Alright, everyone. Let's move on here. So solving equations is going to be a super important part of the course. Throughout the course, you're going to be solving lots of different variables and lots of different equations, so it's good to have a good handle on that. Basically, the way we're going to do that is we're going to use different operations like addition, subtraction, multiplication, stuff like that. And the whole thing we're trying to do is isolate variables. Sometimes it's x. Sometimes it's going to be other variables. We're going to use all different kinds of letters throughout the course. And, basically, what happens is whenever you do an operation to some kind of an equation, you always have to perform operations to both sides of the equation. So, for example, if I've got 2x-3=0, I know I solve for x. What I have to do is I want to get x by itself, and I have to do a bunch of operations. So the first thing I have to do here is I have to distribute the 2 to everything that's inside the parentheses, and you end up with 2x-6=0. Now I want to solve for x in this equation, so I want to get x by itself. And I have to isolate it by moving stuff to the other side. And whatever you do to one side, if I add 6, then I have to add it to the other side. You always have to do that. So I get, just 2x6 over here, but I'm not done yet. I have one last thing to do and I have to divide by 2. But remember, whatever I do to one side, I have to do to the other. So always just, you know, be really, really careful that you do that. When you work this out, what you're going to get is x equals 3. So the general steps to do this is you're going to sort of distribute any constant inside of parentheses. You're going to combine like terms, and we've already seen how to do that. Then you're going to group the x terms and constants on opposite sides, isolate and solve for x, and then you can always just check your solution by replacing this solution that you get back into the original equation and make sure that you get the right answer. Alright? So let's just do the second example again. I've got 1x4 + 5 = -3. Again, I want x by itself, so I want to sort of move everything that's a constant over to the other side. What I end up with is 1x4, and then when I move the 5 over, I have to subtract 5 and then subtract 5. And this actually ends up becoming negative 8. Now what happens is, in order to get rid of the 14, what I'm going to have to do is I'm going to have to multiply by 4 on this side of the equation. So I have to multiply by 4, but then I also have to multiply by 4 here as well. What you'll end up getting here is x equals negative 32, and now you've solved for this equation. If you want to check, you can always just plug this negative 32 back inside of this equation just to make sure that it makes sense. Alright? That's how to solve equations for different variables. Super important skill. We'll be using it a lot throughout the course. Let's move on here. Now a lot of the course will also just involve graphing. So graphing was usually going to involve something like plotting points or equations on a rectangular coordinate system. This is basically just sort of like the 2-dimensional plane that we have over here. And the way that we do this is you're going to be given some kind of an equation, and we're going to have to just plot a bunch of points. So the way we do this is we isolate y to the left side. So for example, if we have an equation like this, you want y by itself on the left side because then what you're going to do is you're just going to plug in values of x anywhere you see x inside, and you're going to calculate a bunch of values, and then you can basically just get ordered pairs. So for example, if you plug 0 into this equation, 0x2-3∙0+2, both those things go away, and you end up with just 2, so your y-coordinate is 2. So the ordered pair is 0 comma 2, and we can plot that here on the grid. So 0 comma 2 is going to be this point over here. Remember, ordered pairs are always x comma y. You do the x value first and then the y value. So just a couple of more, if you plug in 1 1x2-3∙1+2, you actually just get 0, so the ordered pair is 1 comma 0. And actually, the same thing happens if you plug 2 into this equation. You're going to get 22x,4-6+2, also works out to just 0. So this is actually 2 comma 0. So it means we can plug the next two points in. 1 comma 0, which is right over here, and then 2 comma 0, which is right over here. And then finally, if you plug in 3, what you're going to get is 2, so the coordinate is 3 comma 2, and then 4 comma 6. If you plug all these things in together, what you're going to see here is you're going to get these points, 3 comma 2 and then 4 comma 6, which is all the way up here. So, basically, what you're going to do is you're just going to isolate. And then when you calculate a bunch of y values, you're just going to pick a bunch of x values to sort of plug into the equation, and then you're going to plot those points that you've gotten from step 2. And then finally, you're just going to connect those points with a line or a curve or something like that. In this case, this isn't a line. In fact, it's actually going to look something like it looks like this, which you may remember is actually the shape of a parabola or a quadratic equation. Alright? So that's basically how to plot equations and plot points and things like that. Alright? Let's move on.
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0. Math Review
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