Two-Variable Equations - Video Tutorials & Practice Problems

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Equations with Two Variables

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Hey everyone. So up until now when we've solved equations, they've always been of one variable like X plus two equals five. But equations in this course won't always be so simple. Instead of just one variable, a lot of equations in this course will now start to involve two variables in the most common ones you'll see are gonna be X and Y. So for example, instead of X plus two equals five, now we're gonna have X plus Y equals five, I'm gonna show you the main difference between these two types of equations and how to solve them and more importantly how to visualize them. So let's go ahead and get started here. So with equations with one variable, like for example, X plus two equals five, you really just trying to find the value of X that makes this equation true. So for example, you would isolate and solve for X, you would subtract you from both sides and your answer ends up being three. That's the number that makes this equation true the way we visualize it. And just by plotting it on a one dimensional number line like this point over here, now let's take a look at X plus Y equals five. So now we actually have two variables, X and Y both are letters that can be replaced with numbers. So how do we solve this? Well, one thing you might notice is that if you try to isolate one of the variables, it's not really going to be much help because X equals five minus Y doesn't give you any information about what X or Y could be. But let's think about what this equation means. Can I think of two numbers that when I add them together, I get five? And actually, yes, I can because for example, if X equals one, what do I have to add to one to get to five? Well, why could be four? So that's a solution to this equation? Both of these numbers here, this combination makes this equation true. But is this the only combination of numbers that makes this true? Well, actually, no, because what if X is equal to two? If X is two, then Y could be three. And that also satisfies this equation X plus Y equals five. In fact, you could have another one, X equals five already and then Y could just equal three. So the whole point here is that with equations with one variable, the solution was always just one number. It was a single points on a one dimensional plane versus when you have equations with two variables. As we can see, you end up with many solutions that satisfy this equation. And the way that we represent represent them is not on a number line, but actually as points as ordered pairs X comma Y on a two dimensional plane. So basically what happens is I can take this X equals one, Y equals four and turn it into an ordered pair, one comma four. This becomes two comma three and this becomes five comma zero. And we can plot these on a two dimensional number or a two dimensional plane. So for example, one comma four, this point satisfies this equation uh two comma three that also satisfies this equation and five comma zero also satisfies this equation. So all these things actually satisfy, in fact, there's actually an infinite number of solutions because if you sort of keep this pattern going on here, any points that is basically on this little, little line over here will actually satisfy this equation. So any points along this line is a solution to X plus Y equals five. All right, that's the main difference between us between these two types of. Now, how do we actually use this in problems? Let's take a look at our example here. In our example, we have this equation X plus Y equals five. So this is the same exact equation we've been using before. And we want to first determine if these points satisfy the equation we've got three comma +24 comma +10. Comma zero and negative one, comma three. So in order if you're, if you're ever asked to determine whether uh points, xy points satisfy an equation, what they're really just asking you to do is they're asking you to replace the X and Y values to check if the equation is true. Remember that's what satisfying equation means. So what we're doing in part A is we, when we have a coordinate like three comma two, this is really just giving us an X and A Y value and we just plug it into this equation X plus Y equals five. And we just figure out if that makes the equation true. So does it, well, if I have, if I basically just replace X with three and Y with two, then three plus do two does in fact give me five. So this definitely does satisfy the equation. Let's get started with the next one which is four comma one again, does this satisfy X plus Y equals five? Well, if I replace X with four and Y with one, this does indeed get me five. So both of these points do satisfy the equation. What about zero comma zero? I do zero, comma zero into X plus Y equals five. And I'm just replacing X with zero, Y with zero and zero plus zero does not give me five. So this actually does not turn out to be a solution of this equation. And last but not least What about negative one? What about negative one comma three? So into X plus Y equals five. Well, this is gonna be negative one plus three and this also does not give me five, it gives me two. So it turns out that not all of these points are gonna satisfy the equation. Um The first two work, the second two didn't work and now we're gonna go ahead and plot them right. So as we can see the equate, the 0.3 comma two is gonna be here. That's the 0.3 comma two, right? That's this point over here. We've got four comma one. This is also going to be um a solution to the equation and then we've got zero comma zero, which is over here and then negative one comma three. So if you've seen here, there's a pattern that happens basically when points do satisfy an equation, when they do make the equation true, then they are on the graph of that equation versus when they do not satisfy an equation. As we've seen with these points over here, they are not on the graph of that equation. So that's the basic difference between uh equations with two variables versus one variable. All right. So hopefully, that made sense. Thanks for watching.

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concept

Graphing Equations of Two Variables by Plotting Points

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Hey, everyone. So up until now when we've seen equations with two variables, their graphs were already given to us. For example, we saw that X plus Y equals five was a line that looks like this. But sometimes the graphs aren't gonna be given to us and you're gonna have to go graph them yourself. That's what I want to show you how to do in this video. And basically the way we're gonna do this is by plotting points, the graph an equation we're gonna calculate and plot a bunch of ordered pairs that make the equation true. And the way we do that is by plugging a bunch of values into this equation and we're gonna get points out of it, those points we can take and plot them on a graph and then connect them with a line. That's basically what I'm gonna show you how to do in a step by step way. Let's get started. All right. So we're just gonna actually dive right into our equation or our example here. So I can show you how this works. So we're gonna graph this equation negative two X plus Y equals negative one and we're gonna do this by creating a bunch of ordered pairs using some X values that are already chosen for us. So here's the first step, you're gonna take an equation. And if it's not already isolated to one side, you're gonna have to isolate Y to the left side. So basically, you always want Y equals and then the rest of the equation on the right side. And the reason to do this is because you're gonna plug X values into this equation and it's just gonna be way easier to see what Y is if it's isolated. All right. So how do we do this? How do we take this equation over here and isolate Y? Well, basically, all I have to do is just take the two X, the negative two X and move it to the other side. So I'm gonna add two X over here and add two X to both sides. And basically what I end up, end up with is when I bring it down here, this is actually just gonna end up being two X minus one. So now that Y is isolated, I can move on to the next step. What I wanna do is I want to calculate Y values for a bunch of a bunch of X values that I choose. And usually 3 to 5 is enough. So I've got my table here that we've got a bunch of X values that are already chosen for me negative two and so on. And what I'm gonna do here is I'm just going to replace the X inside of this equation with whatever that X value is. And then I'm basically just gonna get a Y value out of it. So for example, I'm just going to take this negative two and I'm going to plug it in, right? So this is going to be two times negative two and then minus one. So what does that give me? What y value does that give me? Well, two times negative two is negative four and negative four minus one is actually just gonna give me negative five. So this is going to be negative five over here. What that means here is that I plugged in two, negative two and I got out of it negative five. So that's an ordered pair. The ordered pair that satisfies this equation is negative two. Comma negative five. You're just gonna repeat this a bunch of times over here. So that's what we're gonna do. So this Y equals negative or sorry two times negative one minus one. So this is gonna be negative two minus one which is gonna be negative three. So this is gonna be negative two. Comma negative 30 is always a pretty easy point to do because usually what happens is you just replace X with zero and one term just kind of goes away. So this whole term just cancels out and then you get the negative one, which is just negative one. So the ordered pair is zero, comma negative one. And then over here we're gonna get two times one minus one. So that's two minus one and that's just gonna give us positive one. So this is gonna be one comma one. And then finally, we've got two times two which is four and then four minus one is equal to three. So the ordered pairs, two comma three. All right. So you just have to basically just plug and chug a bunch and fill out your table. A tables are gonna be really good at organizing your information. And the last thing you're gonna do or sorry, the third thing you're gonna do is you're gonna plot these xy points that you've just gotten in this table over here. Um And you're gonna plot them on your graph. So let's go ahead and do that. So I'm gonna take the ordered pair negative two comma negative five and that's gonna be right over here. I'm gonna take the point. I'm sorry. This is gonna be whoops, that's gonna be negative one. I'm sorry. Hopefully, hopefully you caught that. So negative one comma negative three is gonna be this point over here. Then we've got zero comma negative one. So that's over here and we've got one comma positive one which is over here. And finally we've got two comma three over here. So I just end up with a bunch of points with a plot them all. And the last thing I do is connect them with a line or sometimes a smooth curve. In this case, what happens is I can actually I and actually plot a line that goes through all of these five points. And so this is the graph of the line. So this is the graph of the equation Y equals two X minus one or negative two X plus Y equals negative one. That's how to graph these types of equations. The very last thing I want to sort of mention here is that most of the time these X coordinates won't be given to you. And if you're not given X values to evaluate, then you can always just feel free to choose your own and usually 3 to 5 is enough. All right. So thanks for watching. That's it for this one.

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Problem

Problem

Graph the equation $y-x^2+3=0$ by choosing points that satisfy the equation.

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Problem

Problem

Graph the equation $y=\sqrt{x}+1$ by choosing points that satisfy the equation. (Hint: Choose positive numbers only)

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concept

Graphing Intercepts

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4m

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Everyone. So throughout some of your graphing problems, you may be asked to identify these things called intercepts. And that's what I want to show you how to do in this video. And basically what we're gonna see here is that intercepts are really just special places or special points where the graph crosses either one of the X or Y axis. So I'm gonna show you how to identify those types of points and also some special things about them. And we'll do an example. Let's get started. All right. So in this equation, I've got, or in this diagram here, I've got two lines, I'm gonna call this one A and this one B, both of these graphs actually cross the X and Y axis. If you look at line a line A crosses the X axis over here, line B crosses the X axis over here. These things are called the X intercepts because that's where they cross the X axis. So the X intercept is where the graph crosses the X axis. Now, similarly, these graphs also cross the Y axis over here and over here. So these points over here are called the Y intercepts that is basically the Y value where the graph crosses the Y axis. All right. So pretty straightforward here, if you take a look at line, a line A crosses over here. So this is an X intercept uh where X is equal to negative two. And over here, this is where X is equal to negative four. So it's basically just the X value where that graph crosses the X axis. All right. And over here, what we've got is we got Y equals four. And over here we've got Y equals negative three, right? So that's what the Y intercepts and X intercepts are. Now what's sort of sort of confusing about this is what happens to the other value. So for example, in the X intercepts, what you're gonna see here is that the Y value at any point along this line is always equal to zero because you're basically right on the line, you have no height above it. So the X intercept is where across the X axis, but it's also where the Y value is equal to zero. Now, similarly, what happens for the Y intercept is that the X value is going to be zero. So notice how here any time you're on the Y axis, you're not to the right or to the left of the origin. So the Y intercept is where it crosses the Y axis, but the X value is always zero. So it's kind of like the opposite All right, that's really all there is to know about X and Y intercepts. So let's go ahead and do a quick example. So here we're gonna take this graph and we're gonna write the X and Y intercepts of the graph that's shown below here. So remember the X intercepts where it crosses the X axis. If you look at this line or this sort of curve shaped over here, this curve crosses the X axis twice once over here. And what's over here? So what happens here is that the X intercepts are just the values is just the number of the X coordinate. So in other words, our X intercepts are negative three and positive five, you don't have to write the ordered pair, you just have to write the numbers. And what happens for the Y intercept. Well, this graph also crosses the Y axis over here. That's gonna be a Y intercept. So your win intercept over here is gonna be Y equals negative four. So that's really all there is to it, right? So let's take a look at another example here where now the, the question is written a little bit differently because instead of having to write the X intercepts and Y intercepts, we just asked to find the intercepts of the graph below. OK. So let's take a look at that. So in this graph here, we've got this sort of like circle looking thing um and notice how this graph actually crosses the X axis right over here. So basically what happens here is when they're asking for the intercepts, they're actually not asking for the X value, they're asking for the ordered pair. So that's sort of the, the real confusing thing with these sort of problems. Um But basically, if they ever ask for an X or a Y intercept, whenever they actually reference the letter by name, then all you actually have to do is just write the X or Y value. But if they just ask you to find the intercepts, like they have like they're asking you for in the second example, then you're gonna write the ordered pair. So really all you have to do or the second example is just write the ordered pairs for these two numbers. And in this case, it's just gonna be two comma zero and then also so to write and, and then it's gonna be four comma zero. So these are the intercepts of the graph, but these are the X and Y intercepts of this graph, right? So that's really all there is to intercepts. Thanks for watching.