Two-variable linear equations intro
- [Voiceover] What I'd like to introduce you to in this video is the idea of a Linear Equation. And just to start ourselves out, let's look at some examples of linear equations. So, for example the equation y is equal to two x minus three, this is a linear equation. Now why do we call it a linear equation? Well if you were to take the set of all of the xy pairs that satisfy this equation and if you were to graph them on the coordinate plane, you would actually get a line. That's why it's called a linear equation. And let's actually feel good about that statement. Let's see, let's plot some of the xy pairs that satisfy this equation and then feel good that it does indeed generate a line. So, I'm just gonna pick some x values and make it easy to calculate the corresponding y values. So, if x is equal to zero y is gonna be two times zero minus three which is negative three. And on our coordinate plane here that's-- we're gonna move zero in the x direction, zero in the horizontal direction and we're gonna go down three in the vertical direction, in the y direction. So, that's that point there. If x is equal to one, what is y equal to? Well two times one is two, minus three is negative one. So we move positive one in the x direction and negative one, or down one, in the y direction. Now let's see, if x is equal to two what is y? Two times two is four, minus three is one. When x is equal to two y is equal to one. And hopefully you're seeing now that if I were to keep going, and I encourage you though if you want pause the video and try x equals three or x equals negative one and keep going. You will see that this is going to generate a line. And in fact, let me connect these dots and you will see the line that I'm talking about. So, let me see if I can draw, I'm gonna use the line tool here. Try to connect the dots as neatly as I can. There you go. This line that I have just drawn, this is the graph, this is the graph of y is equal to two x minus three. So if you were to graph all of the xy pairs that satisfy this equation you are gonna get this line. And you might be saying, "Hey wait wait, hold on Sal, you just tried some particular points, why don't I just get a bunch of points, how do I actually get a line?" Well, I just tried, over here I just tried integer values of x. But you can try any value in between here, all of these, it's actually a pretty unique concept. Any value of x that you input into this, you find the corresponding value for y, it will sit on this line. So for example, for example, if we were to take x is equal to, actually let's say x is equal to negative point five. So if x is equal to negative point five if we look at the line when x is equal to negative point five it looks like it looks like y is equal to negative four. And that looks like that sits on the line. Well let's verify that. If x is equal to negative, I'll write that as negative one half, then what is y equal to? Let's see, two times negative one half, I'll just write it out, two times negative-- two times negative one half minus three. Well this says two times negative one half is negative one minus three is indeed negative four it is indeed negative four. So you can literally take any, any-- for any x value that you put here and the corresponding y value it is going to sit on the line. This point right over here represents a solution to this linear equation. Let me do this in a color you can see. So this point represents a solution to a linear equation. This point represents a solution to a linear equation. This point is not a solution to a linear equation. So if ex is equal to five then y is not gonna be equal to three. If x is gonna be equal to five you go to the line to see what the solution to the linear equation is. If x is five this shows us that y is going to be seven. And it's indeed-- that's indeed the case. Two times five is ten, minus three is seven. The point-- the point five comma seven is on, or it satisfies this linear equation. So if you take all of the xy pairs that satisfy it, you get a line. That is why it is called a linear equation. Now, this isn't the only way that we could write a linear equation. You could write a linear equation like-- let me do this in a new color. You could write a linear equation like this: Four x minus three y is equal to twelve. This also is a linear equation. And we can see that if we were to graph the xy pairs that satisfy this we would once again get a line. X and y. If x is equal to zero, then this goes away and you have negative three y is equal to twelve. Let's see, if negative three y equals twelve then y would be equal to negative four. Nega-- zero comma negative four. You can verify that. Four times zero minus three times negative four well that's gonna be equal to positive twelve. And let's see, if y were to equal zero, if y were to equal zero then this is gonna be four times x is equal to twelve, well then x is equal to three. And so you have the point zero comma negative four, zero comma negative four on this line, and you have the point three comma zero on this line. Three comma zero. Did I do that right? Yep. So zero comma negative four and then three comma zero. These are going to be on this line. Three comma zero is also on this line. So this is, this line is going to look something like-- something like, I'll just try to hand draw it. Something like that. So once again, all of the xy-- all of the xy pairs that satisfy this, if you were to plot them out it forms a line. Now what are some examples, maybe you're saying "Wait, wait, wait, isn't any equation a linear equation?" And the simple answer is "No, not any equation is a linear equation." I'll give you some examples of non-linear equations. So a non-- non-linear, whoops let me write a little bit neater than that. Non-linear equations. Well, those could include something like y is equal to x-squared. If you graph this you will see that this is going to be a curve. it could be something like x times y is equal to twelve. This is also not going to be a line. Or it could be something like five over x plus y is equal to ten. This also is not going to be a line. So now, and at some point you could-- I encourage you to try to graph these things, they're actually quite interesting. But given that we've now seen examples of linear equations and non-linear equations, let's see if we can come up with a definition for linear equations. One way to think about is it's an equation that if you were to graph all of the x and y pairs that satisfy this equation, you'll get a line. And that's actually literally where the word linear equation comes from. But another way to think about it is it's going to be an equation where every term is either going to be a constant, so for example, twelve is a constant. It's not going to change based on the value of some variable, twelve is twelve. Or negative three is negative three. So every term is either going to be a constant or it's going to be a constant times a variable raised to the first power. So this is the constant two times x to the first power. This is the variable y raised to the first power. You could say that bceause this is just one y. We're not dividing by x or y, we're not multiplying, we don't have a term that has x to the second power, or x to the third power, or y to the fifth power. We just have y to the first power, we have x to the first power. We're not multiplying x and y together like we did over here. So if every-- if every term in your equation, on either side of the equation, is either a constant or its just some number times x, just x to the first power or some number times y, and you're not multiplying your x's and y's together you are dealing with a linear equation.