Variables, expressions, & equations

when we're dealing with basic arithmetic we see the concrete numbers there we'll see 23 plus 5 we know what these numbers are right over here and we can calculate them it's going to be 28 we can say 2 times 7 we could say 3/4 in all of these cases we know exactly what numbers we're dealing with as we start entering into the algebraic world and you probably have seen this a little bit already we start dealing with the ideas of variables and variables there's a bunch of ways you can think about them but they're really just values in expressions where they can change the values in those expressions can change so for example if I write if I write x plus 5 this is an expression right over here this can take on some value depending on what the value of x is if X is equal to so if X is equal to 1 then then X plus 5 our expression right over here is going to be equal to is going to be equal to 1 because now X is 1 it'll be 1 plus 5 so X plus 5 will be equal to 6 if X if X is equal to I don't know negative 7 then then X plus 5 is going to be equal to well now X is negative 7 negative 7 is going to be negative 7 plus 5 which is negative 2 so notice X here is a variable X here is the variable and it's value can change depending on the context and this is in the context of an expression you'll also see it in the context of an equation it's actually important to realize the distinction between an expression and an equation an expression is really just a statement of value a statement of some type of quantity so this is an expression an expression would be something like what we saw over here X plus 5 the value of this expression will change depending on how this what what depending on what the value of this variable is and you could just evaluate it for different values of X another expression could be something like I don't know Plus Z now everything is a variable if Y is 1 and Z is 2 is going to be 1 plus 2 if Y is 0 and Z is negative 1 it's going to be 0 plus negative 1 these are these can all be evaluated and they'll essentially give you a value depending on the values of each of these variables that make up the expression an equation you're essentially setting expressions to be equal to each other that's why they're called equations you're equating two things an equation you'll see one expression being equal to another expression so for example you could say something like X plus 3 is equal to 1 and in this situation where you have an equation with the only one where you have one equation with only one unknown you can actually figure out what X needs to be in this scenario and you could might even do it in your heads what plus 3 is equal to 1 well you can do that in your head if I have negative 2 plus 3 is equal to 1 so in this context an equation is starting to constrain what this what value this variable can take on but it doesn't have to necessarily constrain it as much you could have something like X X plus y plus Z is equal to 5 now you have this expression is equal to this this other expression 5 is really just an expression right over here and there are some constraints if you if someone tells you what Y and Z is and you're going to get an X if someone tells you what X and Y is and that constrains what Z is but it depends on what they what the different things are so for example if if we said Y is equal to 3 and Z and Z is equal to 2 then what would be X in that situation so if Y is equal to 3 and Z is equal to 2 then you're going to have this the left-hand expression is going to be X plus 3 plus 2 it's going to be X plus 5 this probably right over here is just going to be 5 X plus 5 is equal to 5 and so what plus 5 is equal to 5 well now we're constraining that X would have to be X would have to be equal to 0 but the important point here 1 do you hopefully you realize the difference between expression and equation equation essentially you're equating two expressions the important thing to take away from here is that of variable can take on different values depending on the context of the problem and to hit the point home let's just let's just evaluate a bunch of expressions when the variables have different values so for example if we had the expression if we had the expression X to the X to the Y power if X is equal to if X is equal to five and Y is equal to two Y is equal to two then our expression here is going to evaluate to well X is going now going to be five X is going to be five Y is going to be two it's going to be 5 to the second power or it's going to evaluate to 25 if we change the values if we said X if we said X let me do that same colors if we said X is equal to X is equal to negative two and Y and Y is equal to three then this expression would evaluate two it would evaluate to let me do it in that so it would evaluate to negative two that's what we're going to substitute for X now in this context and Y is now three negative two to the third power negative two to the third power which is negative two times negative two times negative 2 which is negative 8 negative 2 times negative 2 is positive 4 times negative 2 again is equal to negative 8 is equal to negative 8 so you see depending on the values of these aquarii we could do even you know we could do more complex things we could have an expression like the square root of x plus y and then minus minus X like that if X is equal to let's say that X 0 is equal to 1 and y Y is equal to 8 then this expression would evaluate to well where every time we see an X we want to put a 1 there so we would have a 1 there and you would have a 1 over there and every time you see why you would put an 8 in its place in this context we're setting these variables so you'd see an 8 so under the radical sign you would have a 1 plus 8 so you'd have the principal root of 9 which is 3 so this whole thing would sim both I in this context when we set these variables to be these things this whole thing would simplify to be 3 1 plus 8 is 9 principal root of that is 3 and then you'd have 3 minus 1 which is equal to which is equal to 2