Video transcript
When we're dealingwith basic arithmetic, we see the concretenumbers there. We'll see 23 plus 5. We know what these numbersare right over here, and we can calculate them. It's going to be 28. We can say 2 times 7. We could say 3 divided by 4. In all of these cases, weknow exactly what numbers we're dealing with. As we start entering intothe algebraic world-- and you probably have seenthis a little bit already-- we start dealing withthe ideas of variables. And variables, there'sa bunch of ways you can think aboutthem, but they're really just values in expressionswhere they can change. The values in thoseexpressions can change. For example, ifI write x plus 5, this is an expressionright over here. This can take onsome value depending on what the value of x is. If x is equal to1, then x plus 5, our expression right over here,is going to be equal to 1. Because now x is 1. It'll be 1 plus 5, so xplus 5 will be equal to 6. If x is equal to, I don'tknow, negative 7, then x plus 5 is going to be equal to--well, now x is negative 7. It's going to be negative 7plus 5, which is negative 2. So notice x here is avariable, and its value can change depending on the context. And this is in thecontext of an expression. You'll also see it in thecontext of an equation. It's actuallyimportant to realize the distinction between anexpression and an equation. An expression isreally just a statement of value, a statement ofsome type of quantity. So this is an expression. An expression wouldbe something like what we saw over here, x plus 5. The value of thisexpression will change depending on what thevalue of this variable is. And you could just evaluateit for different values of x. Another expression could besomething like, I don't know, y plus z. Now everything is a variable. If y is 1 and z is 2,it's going to be 1 plus 2. If y is 0 and z isnegative 1, it's going to be 0 plus negative 1. These can all be evaluated,and they'll essentially give you a value dependingon the values of each of these variables thatmake up the expression. An equation, you'reessentially setting expressions to be equal to each other. That's why they'recalled equations. You're equating two things. An equation, you'llsee one expression being equal toanother expression. For example, you could saysomething like x plus 3 is equal to 1. And in this situation where youhave one equation with only one unknown, you canactually figure out what x needs to bein this scenario. And you might evendo it in your head. What plus 3 is equal to 1? Well, you could dothat in your head. If I have negative 2,plus 3 is equal to 1. In this context, anequation is starting to constrain what valuethis variable can take on. But it doesn't have tonecessarily constrain it as much. You could have something likex plus y plus z is equal to 5. Now you have this expression isequal to this other expression. 5 is really just anexpression right over here. And there are some constraints. If someone tells youwhat y and z is, then you're going to get an x. If someone tellsyou what x and y is, then that constrains what z is. But it depends on whatthe different things are. For example, if we said y isequal to 3 and z is equal to 2, then what would bex in that situation? If y is equal to 3and z is equal to 2, then you're going to havethe left-hand expression is going to be x plus 3 plus 2. It's going to be x plus 5. This part right over here isjust going to be 5. x plus 5 is equal to 5. And so what plus5 is equal to 5? Well, now we'reconstraining that x would have to be equal to 0. Hopefully you realizethe difference between expression and equation. In an equation,essentially you're equating two expressions. The important thingto take away from here is that a variable cantake on different values depending on thecontext of the problem. And to hit the pointhome, let's just evaluate a bunch of expressionswhen the variables have different values. For example, if we had theexpression x to the y power, if x is equal to 5and y is equal to 2, then our expression hereis going to evaluate to, well, x is now going to be 5. y is going to be 2. It's going to be 5to the second power, or it's going to evaluate to 25. If we changed the values-- letme do that in that same color-- if we said x is equal tonegative 2 and y is equal to 3, then this expressionwould evaluate to-- let me do it inthat-- negative 2. That's what we'regoing to substitute for x now in this context. And y is now 3, negative2 to the third power, which is negative 2 timesnegative 2 times negative 2, which is negative 8. Negative 2 times negative 2 ispositive 4, times negative 2 again is equal to negative 8. We could do evenmore complex things. We could have an expressionlike the square root of x plus y and then minus x, like that. Let's say that x is equalto 1 and y is equal to 8. Then this expressionwould evaluate to, well, every time we see an x,we want to put a 1 there. So we would have a 1 there,and you'd have a 1 over there. And every time you see a y,you would put an 8 in its place in this context. We're setting these variables. So you'd see an 8. Under the radical sign,you would have a 1 plus 8. So you'd have the principalroot of 9, which is 3. This whole thing would simplify. In this context, whenwe set these variables to be these things, this wholething would simplify to be 3. 1 plus 8 is 9. The principal root of that is 3. And then you'd have 3 minus1, which is equal to 2.
