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Pre-Algebra - 4.5: Solving 2-Step Equations

Added Jan 05, 2021 •

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Solve the Algebraic Equation (Isolate the variable) 1) Zero out the Constant (add or subtract to both sides of the "equal sign") 2) Convert the Coefficient to One (multiply by the multiplicative inverse)

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alright so 3.5 solving two-step equations but what is a model 445 but I'm going to do it first related to it so it's like a little warm up that goes with it so algebra tiles I could not find my algebra tiles so I have to use this right here so variables are rectangles and numbers are units of one so let me show you this right here so let's say I have the equation can you rent right now all right let's say we have the equation 3x + 2 equal 8 now is not nice way to show work is because it takes too much to draw but I'll show you cuz it's something simple so variables I have three X's exes are always rectangle tiles that's One X two x's three x's and this will also help us review like terms and unlike terms understanding why you can combine them no it's called an ex because look at this right here the length of this is X and the length of this is one the area multiplication is an area model length times width of a giraffe to the ride on a separate piece of paper so you can see it think about this I have a I have a let's call this a 2 by 3 does a 2 X 3 rectangular area is 6 this would be a 6 unit block because I cut the side into two and I cut this out of the three I have six little one unit blocks and the idea is in a variable a unit block is always one unit in wits I don't know how long this is that's why it's called the variable length x width x x one that's why this is called an Xbox Annex unit but we don't always have number units so that's how we go with simple basic units I have three X's + 2 A plus 2 so each one of these little squares represent a once unit Square it's a one by one square one plus one is two that equals 8 and this is why we don't like that model algebra tiles I got to do this I can do for 8 123 each one of these represents one but I got four runs and four ones that's eight one so 3 x + 2 equals 8 understanding how this works so the whole thing is to figure out what does your main goal your main goal is to figure out what does 1x equal what's the value of 1x cuz we got to figure out what it's worth that's what a variable is we don't know what this number is but we want to figure it out these are numbers they're Constance they're literally numbers Even though three is a number it's a coefficient it tells you how many variables you have I got three exit I will get this I could combine my ones and what 1 + 1 is the two that's why they're called like terms I can't combine constants variable that's what's 3x + 2 as an expression and an equal sign is like a wall I cannot combine things across the wall but I could move things across the wall yes thank you to show me the work will do it so this right here the next step we got to get so if you look at this right here and we had to move this over and I move this over but then you look at this right here how do we show subtraction do most people get stuck if you look at this right here I'm going to circle what matches on both side I got eight units of one I got two units of one and a 3 but if I have this right here and I got this right here I'm going to sit all they match they meant I have two that are here into that are here so let me subtract those away and if those are incoming notices right equal equal means balance that means I know these two are equal visually I can see them the same thing that means if I throw them away how do I throw away twos I subtract it so 3x + 2 - 2 + what you do on one side you do to the other 8 - 2 when I have left I got 3 algebra tiles of exes X x 3 X's equals look what one away is that one away this one away so I'm left with this so if you want to rewrite this expression 3x equals 6 what is 1 X so let me realign all of this stuff over so this right here I wrote too early if I had reached if I turn them around and I went this way here is One X 2X 3x another way of writing 3x + 3x equals 3x equals 6 and that's what's great about algebra tiles that you can rearrange instead of going up and down I turn them to the side cuz now each X bar lines up with two units so rearrange my 6 into this right here so 1X this is 1 x 1 x bar is equal to True units what's 2 units one unit 1e that's two 3x equals 6 each X is equal to how do I show the work / 3 6 / 3 that's 2 that is pretty much a basic two-step equation showing algebra tiles so pretty much like we have this right here and let's go with this let's do a quick review a simpler one x + 4 equals 9 x + 4 equals 9x + 4 equals 9 and again this is why no one wants to do algebra tiles or drawings of owls because it just up so much time Sofia matching both sides this would match up with this what's left over I got an ex block what's left over is five of those x equals 5 what should we do with this for we had to get rid of it we threw them away we subtracted for and yeah then that means we bringing those negative tiles I hate being in the negative house but you get the idea we subtracted rid of it and if we had something like this let's go keep it simple cuz I don't want to keep on trying all this stuff and let's go 3x equals 9 you know I got three exes that is equal to 9 units division is about dividing things equally 3x equals 9 what are whole goes to figure out what One X is what is 1x equal to so I don't care about these if I divided equally can see that one x equals 3 so this whole ideas figure out what One X that is the main premise of algebra and two step equations and whatnot sometimes you don't got fancy stuff but negative numbers blah blah blah all right so let's go looking at this and it helps you cuz you going to have to do word problems and set up an equation that's the next thing will do but let's get on with the basic stuff first so let's read this quickly so Santos planning a hot air balloon ride the cost of the ride is $125 plus $50 for each hour in the air I'll look at this the equation see which I was the variable speed because that's the car cost of a hot air balloon you no matter what you got to pay $125 plus you decide how much time you want to spend up there before the the balloon blows up in the club $50 for each hour so cost is $50 in an hour but no matter what the constant got to pay 125 no matter what so represents the total cost to ride a hot air balloon find the total cost if Santos rides for 2 hours be kind of do it in your head if it's two hours $50 an hour to $100 but just got to pay that 125 a hundred 125 $225 how long could Santos ride if he had 325 so goal to create an equation to saw do something and then if you're given another bit of information how do you use a set up an equation so we'll do all these so it's going to be I'm going to do one you're going to do one this modeling so showing work there two different ways to show work there's a vertical method vertical means up and down and then there's a horizontal which technically is still the same thing as just written differently but let's start with this equation right here we got 3A + 9 equals 33 + 9 equals 33 the work is exactly the same it's just how you want to see it visually so there's a horde vertical methods just straight up and down I prefer this method because it keeps you in check I'm going to rewrite the problem just so I can start the work so you got imagine I got three a rectangles and 9 little squares and 33 little squares that's why I don't want to do the unit squares you got a thing I have 9 matching squares here in 9 matching squares here so if I get rid of it what am I doing I'm subtracting so the vertical method vertical meet you on a line things up and down so I want to subtract this 9 right here and I want to subtract 9 right here notice this right here my work is equal If I subtract 9 from one side I got to subtract 9 from the other side you think about like a seesaw is a boring seesaw cuz you're going nowhere it needs to stay level that's what's called vertical 9 - 9 0 what are you left with 3 a equals and then you just do this I'd how do you show work in adding subtracting you lining up top and bottom so 33 - 9 1414 2424 now look at this I got three as you got it 3 A's how do you want to rearrange these 20 festival thing I don't want to do 24 little squares you going to find what three that means I want to divide them equally so I got this 3A what's that one anime I don't care about 3 I'm going to divide I just want one a but what's my share out of 24 you on one side you always do to the other to keep it balanced because I want one day I want my equal share I just want to be greedy my own share you get yours we get mine 2 out of 24 I can't get all of it since there's three of us we getting eat why are we divided by 3 so a equals 24 divided by 3 which is 8 this is vertical method top-down there's no real difference in the horizontal method except how you want a visual is it work or is it all just left or right the only difference and I'll show you is right here I personally don't like this method because you get lost easily this right here lines up all my work I see exactly when I'm combining my lifetime this right here you just do side-by-side so that's the only different that's literally the only difference between vertical and horizontal so 3 a + 9 - 9 what's 9 - 9 get the same thing people like this method cuz I think it's less writing it's the same it's just how you organize it I notice when you get to hear literally it's the same thing no difference so I don't care which method you choose but here's one thing that they'll always ask you to do on tests when they say check your answer here's how to check your answer how to check your answer you just plug it in take the original 3/8 + 9 + 33380 + 9 equals 3323 and then we're supposed to put substitute in check text the term check your answer check 3 * 824 24 + 9 whats 24 + 933 equals 33 check if that's true you get a check mark if it's false you messed up on your work somewhere take a second out both of these so let's do one at a time if you want to do the second one go ahead and do the second one but I'll just write this one down first do this one take 30 seconds do this will go over it and then we'll do another problem after this make sure you know what you're doing a lot of guys are just watching me take time out to do this and for those of you guys are actually doing this well that's true give me the answer tell me what x equals typing in the chat that lets me know you're done I want at least six people so I can see who knows what the heck they are doing I got one person two people 34 good correction so it's two people gave me something really good one person corrected it but this is little things you got to watch out for good keep on typing when you guys go too fast you're going to make simple mistakes a lot of guys should have got a perfect score on your last couple test but went so fast you forget things you write it wrong and it hurts you are you guys got to write the job so we slept so good what are you want to do want to get X by itself our whole goal is to subtract 1 subtract 1 and then 6X equals 24 and then I'm going to / 6 x equals 24 / 6 4 that's how quick and easy all right everyone you got it right even though had to fix it try this one out on your own I will go over in 30 seconds 4x - 5 equals -33 with negative numbers now and I need you guys that just understand negative numbers think they cancel each other out think about how it's equal to each other what do you have what's left negative what messing up a lot of you guys type in your answer so I could see what you got then we can move on excellent excellent excellent a lot of guys are saying this is easy yes that's why you show your work most he doesn't try and do the stuff in your head that's where you mess up just trust show the word it doesn't take that much good good oh I got two different answers here it's split it's split between two number are you guys alright alright if you're working fall for those you guys that did it all right let's go so look at this right here our whole goal is X so in order for X I got negative 5 how do you get -5 to make it a zero you got to add what you doing one side you do to the other so + 5 so we can Forex bring that down equals I remember when the signs are different when things are different you subtract you always keep the number the sign of the bigger integer so -33 + 5 just imagined 33 - 5 but since the number the bigger number is the negative the bigger we keep it negative and that's going to be negative 28 and then we got to / 7 I'll do it on a separate I do an extra step divided by 4 4 / 4 that's one you do not need to write 1 x when X is the same thing as just X but negative divided by a positive why do you guys have the seven perfect you guys at the Edition don't forget the negative how to sign these things will kill you every single money guys I know wants to go to college one of the requirements that I messed up in and I left out and had to go to communicate don't that's going to communicate is not locking out I messed up in chemistry chemistry it was very important for going to college messing up messing up on a decimal you're going to be so pissed because you spent all that work chemistry is like the very beginning of like calculus type work when I say calculus time I mean that you're going to take a whole page to do a problem that's why that's why if you go to when you take mr. Sino in chemistry in a couple years you want to get a home from this Racino when he says alright dude 15 problems no in chemistry you'll see that you would rather do 15 problems and most mr. Sinister all right I got four problems for you to do when he says it's for promise because each prom takes a page and what happens if in the middle of your work you messed up on a negative the rest of your word pour a decimal the rest of your work goes up in smoke literally is chemistry pun intended so that's why you guys got to show work please show your work so you don't mess up these little points keeping you from 100% And instead now you messed up and you're at 78% watch out watch out watch out said it three times so that you can watch out next problem slightly different looking but the same idea POV 5 - 12 equals 20 I rented SP over 5 I want you to type it in what's another way of saying P / 5 what's another way any over type it in I want to know who knows this what's another way of Reading PA over 5 minus 12 equals 22 minus 12 equals 20 do we know it's 3 we don't know the three correct correct well you know that's an eight we don't know what that number is what's another way forget the rest of this. There's only one way of reading - other ways but I read it as P / 5 what's the other way of reading p over 5 you had it right the first time so a lot of people get stuck when they see a variable in a fraction yes divided divided / you a complicated way of doing it in a second but we'll do it the easy way first but download you're going to actually like the complicated way more his when you start getting too complicated looking problems no one ever likes to do with a fraction and a variable but we'll deal with that later this right here so you are whatever is connected to the variable that's the last thing you get rid of see where the parable is is connected to this number get rid of the things the constants first so let's add 12 P / 5 equals 20 + 1232 how do you get rid of a divided by 5 what is the opposite notice everything we're doing when we bring back the old ones don't type anything yet don't type anything notice everything we're doing we're always doing the opposite look at this 6X + 1 I don't care 25 x by itself always do the opposite how do you get what's the opposite of a plus one negative so we subtract one boom and then we ended up with just 6X it such a success 6 * X how do you get rid of what's the opposite of 6 * x / 6 always doing the opposite look at this I don't care about the 33 yet what is the 4x - 5 I just want One X how do I get One X like I'd get rid of the -5 first what's the opposite of -5 + 5 but what you do on one side to do on the other always stay balanced so I do the opposite when I come I'm back. I had four times x equals negative 28 what's the opposite of 4 * 4 divided so same concept right here what is the opposite of -12 + 12 tomato zero and then I hear P / 5 the opposite would be x v I'll switch it is an extra step just so you could see what you doing one side you do to the other this is an extra step so I'm going to rewrite this in French rewrite it like this these next couple bits are extra steps how do you multiply a fraction with something that's not a fracking turn it into a fraction and this doesn't change I'm just doing these obits for you to understand multiplying fractions so P / 5 * 5/1 P * 5 is 5 P 5 * 1 is 5 and I'll leave this finish this later what's 5 / 5 that's one that's a 1p and then 32 * 5 is 160 PT equals once that's more work than you need I'm showing you where most people get lost in the fractions now I'm going to show you a more complicated way but it's not more really most people don't like fractions this is called clearing the denominator clearing the denominator means I'm going to get rid of the fraction cuz I don't want to do the fraction to get rid of the fraction first this right here G / 5 - 12 I can actually do X but this involves distribution so let me multiply this side so this is p / 5 and this is why I don't like teaching this some of you guys asked about it I'm going to show it to you do not do this mess if you're not comfortable what you do when you multiply you multiply to everything if you want to start out like that so it's a giant distribution to get rid of the fraction multiplied by the denominator so we multiply everything I-5 to stay balanced do we distribute notice this understand what multiplication is if I only * the P / 5 and the 20 that means only these to change but I have to multiply this with the five also cuz ever I need to stay balanced I'm doing everything five times bigger cuz if I get five times bigger with the p / 5 I just end up with the p but that means everything else needs to be five times bigger so this right here we saw that's a p that's a plus it's a minus 60 and then 5 x 20 that is 100 so now look at this here we may P / 5 - 12 equals 20 into an equation that doesn't have fractions looks looks totally different but look what happens now we got to add 60 all sides he 60 - -60 + 60 + 0/100 + 60 160 which method do you like better that's a personal choice I would prefer you not to do this right now but Sunday guys asked you know what Clary the denominator cuz you got she done your I ready thank you this is what that is it's the same it's just you do the other stuff first but ignore this second way try this one on your own make sure you know what you're doing eight equals 15 + n / 3 notice the variables on the other side now and it's in back instead of the front of the 15th it doesn't matter do the same thing take 30 seconds your answer to let me know you're done alright plenty time playing time I'm almost almost there alright alright I got three people sup please extinguish look at this 15 little units and I got eight units right here you just said imagine if I got to move over I got to subtract 15 subtract 15 and look at this right here the bigger number should be on top so let's go rearrange this let's go -15 / 8 so 50 - 15 is a zero and then we bring this all the way down and / 315 - 8 because the signs are different is 7 but the sign they were nice or -7 think about this some number divided by 3 is -7 what divided by positive gets me Negra So my answer needs to be negative so let's think about this and divide I3 how do you get rid of a / 3 x 3 how much do an extra line of work 3 / 3 that's a 1 please me with just n 3 * -7 negative 21 watch out watch out watch out oh my fault that was a 15 sorry I didn't get what you were saying all right one more before we move on something but we different can you do this one right here -1 6x - 3 equals to A lot of people freak out when the coefficient is a fraction I'll show you it's no different than anything we've done you're off check your work type it in let me know you're done good yes okay seems like a lot of people are stuck cuz I'm getting slower amount of people typing in all right I'm going to show you one extra little bit right here forget this as just asked see that this is an X over one let's multiply remember ignore this multiplication comes before addition subtraction 16 * X I put this over one so you can multiply -1 * X where I can put at the bottom or One X x s x over negative 6 minus 3 equals 2 it's the same thing as this right here except that I think cuz this right here and over 3,000 a same thing as 1/3 so -1 6X is the same thing as X over negative 6 or you put the negative in front but I'll just leave for now X over negative 6 and then the -3 so why do people freak out start what to do this first always start somewhere why do guys freak out staring at this and you didn't even try and do this so let me rewrite the equation and start over -1 6x - 3 equals to why did I started this didn't even move but you could have done this right here so let's add five and he had three so now I got -1 6X equals 5 at least you would have got this far don't just give up because I can finish it get to this know this right here would have helped if you saw another way of writing this as for the negative in front x / 6 equals 5 so ignore the negative for now we're going to ignore the negative for now x divided by 6 equals 5 it's the same thing we did there what's the opposite of / 6 * 6 we got this negative stuff will do it 6 / 6 that's a one- x equals 5 * 6 which is 30 the opposite of X is 30 so that means the opposite of that is -30 watch out for the signs negative negative so this one we're definitely going to do more hint hint I'm not going to give you some reason why I'm not going to give you something super easy like that if I was going to give you a test I would literally give you something like this just to do your daddy all right next to mine what's a good list which leads us to this right here from here what if you have a negative variable no difference who cares if you have a number in front some reason of a dicl it's a minus I got a mind is to know let's get rid of the constant first that's a plus nine right here it's not a - 9 that's one of the biggest mistakes in prom that looks like this why did I see I got to think this man ghosts - goes to the 9 no it's a negative T that's a positive 9 so to do that would subtract 9 9 - 9 is 0 and you got a here's the thing this + -10 I mean not negative t and then right here the signs of the same negative and negative combining two negative numbers by subtracting we add them and keep the sign at yes you are correct so that right there for 4 + 9 is 13 karaoke 43 and since my signs are the same if the signs are the same when you're adding and subtracting you keep the sign it's at the same signs you add your digits and you keep the sign and look at this negative T equals -43 the show the extra little bit of work let's divided by -1 negative divided by negative is positive extra work just so you could see negative divided by negative is positive anything over one is everything T / 1 T 43 over 143 I'm just showing you the extra bit if you were going to work this is all you need to show give me the show that that's good enough right there boom boom boom that little bit from there is extra work all right let's do two prongs and make sure you know what you're doing then move on here try this one first 30 seconds -15 - b equals 44 try this on your own this stuff is just following the steps of a puzzle all the stuff is the only hard thing you need to do is think about the calculations momentum app yes this is easy alright so first let's 15 so sad 15 class 15 which pretty much at the very least I'm going to say the same. so I got a couple answers were people kind of messed up on but you still chose the right numbers to move that's the first that don't say a lot of the same man I messed up but no don't ever type that I'm stupid no what you did correctly you chose at least to move the 15 you saw that where you messed up is just the arithmetic like I told you 80% of all our math mistakes is the arithmetic -15 + 15 that's the 0 don't forget to bring down the negative on the B that's where the next mistake a lot of guys made that let's do this 44 + 15 59 that's what happened right there watch out but that's a negative B negative B equals 59 B we got to take. We want be not negative be so if the opposite of B is 59 what's the opposite of 59 -59 that's the least amount of work all right and then we got one more let's do decimals that's an end -6.5 equals -4.3 - 10 I know that's why you got to watch out meth the next forget your arithmetic keep on working the next biggest thing that we mess up in math is when we actually just right things wrong and I did a writing so fast so there's two of you I'm going to call you out cuz you guys do Orkin get perfect work but because you're too cheap and you want to say paper and right he was small Diego Stella yeah I called you out you use the paper spacer block it out cuz a lot of guys are trying to say paper you squeeze it in your run over other stuff you got your work mixed up and other stuff there was some yes I forgot who that that that did that when you you know just block off if you're going to write small block off block off sore right here always deal with the side with the variable first so let's get rid of this -4.3 what's the opposite of -4.3 positive 4.3 so I added 4.3 to make that a zero what you on one side of the equal sign you do to the other so it's equal this becomes a zero and I bring this down I got negative and this right here who cares if his decimals when you add two that's what you make sure to line them up the decimals and since my signs are different I'm subtracting and I'm going to keep the sign of the top bigger integer 5 - 3 is up to 6 find a sport is a two- 2.2 equals negative n but we want a positive and so the Bible sides by -1 and equals 2.2 couple of you guys wrote down on 2.2 or -2.2 no choice one answer we're not at the point where we got multiple answers all right example then two more you tries and then we are done all right let's go look at this combine like terms before solving so here's an example race listen to X Plus x minus 27 equals 3 all right sometimes you have an expression and equation where one of one side of the express one side of the equation expression can be simplified always looked at your wall if I have a constant by itself deal with this last always clean up always combine like terms always simplify before you move things left and right before you do anything good so pretty much two people said come back so we should come by so I like one person so we combine like terms first and then someone said combined the exit and then someone said it's 3 x - 27 correct so let me slow that down before you even do anything to this three like moving things over safest thing to do is always come by and like terms first understanding what like Tumblr that's why I did the algebra tiles to understand that you have two x's right here and One X right here I got two x's in one X's that mean have three X's I can't do anything these are just twenty-seven -1 little unit chips so that does nothing notice all I did was rewrite simplify one-sided I don't care about this down now I get to something we just did so we add 27 notice I don't draw that line under the unless I'm combining terms I mean unless I'm you know simplifying those things I didn't do anything all right but 3x + 27 + 27 - 0 we get 30 this I could do mental math x equals all right try this one on your own actually no I messed up facts and sometimes this can even be longer I got I totally 3 turn but sometimes you got like five terms six terms in there and you got this lonely little 60 by itself always clean up I need to fix the screen all right I got a couple answers let's check so understand what like terms are notice where is this right here I can combine the very what is at two x's and One X but this I don't got this is the only K what I could come by and are these two right here the negative 8 in the negative 12 so when I rewrite this this is the part you need to watch out for what's -8 -12 it is -20 watch out for that that equals 60 that right there was the hardest part and then we add 22 bolt size 4 k equals 80 divided by 480 divided by 4 k equals 20 correct you guys are getting the idea you need to practice combining like terms and understanding what those are that is the biggest thing next thing you need to understand the hard thing is the fractions how to deal with a fraction besides that I think that's Italy what time is it in 10 minutes classwork I think I put it as 3 to 30

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Pre-Algebra - 4.5: Solving 2-Step Equations

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