going to take the numbers to the other side. I'm going to go minus four So right now what I have Is a half not to x half x plus x Is equal to 13 minus four is nine What's a half x plus x One half 1.5 x right you got a half of something plus one of something Is one and a half of that thing so you got 1.5 Can you see this if I write it here here? Let's write it here So you got 1.5 x plus nine So how do you find out what x is divide? Divide by 1.5 now the way I do division you divide this side and this side But what I'm doing is just drawing a big line and dividing the whole equation by 1.5 Saves me a little bit of work right or writing down anyway So x is equal to nine divided by 1.5. I'm just going to do this manually Three over two divided by so it means nine times two over three three goes into nine Three times three times two is six. So this should be six Okay, you can do it with your calculator. It doesn't make a difference. So x is six You're okay with this? So this value here let's do this in pink So that's the x part of our answer, right? So this value here is six this x is six Right We need to find the y. How do we find the y? Let's just leave it there. So it's not confusing So we have x is equal to six. So we just found this value here x is equal to six All right x is equal to six All right, is that approximate what it should be one two three four five six sure Oh, our graph is very generic. It's like very it's not very accurate, right? But it looks to be right, right What do you think the y is going to be one two three four five six seven seven or eight or something? How do we find yeah, how do we find the y now this point? How do we find the y? We have x is equal to six. How do we find it? What's y? No, we just found the x to find yourself on a Cartesian coordinate system You need the x and you need the y you need the y associated with the x Think of the x and the y as married They're joined at the hips, right If you have an x you got a y if you got a y you got an x Right, so how do you find out what the y is? What's y equal to? What's y equal to we've got two equations. What's y equal to at this point? One half x plus four it's also equal to negative x plus 13, right? So How do you find what y is the equations tell you what to do? Just take x and plug it into here or here you're going to get the same answer All right, you just plug it back into the equation Does that make sense? Okay, so how do we find the y how do we find out what the y is? We plug Six in here or in here it doesn't make a difference. Let's plug it in here first This becomes I'm going to erase these guys. Okay We got x is equal to six, right? So we have Our solution is x equals six Let's take the six and plug it in this equation and this equation just to confirm That we get the same y we have to get the same y if you get different y's Then you did something wrong if we plug it in here In equation two you're going to get y is equal to negative six plus 13 negative six plus 13 is seven So this equation equation two Is saying y is seven This point here is seven so that's seven Right this equation that one equation one Equation one is y is equal to one over two Times six because we're plugging the same six plus four two goes into six three times three plus four is seven Equation one gives us the same seven seven y is seven And Yeah, we just have to find one but I usually if there's a system equations I plug the x into both of them Just to confirm as a check to make sure you did it right Yeah, yeah for sure right If you're writing a test or handing an assignment you do want to get the marks Right, and you you do want to make sure you did it right because there's a lot of places where Here you could make simple little silly mistakes and Have the wrong graph or the wrong Equation written down or something right so it's always good to Run a little minor little detour. It's not even a detour. Just check just to make sure you're going You're doing the steps properly. You're getting the right answers right easy Easy Okay, you need you need to be very very familiar With the Cartesian coordinate system how to graph lines on an x y axis You really need to be comfortable with that. Okay Super super important. That's what economics is taking equations taking systems and graphing them And trying to find out when you're going to be and we wanted to one of the best examples. Can I erase this? You sure you got it copied you got a screenshot Yeah, take a screenshot of it. Okay Okay, now take a look at this if you want to know where this comes into play for economics So just imagine if we had You know, if you're we're doing economics, right? That's this is what you're learning right now, but basically economics. They're telling you it's math 10 right now, right? So just imagine if you had a company that Your expenses were like this, right? That's an equation for your expenses Right, sometimes you're paying more depending on fuel costs electricity costs or whatnot sometimes products They're in high demand. So the prices goes up. Sometimes they come down or whatever it is, right? Let's say you start it off here time zero This is when your company started and this is I don't know A few years projection. Let's say this is a five-year projection if someone comes up to you and said, hey, when did you become profitable if these this graph is your expenses And your revenue coming in let's say you started selling stuff right off the bat day one You wouldn't have that much revenue, right? You wouldn't have that much revenue revenue would be very low, but hopefully over the years Your revenue would increase right Now these aren't lines they're polynomials right defined as polynomials, but basically you could tell Any potential investors or if you're looking at this if you're doing financial reports or something like this You could come out and say at this point In this year, right? What year would that be? That's five. That's 2.5. That's about three years You could say Three years into the company's life. You became profitable right because Your profit or your revenue. This would be your revenue your revenue Increased crossed over your expenses Does that make sense? That's what you're doing, but right now Because they're just teaching you the stuff. They're saying think of this as a line and Think of this as a line Where do they cross? When did you become profitable? Okay, is that cool? Does that cover basically get you going to where you want to be? Yeah Yeah, I know we just did what hour and a half of mathematics basically But all of it review from grade 10 that you should have known and if you had If you took this course That you're signed up and is it econ 101? What is this course? econ 101 So just imagine if you took econ 101 in grade 10, you would have already finished off These two assignments Right Grade 11 you would have laughed at this assignment. You would have gone. They want me to do what? Right, so the mathematics you learn in high school is very very important It applies many places. They're just beginning to teach you where it applies They should have taught this to you in grade 10 and grade nine Or a grade 11 10 and 11, okay So basically, uh, send me a message when you're ready to do more Oh, you're welcome, man Oh, yeah Refresher refresher It's good though Now that you learn it, you're not going to forget man You won't you won't when you learn this stuff mathematics when you're a little older when you have more appreciation for it You don't tend to forget it Yeah, yeah Good so listen, uh, send me a message tomorrow next day whenever and We'll do another session. Okay this week for sure Okay Good luck brother Bye Real problem. Okay. Bye