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everybody mystical here with simple math
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today we're gonna dip our toes in the algebra waters by looking at
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very simple equations what we call one-step equations
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when I started teaching my students pretty good middle school you know they
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measure angles calculate area no problem the wanna put this on the board
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I could ever really figure out why it's just a variable
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and a simple definition for variable is a letter that represents an unknown
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you know X or Y your a RBI
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and these variables will they're just mysteries
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they're just puzzles and when you see
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in Equation like this you should really start thinking about it like
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a sentence like this when you see that sentence you've already solved the
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your brain just doesn't automatically it's something you like to do it
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something or do naturally
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so here we have to make these X's seem a bit more
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if you were getting ahead of ourselves a little bit let's take a look at a really
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easy equation so what does this mean
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what are those two little lines in the middle trying to tell me
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while historically people have thought about equations like balance scales
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you could see one over here when the two sides are the same
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its green when weight is added to one of the platforms
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skill turns red indicating that the equation is unbalanced
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when you remove that Wade goes back to being an equilibrium
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so let's take a look back at our equation over here
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three plus three equal sex
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the first part of that equation is three plus three so let's go ahead model that
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you can see the skills become unbalanced we have to finish the equation to make a
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on the right hand side
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we have sex so let's go ahead and place a six
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on the right hand side of our scale and you can see they balance each other out
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three and three is six though the same
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poor actually saying something a bit bigger here see that six on the right
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that's not really a sex that's six
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is actually six little one blocks and the three blocks on the left
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will each of those blocks is actually worth 31 blocks
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so instead of thinking about it as adding two threes and getting 6
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you can think about as combining two sets up ones
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so what this equation is really telling you is that
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3 ones plus three ones equal sex once
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this may seem kinda trivial but remembering
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inner jours as groups have ones is really helpful for algebra
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and that is the mystery of X we're trying to find out how many ones are
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inside that axe because it's different every problem
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it might be one in one problem the next problem X might have three ones inside a
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and the next problem X might have well
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a buncha ones in it so that sorta but we're trying to figure out how many ones
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axe so let's look at this really easy algebra equation
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you might be able to do this in your head let's examine some other concepts
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will start by modeling on the scale we have 1x
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and three ones on one side and on the other side
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we have seven ones so to solve this
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when I might do is try taking one from each side at a time
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I'll take one from both sides
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and it still imbalance I take the same thing from both sides again
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it still amounts not try that one more time
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and I'm still in balance look with the skills telling us 1x
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X equals four and how do we get that
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will be subtracted three from both sides let's take a look at the numbers
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see what we did was we subtracted three from both sides
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effectively cancelling out the plus three on the left and that left us with
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X equals four so algebra is sorted the process
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love removing clutter from the side with axe I want to make this equation tell me
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equals and the way I do that is by removing the plus three by subtracting
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three from both sides
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this is the basic process of solving algebraic equations
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getting excites elated on one side okay here's another equation
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expose D'Qwell 16 we want to deal with that plus 8
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so we're gonna deal with the by removing it from both sides we get
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X equals eight K pretty simple
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let's take a look at a different kind a problem
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markers is going to be a little different well it's
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head on over to scale star modeling so we've got an X in there
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on the left hand side and we've got that
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-2 who had been a deal with negative numbers
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some negative to I mean we know that one's way down the side
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that a end up on may be negative ones lift up the side
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and actually can model that using this to -1 balloons
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the scale is still unbalanced because we haven't completed the equation
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on the right hand side we have three ones there
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now a balanced but what we gonna do with those -1 balloons
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man actually believe it or not a negative one balloon can be popped
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by a one-block if you throw it out it just right
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they actually meet and cancel each other out so let's add another one
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to their side okay but now we're on balance because we removed those two -1
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balloons that were letting us up
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and we did this before but we didn't explicitly mention it to an old rule for
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equations what you do to one side you must do to the other
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so taking a look back at this scale we
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added to once the left side we're gonna have to add to once the right side
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and look at what the equation is telling us X equals five
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alright so if we go back to the math on the board here
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the I didn't clear from the Exide is the -2
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I can't do that by adding to cancel it out
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so I had X equals five
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the one I have an ex -5 here to deal with their -5 by
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adding five to both sides and that tells me that XE cool sex
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all right let's take a look at another type your here we have 3x equals nine
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oh man well that three out in front there
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that three is what we call a coefficient and a coefficient is a number that
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multiplies a variable
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so in the case of three acts the coefficient is three
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and by now you've seen many different ways to show multiplication
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the old ext I'll the dot style in the coefficients style
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the thing is all three of these things are worth the exact same there are three
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we don't really use the first al anymore because the multiplication symbols easy
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isn't that just like math teachers to teach you something and then later tell
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you not to do the thing they taught you to do
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I guess I probably would have used the dot but you might have gotten that
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confuses the decimal point
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well anyway let's move on so 3x
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equals nine from was won over the scale try modeling manner
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so on the left side we have three axes okay
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and on the right side we have nine or nine ones
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and that balances out as an equation
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so I guess what I would do
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if this was real would be take
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to the exits away and then start removing ones from the other side until
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but the problem is you can't do that with a math test you don't have a scale
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from any certainly not have an x-box
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so we have to accomplish the same effect conceptually
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cell from let's put the left side appear
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in a3 and I can actually do that to the right hand side too because there's nine
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women what if I divide both sides
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by three huh well economic sense
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let's take a look at it on the board so I've 3x equals nine
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divided by both sides by three well that leaves me with XE calls 3
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cool huh we can actually think about this another way
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go back to the equation and think about breaking these up and threes
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we're actually taking one-third up both sides
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and if we go back to the blackboard here we can take a look at the equation
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3x equals nine will multiply both sides by one over three
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we also know that's true because the rules a multiplication
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one over three times X is the same thing is dividing expires 3/
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and a more general sense one over any number
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times a variable equals that variable divided by their number
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knows the numerator is always one in that case
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okay so we have one third time's both sides the question let's go ahead and
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rate this outcome long in style 3x equals nine
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but multiply both sides by one-third and to do that we're gonna have to convert
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and nine into fractions so we'll get 3x over one equals nine over one
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now and multiply both sides by one over three follow the rules from multiplying
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and then reduce those fractions and we figure out that
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X equals three and it's probably good to talk about the work we've done here
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each new line represents one mathematical move that we did to move to
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when we chart our math work this way someone can follow and see the exact
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motions we made to solve the problem
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UK all this problem should be easy now
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we have to deal with that six out front the way we're gonna deal with that sex
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is by dividing so we can divide both sides by six
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6x over six equals 36
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over sex and I we reduce those factions to find XE cool sex
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really simple what's goin' think about the other way
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so instead of dividing by six will think about it is multiplying it by 16
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so 6x it was 30 secs and we follow through the math process just like we
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I love my students with complaint said why are you doing this you're confusing
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man just wanna divided by sex
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well that's fine she just one by by sex go and do that
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okay wow 1/2 X equals three
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house is gonna be a little tougher here on how we gonna model
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missiles what's move over to the scale here in attempt to model the so
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hafer than tax equals three ones
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so I have happen axe
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how am I gonna figure out what a whole access okay well here's an idea
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if I double what's on the left side ago from having happen axe
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to having whole axe what's going on balance me again as in
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more no idea the other side alright you remember the rule
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what i do to one side I must do to the other source inside
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doubled the left hand side would have to go ahead and double the right hand side
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and we go back to math you'll see 1/2 X equals three
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we took both sides and we multiply them by two
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let's go unsolved that out so I have 1/2 X equals three
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multiply both sides by two convert everybody in the fractions
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to the fraction multiplication hand then reduce your fraction
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giving us the answer that X equal sex but we can also think of this problem in
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instead of thinking about it like doubling the left side
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let's think about it like adding happen X
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so I've got 1/2 X plus another half for the tax
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and on the right hand side all I would have to do is that happen acts as well
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and this may seem kind of foolish but let's take a look at the math
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and look at what it shows us so here's our original equation
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half with an axe equals three ones we're gonna
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add half an axe to both sides here's what that looks like me read out an
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okay but we know something special about half an axe
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we know that one happen X equals
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3 ones so we can replace that happen the next in the top right corner there
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with three once and this may seem like a ridiculous way for solving this problem
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what it does show you that as long as you follow the rules and algebra you
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really can't mess up
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even if you thought this had to do with adding rather than multiplying still
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have gone the correct answer
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let's look at another problem one-fifth X
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equals four need to deal with that one-fifth out front from
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let's think about this in a different way so what the problem is told me
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is that one fifth tax equals four but I need to know is what is a whole ex worth
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so what I need to do is convert that one over five into one
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okay so what multiplied by one over five equals 1
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well here's a little him for your okay
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so hope you got net will be five over one
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one over five times five over one equals five over 5
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which is one and there's are correct multiplier so is multiply both sides by
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and follow up the mathematics follow the rules and multiplying fractions and
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you'll get 5x over five Eagles 20 over one
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or exceeds twenty circle your answer
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the understood why kids don't like circling the answer at the bottom
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sorta like clinton flag on everest well
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covered a lot of different kinds a one-step equations here I think on are
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was to stretch a question here before we finish up okay for over 3x
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Eagles 8 hot all different attractions but scarier than the ones who looked at
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let's think about it like we thought about the last problem K
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we have therefore over 3x Eagles a we won final 11 X is worth
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so we have to turn for over 3 into one and the way will do that is by finding
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forever three is reciprocal
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or its multiplicative inverse K like that one for free
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so through a hint in here okay for over three times 3 over 4
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is gonna give you twelve over twelve which is one so there's the reciprocal
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okay let's use our equation here follow now
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for over three axes was a multiply both sides by 3 over 4
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convert those guys infractions so we've got four x over three
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an 8-over one times 3 over 4 do the multiplication
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to the reduction we find out that axe
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okay not too bad so this is just one step equations really easy ones
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next time we'll delve into things that are a bit tougher so
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have yourself a great day
