Grant consults his books from the first four days of manufacturing. Everyday, the total number of sest of wiper blades that have been produced up until that point is logged. So after one day, 23 wiper blade sets have been made. After two days, there are 46 total. Three days, 69. And so on and so forth. When Grant looked at this, he noticed a pattern. Let's help Grant continue the table. How many sets of wiper blades will have been produced by the end of day 5?
The trick here was to identify the pattern. If we look at the total number of wiper blades that have been made at the end of any one of the days on this table, and then we look at how many have been made by the end of the next day, that is where we see the pattern. After the first day, there are 23, but after the second day, there are 46. The difference between 46 and 23 is 23. To get from 46 to 69, we have to add 23 again. And we do the same thing to get from 69 to 92. So, you can see that as each day goes by, we're adding 23 wiper blade sets to the total number of wiper blade sets that have been made. So, logically, we do that again when we move from day 4 to day 5. This means we need to add 23 to 92 to get the number that belongs in the teal box. 92 plus 23 is
Pattern finding, just like we did in the last quiz is a big part of what makes math so interesting. Now that we know that each day we make 23 more units total, what's the best way to find out how many we'll have by the end of day 14, which is when the expo is? Should we complete the table down to 14 days? Should we find a mathematical relationship between days and total number made? Or, last but not least, should we make an estimate about how many wipers will be made by the end of day 14? This a little bit subjective, and more than one of these might actually work, but please pick what you think would be the most.
The best and fastest way is to find a mathematical relationship between the number of days that have passed and the total number of wipers said that I've been made. Completing the table would work, but it would be pretty time consuming. Estimating is often a good way to begin thinking about a problem, but it doesn't give exact answers. We want to get information to Grant quickly but we also want to be really accurate about. it.
For this situation, there's a simple relationship between the day and the number of units that have been produced up till' that point. I'll begin by telling you that the total number of wiper blades that have been made, which I'm going to label n right now, is equal to some number times the number of days that have passed. And from now on, I'm going to call the number of days that have passed d. So, what I just said is that n equals some number times d. Remember, n refers to the total number of units that have been made, and d refers to the number of days that have passed. So, fill in the blank right here with the number you think belongs in that spot.
The number that belongs here is 23. So, how did I figure that out? Well, one way to check if the solution is correct is by writing this equation, and then substituting in values for d and for n, and seeing if they match our table. We know that,= after 3 days, for example, 69 wiper sets have been produced. So, that means that if we take d and we replace it with 3, we should get that n equals 69. And sure enough, 23 times 3 is 69. That checks off their table. So, we can see that our equation fits the situation that we were trying to make it fit. You could do a similar thing with any of the other sets of entries on this table. When you plug in a number for d, you should end up with a corresponding number for n when you simplify. But checking if a solution is correct is pretty different from creating the solution yourself. So, how did we come up with these mathematical descriptions, these equations on our own? That's what the next few lessons are going to be about.
We've now spent quite some time looking at expressions, thing like 2 x cubed minus 7 y squared plus 28. Now expressions are great and useful and all, but we can't actually solve anything when we just have an expression. To be able to find solutions, we need equations. We've already looked at how to set up equations, but just to remind you, what is an equation actually? Here are a bunch of choices for what an equation might be, and I'd like you to pick the best answer.
The best answer of these five choices is the third one. An equation is two expressions that have been set equal to one another. Although this second choice, something that has an equals sign, is partially correct. There are more things involved in creating an equation than just an equals sign. We'll talk more about that in just a moment.
Just before we go on, these answer choices for the last quiz that now have these blue arrows next to them, are definitions of other things that you met a few lessons ago. Please fill in each teal blank with the word that you think properly completes the sentence. Remember, you might have to think back to a few lessons ago to remember what these words are.
Thinking way back a few lessons ago, a combination of terms that are added together is an expression. A letter or symbol representing values that vary, keyword vary here, is a variable. Remember, you can really see and hear the word vary inside variable. So, that's just a good way to remember what the function of these things is. And finally, the product of variables and numbers is a term. A term is just a set of variables and constants that have been multiplied together. I hope this was a good review of some of the algebraic form we've learned so far.
Right. Now, back to equations. I mentioned a moment ago that an equals sign isn't enough for something to actually be a true equation. Which of these choices don't make sense, just aren't right from a mathematical point of view? Remember, I want you to check off the ones that don't make sense, not the ones that do make sense.
We could jokingly say 2 plus 2 equals 5, but this is not actually an equation. So I'm going to check it off. The reasons it is not an equation is that the quantities on either side of the equal sign are not actually equal to one another. We know that 2 plus 2 equals 4, and we know that 4 is not equal to 5. So this is not a true equation. When you do the same check on this equation right here. Once again, since 2 plus 2 equals 4, that means we have 4=4, which is an equation. So we won't check this answer choice. n plus 2 equals n plus 3 is a little bit trickier. Now, what is the problem here? Although I didn't tell you that this is an answer that you should have selected. The issue is that, no matter what value you substitute in for n, the two sides of this equation will never be equal to one another. The right-hand side, that says n plus 3, will always be 1 larger than the left=hand side. So for example, if we let n equal zero, we have 0 plus 2 equals 0 plus 3, which leaves us with 2 equals 3, and that is not true. A similar thing will happen no matter what number you plug in for n. The same thing is not true of this equation, 1 plus n plus 1 equals n plus 2. In fact, the opposite is true. No matter what value we substitute in for n here, this equation will always be true. This is a special kind of equation that we'll talk about in just a second.
What if we have the equation n plus 1 equals 2? Are the two sides of this equation equal to one another?
The answer is, it depends.
And what does it depend on? What determines weather or not this side, n equals
Remember that variables are so great and useful because they can take on different values. We could substitute in a whole bunch of things for n right here, but only one of those numbers would make this entire side of the equation, n plus 1 equal to 2. If we pick n equals 7, for example, we'll end up with 7 plus 1 equals 2. Or, if we simplify, 8 equals 2. But that definetely can't be right. Either way, the answer to this question is, the value of n.
We know that some value of n makes this equation true, but, what is that value? Fill in a number that you think belongs in this equation for n into this box.
This equation tells us that 2 is a number that is 1 greater than n. So that means that n is a number that is 1 less than 2, so n must be equal to 1. We can check this by plugging 1 back into our equation, or using substitution. If we do that, we get 1 plus 1 equals 2. Since 1 plus 1 does equal 2, that checks out. And our answer for n must be correct.
There are several different situations in which we might see equal signs. The last case you did had an example of a conditional equation in it. Conditional equations are true for some values of the variable, but not all. So, like we saw earlier, this left side of the equation will be equal to the right side, depending on what we plug in for n. Some numbers will work, but some will not. Some things with equal signs will be identities. In an identity, no matter what number you plug in for the variable, you will get a true equation. The third case is that we might see an equal sign in something that is not actually an equation. You've seen a few of these before. No matter what number I plug in for y, over in this equation, I will never be able to make the left side equal to the right side. We'll talk more soon about the situations in which these different types of equations are used or at least, where these top two are used. The last one is usually something that comes up if a mistake has been made somewhere in our math.
Which of these equations over here on the left are identities? Which ones are conditional equations, and which ones aren't equations at all? For each equation, pick the category that it falls into.
For this first equation, x plus 5 equals x plus 7, the right side of the equation is always going to be 2 greater than the left side, since 7 is 2 greater than 5. Since no value of x can make these two sides equal to each other, this is not an equation. There is some number that we can substitute for x though, in the second equation, that gives us 27 when we multiply it by 3. So, what times 3 equals 27? 9. So, this is an equation. However, other numbers that we can put into x like zero would not make this true, so this has to be a conditional equation. Our third equation is the same type as the second one. Since some values of x make this true. And our last equation is an identity. If we simplify the expression that we have on the left-hand side, we can combine like terms, since we know that x and 2x are like terms. And we can make the left-hand side equal to 3x plus 3, which is exactly the same as the right-hand side. No matter what we plug in for x, these two will be equal. So this is an identity.
Now, that we've talked about what an equation really is and what types of equations there are, let's actually build some up. To do that, we're going to keep helping Grant deal with his manufacturing process. Since Grant's Gleaming Glasses has started manufacturing now, they actually have to pay for things, namely, the materials they use to make glasses wipers and nozzles. To buy just the material that he needed to make the wipers that were produced in the first 4 days, Grant spent $598. He's thinking that next week, he'll need to spend around $750 to start the nozzle manufacturing process. We know from that chart we made earlier, that he was able to make 92 glasses wipers in the first 4 days. Oh, and he also told me that he took his friends out to lunch and he spent 54 more dollars on that. To plan for the future, Grant is getting a little bit worried. There are a lot of different numbers he needs to keep track of. He wants to know how much he spends on the materials to make just one set of wiper blades, but he needs our help to sift through all these numbers.
Okay. So, there were a lot of words in that last problem I just said out loud to you. There are a lot of things to look and a lot of numbers to think about. And many people, including myself, end up feeling kind of confused and overwhelmed when starting to work on a word problem. To help you out, here's a list of the facts that I told you in our story. I'd like you to check off all of the ones that we actually need to pay attention to in order to find what we're looking for in the problem, which is, to remind you, the cost of manufacturing just one set of glasses wipers.
The key to figuring out which pieces of information we care about is to keep in mind what we're looking for in the problem. Since we want to find out how much one set of wiper blades costs to make, the total amount of money that Grant has spent on micro productions so far, definitely matters, as does the total number of wiper sets that, that money, $598, funded. Grant's lunch with his friends does not affect the materials that he needs for a set of wipers, so I'm not going to check this one off. It doesn't actually matter for this problem. And this last choice that we have left to consider has to do with nozzles alone, not wipers, and as you can see from the question, wipers are what we're concerned with. We can tell already from this list, and from the fact that half of the answers aren't checked off, that a lot of the times situations that pop up are full of information that isn't relevant to the problem we're trying to solve. And the first step to solving problems is to make sure we identify what we need to pay attention to and what we can ignore, to make our lives easier.
Thinking back to what information we decided was important in the last quiz, we're going to use that to create a word equation for the cost to make 92 glasses wipers. So try to remember which quantities we decided were relevant for figuring out this problem, and then think about how those 2 quantities might be related to each other. Place the letter of each quantity that you'd like to be an equation in these teal boxes, one in each box of course. And then pick which of these four operators you think relates the two of them together. Should they be added, subtracted, divided or multiplied? Remember that what we're going for in the end, we want this side of the equation to equal is the cost of manufacturing 92 glasses wiper sets.
We said earlier that the cost to make 92 glasses wiper sets is going to depend on the cost to make one wiper set, which is choice a, and also the number of wiper sets total that are made, choice c. So, I know that a and c are going to be the letters that go in these boxes, but I need to think about the operator that I want between them before I can decide which one goes in the first box and which one in the second. Or, if that order matters at all. In previous lessons, we thought about how total money spent is equal to the money spent per unit times the number of units that are being made. So, if we want to make two glasses wipers instead of one, we're going to pay twice as much money. If you wanted to make three wipers sets, we're going to spend three times as much money, and so on and so forth. So, we need to multiply these two quantities, a and c, together. I'm going to go ahead and fill in a multiplication sign in this box. We learned earlier from the commutative property of multiplication that when we're multiplying two things together, order doesn't matter. So, it doesn't matter if I have a or c in this box, or a or c in this box. As long as I have one in one spot and one in the other, we are good to go.
At this point, we want to start translating the equation we just came up with into more traditional Math language. As a reminder, here is the information, up here at the top, that we decided before was going to be useful for us. And down here is that word equation that you came up with in the last quiz. We have three quantities related to one another in this equation, and I would like to know right now which one of these you think we need to represent with a variable for the purposes of solving this problem.
We already know how many glasses wiper sets were made in the first 4 days, which is this quantity down here, and we can also see in our given information, how much making these glasses wipers cost, which is this quantity over here. Now the only quantity of these 3 that we don't already have a number for, from our given information, is the cost per wiper set. So this is the thing that we need to represent with a variable, since we don't have any other number to replace it with yet.
Leading up the, the last quiz, we already had a lot of information about our situation. And we'd also picked out what really mattered to us. In terms of how we need to go about finding the cost to make one wiper set. In the last quiz, we added one more piece of information to this list. And I'm going to write that out a little bit more formally now. We decided that we would represent this quantity right here, the cost to make one wiper set, by a variable. I'm going to pick the letter c for that variable. C is going to equal the cost to make 1 wiper set. To move one step closer to having a formal mathematical equation to describe our situation, I'd like to help us get some other numbers in our equation. Looking at this given information up here, can you tell me two numbers that equal, respectively, the cost to make 92 wiper sets, and the number of wiper sets that are made. This is going to help us with substitution, in the next step of the process.
Our first piece of information up here tells us that 92 wiper sets were made in the time frame we're considering. So, number of wiper sets made equals 92. The next piece of information says, that the cost to make all of those wiper sets was $598. So, we can write 598 in the cost to make 92 wiper sets. This may seem sort of silly right now, but we've set ourselves up really well to move forward and come up with an equation that only has numbers and variables. No more words, thankfully. Word equations are really, really useful for helping us conceptualize problems, but they take a really long time to write out. So, we can simplify what we're doing, that's awesome.
Now, we're at our final step in coming up with an equation for Grant. So, we can eventually help him figure out how much it cost to make one wiper set. We have four equations up here in our what we know section. And that may look like a lot if information to handle at once, but using substitution, we can pretty easily create one equation that involves all of these things. So look at this equation right here, the last word equation we came up with, and think about how these pieces of information fit in here. Substitute them into the right spots, and then type your final equation into this box.
You can see here that the cost to make 92 wiper sets is equal to 598. But we also see in the last equation, that the cost to make 92 wiper sets is equal to the cost to make one wiper set times the number of wiper sets made. Now, when two quantities are both equal to the same thing, that means they're also equal to each other. That means we can say that 598 equals the cost to make one wiper set times the number of wiper sets made. That's actually really just another case of substitution. So, we can put a 598 in this spot, and we're going to continue to leave dollar signs out for now just for simplicity. You see the cost to make one wiper set, both here and the last equation, and also here that it's equal to c. So, let's substitute c in and the number wiper sets made is 92. Now, we have an equation. 598 equals c times 92. There's one slight adjustment that I'm going to make. I'm going to say that 598 equals 92c. So, remember that we normally write the coefficients, just plain numbers, at the front of terms and then the variables following them. So, look at that. After a lot of hard work, we finally have a lovely, concise equation. 598 equals 92c. I'm going to put a nice little box around that.
Now we have to think. Why did we want to come up with that equation in the first place? Think back to our original problem. We are trying to figure out the cost to make one set of wiper blades. With our new equation, 598 equals 92c, that means finding out what number c needs to equal to make this equation true. So, what kind of equation are we dealing with? With here. Is this a conditional equation, is it an identity, or is this not an equation at all?
This is a conditional equation. If we were to replace c with 1 for example, then our equation would say 598 equals 92 times 1. We could simplify that to say that means that there are some values of c for which this equation is not true, so we know already that this is not an identity. I'll tell you right now, though, that this last choice, that this is not an equation, is incorrect. There is, in fact, one value for which this equation is true. Our next step is going to be to find that value.
What then does c need to equal in order for this equation to work? What number do we need to put in the place of c to get this side of the equation to equal this side of the equation? Please type that number in right here.
To find out what c needs to equal to make this equation true, we have to modify both sides of the equation so we get c by itself on one side of the equal sign, and everything else on the other side. Now, to insure that both sides of the equation actually remain equal to each other, so that we can legally continue to write equal signs in our equation, whatever we do to one side of the equation, we need to do to the other side as well. If these two things start out having the same value, but I, for example, add 4 to the left side and subtract 1 from the right side, then they can't possible equal to one another anymore, since I've changed them in different ways. We'll do plenty of practice with that concept in the coming lessons. For now though, let's think about what's preventing c from being by itself. We know that we want our final equation to look like c equals blah, blah, blah, blah, blah, something. So, it means we need to isolate c as my petitions like to say. Instead of having just c though, we have 92 times c. Remember, there's an invisible multiplication sign right here between our coefficient and our variable. We basically need to undo this multiplication, get rid of it, and the way to undo multiplication is to do some division. Since right now, c is being multiplied by 92, to have just c on its own, we need to divide by 92. But that also means, since I'm dividing on one side of the equation that I also need to divide on the other side. So, I also need to divide 598 by 92. Looking at the right side, 92c over 92 is, as we wished, just c. And on the left side, 598 divided by 92 is equal to the number 6 and a half. Just to be safe, it's always a good idea to check your answer if you know how to. In this case, we can just use substitution. Starting with our initial equation, 598 equals 92c, we can plug in our new value for c and see if this makes a true equation.
Now, we've finally solved for c in our equation. We found c equals 6.5, wonderful. But, what does this actually mean for a situation with Grant? Please pick the best choice down here for how c equals 6.5 translates into information about our story. Just think back to what numbers in our equation each stood for what, and what we were looking for in the original problem.
This question is basically asking what C stood for in our equation, or what our initial unknown was. And, as you might remember, that was this last choice, the cost for materials to make one set of wiper blades.
You just got a little bit of practice solving for a variable in an equation. Thinking about what you actually did in the last problem or in the last several problems, what does solving an equation really mean? Check all the answers down here that you think are correct.
All of these but one are correct. We saw in the problem we did for Grant that we have to change both sides of an equation in the same way to make sure that we maintain equality. So this last choice down here is correct. The answer we ended up with also was a way of expressing an unknown quantity in terms of other information that we had, which is this third choice. When we found that c equals equation. So, this first answer works as well. We could not, however, change one side of our equation. We could not, however, change one side of our equation in a different way than the other side. And we also didn't have in mind, in advance, what number we wanted the variable to equal. So, this answer right here, the second one, is not something that we want to select.
Let's keep solving equations. When we're doing that, we can add, we can subtract, we can multiply, we can divide, or we can even do other operations. But whatever we do, we have to keep one rule in mind. No matter what, we have to do the same operation to both sides of the equation. If we don't do this, we're going to change the initial relationship that we started out with between our quantities. And the answer we get in the end, won't work with that equation. Let's start out with a pretty simple example, x plus 3 equals 12. Then, I'm going to draw two boxes down here, one for each side of the equation. And in each box, I would like you to write what step you think should happen next, to change either side of our equation, so we can eventually get x by itself. For now though, we're just going to write this one step in. So, for example, if you think that we should add 8 to the right-hand side, then write plus 8 in the box on the right. Or if you think we should divide by 2 on the left-hand side, please write a slash and 2, to symbolize divide by 2. Please remember that if you want to symbolize multiplication, do not use an x, rather use the star symbol and then, of course, whatever number you want.
Right now the thing that is keeping x from being by itself on this side is this plus 3 right here. If that weren't part of the equation, if we just had for example, x equals 12; then we wouldn't have very much work to do. We'd already have a solution to our equation. However it is here, so we need to figure out how to get rid of it. So how do we undo addition? Subtraction. To make it so that nothing extra is added to the left side, we need to subtract x plus 3 minus 3. Now if we saw this as just an expression, we would know how to simplify that. The 3's would just cancel one another out since 3 minus 3 is 0 and we would have x plus 0 or just x. That's a super important step in moving forward in solving our equation. But wait, important. Whatever we do to the left hand side of the equation, we also have to do the right side, so we make sure that these 2 sides are still equal to one another, so I need to put in minus 3 here underneath the 12 as well.
Now that we know we need to subtract 3 from both sides, what does x equal? We saw already, from simplifying expressions, that subtracting 3 from the left side is going to get x by itself. But how does this affect the right side? Please fill in this box right here with just one number that you think the right hand side should be equal to now.
itself. Our equation thus tells us that X equals 9. There is one more step though, we need to check our answer. That basically means confirming that the value of X we found actually makes this original equation, X plus 3 equals 12, true. So you need to substitute our 9 in the place of x in that first equation. And sure enough, if we take one more step, 9 plus 3 is in fact equal to 12. This means that our answer works.
Let's do a few more questions like the last one. For each of these three equations, please take the second line of boxes to fill in the operation that you do on either side of the equation, in order to isolate x in the last step. In each of these teal boxes, please fill in your final answer for what you think x should equal. Then when you're done, make sure that you take that number and substitute it back into the original equation to make sure that it actually is a solution.
In problem number one, 3x equals 12. x is not by itself yet because it's being multiplied by 3. The thing that undoes multiplication is division. So you need to divide by the number we're presently multiplying. So, I'm going to say we need to divide this entire side by 3. Now, whatever we do over here, we also need to do on the right side. Divide 12 by 3 as well. We know that 3x over 3 gives us just x, which is why we wanted to divide by 3 in the first place. So now, we need to carry it out on the right hand side. 12 over 3 is 4. So we get x equals 4. Checking our answer, we have 3 times 4 equals 12, and 3 times 4 does in fact equal 12. So we're in good shape. And number two, x is being divided by undo division with multiplication. To undo division by 3, we need to multiply by that same number 3. You do this to both sides and 12 times 3 is 36. Checking, 36 over 3 is in fact equal to 12. Awesome. Another one done. Last but not least, x minus 3 equals 12. This problem is telling us that we're looking for a number so that if we took 3 away from that number, we would get 12. That means that number is 3 greater than 12 then. So, to undo this subtraction on the left hand side, we need to add 3. And over here, we get x equals 15. We will also confirm that 15 minus 3 equals 12. Great.
The equations we deal with are going to start getting harder. But don't worry we're going to take everything gradually so we make sure that you're all comfortable with the basics. So here's just another equation for you to solve for x. Type your answer into this box.
And the answer is x equals 6. However, there are two different ways that you could have solved this problem. We're going to talk about each of them.
So, the one way you could have done this is started out by subtracting 18 from both sides. That would have left you with negative x equals negative 6. And then if you did the problem this way, what was the next step that you took in order to get x by itself? Please tape what you did to each side or what you would do to each side in these boxes right here.
So there are actually two options here, two right answers. One way that you could move towards having x by itself would be to multiply each side by negative negative 1, we get negative 1 times negative 1, which is just equal to positive could do this. You could also have said that you would divide each side by negative 1. I'm just writing in here what you would type if this had been your answer. The reason this works is that the coefficient here, as we said, is negative 1. So we take negative 1 and divide by negative 1. We're just dividing a number by itself, and we get positive 1. So once again, that would leave us with just x by itself. And a positive 6 on the other side. So, both methods work.
Here's another equation for you to solve, and just do it like you've done all the other ones.
We see here that the term with an x in it has a coefficient of negative 3. So like we've done before, we just divide by that coefficient, and, of course, you do this at both sides. 12 divided by negative 3 is negative 4. so x equals negative 4.
So we just saw the solution to this problem, but there's actually a slightly different way to think about solving it, an equivalent way. Neither one is more correct than the other, but it might be useful for some of you to think about it in this slightly different way. So, before I said that we would divide each side by negative 3, because the coefficient over here is negative 3. What if instead though, I said that I wanted you to multiply by some number on each side, and then you would still be left with the same answer of x equals negative 4. What number would you have to multiply both sides by in order to end up with the same answer?
The number that you would fill in, in these boxes, is negative 1 3rd. So, instead of thinking of dividing each side by negative 3, you could instead think about multiplying each side by negative 1 3rd. Multiplying by negative 1 3rd is actually pretty much exactly the same as dividing by negative 3. The reason for this is because of fraction multiplication, and its relationship to division. So, if we have negative 3 time negative 1 3rd, we convert negative 3 to a fraction of negative 3 over 1. Then, if we just do some fraction multiplication, so you mutiply the top by the top and the bottom by the bottom, we end up with negative 3 times negative 1, or positive 3 over 1 times 3, which is just positive 3 again. And 3 over 3 is just 1. So, multiplying by fractions is just a different way to think about doing division. I want you to decide which way works better for you. Thinking of dividing this side by the coefficient that's in front of x, or multiplying by its reciprocal. Either way is completely fine.
Here is a problem that may look slightly more complicated than what we've done before. But you know everything you need to know to solve it. And I'm even going to give you the first step that I'd like you to do. I want us to start out by dividing each side by 3. If you do this first step, then what do you end up with on either side of the equation?
If we divide this entire left side of the equation by 3, then we just get rid of this coefficient in front of the parentheses and we end up with x plus 2. Over here, things get a little bit more complicated. Since we're dividing the whole side by 3, we have to divide each term by 3. 12 over 3 is 4 and negative 2x divided by 3 gives us minus 2 3rds x.
Now I would like you to figure out what you need to do to either side of the equation to make it so that what we end up with next has all the terms of x on the left side of the equation, and no more terms with x on the right side. You can still have constant terms on the left side, and if you want to combine like terms in places where that's possible, you're totally open to that. But, you don't have to.
In order to get rid of the term negative 2 3rd x on the right side of the equation, we need to, as always, do the opposite. Since 2 3rd x is being subtracted right now, we need instead to add it. This is going to leave us on the right side with just 4, since negative 2 3rd x plus positive 2 3rd x equals side, we now have x plus 2 plus 2 3rd x. If you'd have decided to take this one step further and combine these two like terms, the ones with the x to the 1st in them, you'd end up with 5 3rd x plus 2 equals 4. Just for future reference, I personally prefer to write fractions in their improper form instead of mixed sometimes. Just because that makes more sense in my head. Instead of 5 3rds, you also of course could have 1 and 2 3rds x. This is just my personal preference.
This equation we have now 5/3x plus 2 equals 4, has a similar form to equations you've already solved in the past. So, I would like you to take two more steps or more if you need them, to get to a final answer for what x is equal to.
Since solving this equation is starting to take up some space, I'm just going to do the rest of our work over here. But know that this is a continuation of the work that we've been doing over on the left side of the screen. So, I'll re-write what we had before, 5/3x plus 2 equals 4. We can't deal with this 5/3 yet and we can't get rid of it yet because it's only multiplying x, it's not multiplying 2 as well. Before we get to that, we need to get rid of the 2. To do that, we subtract 2 from each side, and then all we need to do is handle this coefficient. As you've done before, you can either divide by the coefficient itself, so divide by 5/3. But we know that when you divide by a fraction, it's the same as multiplying by the reciprocal. So, this is similar to what we talked about in the previous quiz, where we said that dividing by negative 3 was equivalent to multiplying by negative 1/3. The same thing is true here, division by 5/3 is equivalent to multiplication by 3/5. Either way, we'll get rid of the answer. x equals 6/5. Like with all of these problems, we can take 6/5 and substitute it back into the original equation we had, to check and make sure that both sides of the equation are actually equal to each other when we apply this condition of x equals 6 over
We got some great practice with adding and multiplying and dividing by fractions in the last quiz, or the last several quizzes, but there's actually a different way that you can do this problem that lets us avoid having to deal with a lot of those steps. This time, instead of starting out by dividing each side by 3, I'd like you to use your knowledge of the distributive property to just simplify the left side of the equation. If instead that's your first step, what do the two sides of the equation look like afterward?
Starting out by distributing means that we want to distribute the multiplication by 3 to both terms inside the parentheses on the left side. So we end up with 3x plus 3 times 2 or 6. Since simplifying on the left side of the equation doesn't involve the right side of the equation at all, we don't change anything about this 12 minus 2x over here. We didn't actually add in or multiply in anything extra to the left side, so that means we don't need to change the right side at all.
From here, solving for x is pretty simple. What I would like you to do next is collect all of the terms with variables, so all of the terms with x, on the left side of the equation, and all the terms with constants on the right side of the equation. So, please use paper and pencil to do this. In fact, I'd like you to use paper and pencil to solve all of these equations and write out all the steps. And you can either choose to combine like terms on either side of the equation or you don't have to if you don't want to. And last but not least, I would like you to fill in your final answer for x in the bottom box.
The first thing I'm going to do is get all of the x's on the left side of the equation. So that means getting rid of them on the right. Since 2x is being subtracted, we add, and then we have 5x plus 6 equals 12. Now, I need to move things around, so that all of the constants are on the right side. Subtracting 6 gives us 5x equals 6. All we have to do is divide both sides by 5. And since I ran out of room, I will write our final answer of, guess what, x equals 6 5ths. So in the end, we get the same answer that we did using the first method of starting out by dividing by 3. But as you can see, this takes a lot of your steps. Now, if everything on the right side of the equation had been divisible by three, then that method may have been a little bit easier. But in many cases, using the distributive property is going to make our lives simpler.
Thinking about what lesson you may have learned from the last several quizzes, I'd like you to solve for x with this equation. 3 times the quantity 2x plus 4 equals 2 times the quantity x minus 5. Be sure that you write out everything on paper to make sure that you're not missing any steps.
I'll begin by distributing these coefficients to what's inside the parentheses on either side of the equation. Now that I've done that, we have very pretty integer coefficients. And we don't have to worry about fractions at all. Now I'll get all the variables on the left side of the equation, and then I'll move all the constants over to the right. Last, I just divide by 4, or of course, equivalently, multiply by 1/4. And we get a final answer of x equals negative 22 over 4. But wait, this is not a fraction in simplest form. Both 22 and 4 are divisible by 2, so I can cancel those 2's out, and I end up with negative 11 over 2. So that, in fact, is the correct answer, x equals negative 11 over 2. I can look back at the top and plug this in to check. And in the end, I get negative 21 equals negative 21, so we are good to go with this answer. Awesome job. Just as a side note, you also could have written several other equivalent things for our answer, -11 over 2. You could have had negative 5 1/2, or you could also have written negative 5.5. Any of those three are correct.
Here, we have another equation, and it looks like a situation in which we might want to start out by distributing on either side. However, there's something else that we can start out doing, that might actually make things easier for us in the end. If I want you to start out by doing some operation to both sides of the equation, what do you think the best choice for that would be? Just fill in what step you think we should take first in these two boxes.
What I notice when I first look at this problem, is that both sides of the equation are divisible by 4. Now obviously, we divide the right side by 4, it's going to make things easier, since it's just getting rid of this coefficient. But this is also convenient to do on the left side.
If we start out by dividing both sides by 4, what equation do we end up with?
Since you want to divide this entire side by 4, but we know that 12 is going to multiply each of the terms inside the parentheses, you can conveniently just divide 12 by 4, which is something really nice. That just gives us 3 as our coefficient instead of 12. Over here, the 4 is canceled out and we just have 2x plus 7.
I'd like you to solve this equation on a piece of paper, writing out all of your steps, and then just type in for me what your final answer for what x equals is.
The next step is going to be to distribute the 3 to both terms over here on the left side, and leave the right side the same as it was. And then, I just need to collect all of my variables on one side and all of my constants on the other side. And we get a final answer of x equals 22. In all the problems we've done so far, it just so happened that it was most convenient for us to end up with variables on the left side and constants on the right side. However, that's not necessary. I could very easily have decided that I wanted my variables on the right side, instead of the left side. And started out here by subtracting 3x from both sides. I would have just had, eventually, to divide by a negative number. And that's not always my favorite thing to do. But in the future, feel free to put the constants and the variables on either side of the equation. Whatever is easiest for you.
Now we have 2 3rds x plus 5 equals 17 minus x. I'm not going to walk you through this one step by step and I just want you to type in your final answer for me. But please take the time to use pencil and paper to write out all the steps to solve for x.
The answer I got to this problem was 36 over 5. I can see though two very different methods of going about solving this. So, let's look at each of those. The first method would be to, right away, try to get all the variables on one side of the equation and all the constants on the other side. Then however, we have to deal with having a fractional coefficient for x for pretty much the entire problem. This isn't hard to deal with, it's just a question of what you prefer doing. To get rid of the fraction, I multiply by the reciprocal. This method does give us the current answer, 36 over 5.
There is a possibility though that you really, really hate fractions. I mean, really hate them, like you don't want to deal with them at all. This method over here then is probably not quite your cup of tea. So, there is a different way that you go about solving this problem. And that way would be to start out by getting rid of the fraction that's over here on the left-hand side. So, what do I need to do on each side of the equation right now, in order to end up with the next version of it having no fractions in it, whatsoever? Please fill in that operation in these boxes.
To figure out how to get rid of the fraction over here, let's focus on that term itself, 2 3rd x. I know that 2 3rd x is exactly the same as 2x divided by 3. So, it would be super convenient if I could just multiply this term by 3, since then it would no longer be written as a fraction, and I wouldn't have to go through this whole mess over here. I can't, however, just multiply this term by 3. I need to multiply this entire side of the equation by 3. If I'm going to put in anything from the outside into an equation, I need to do that to the entire side, and I need to do it to both sides. So, I'm going to multiply everything, both sides of the equation by 3. And it leaves us with a very pretty equation that only has integer, coefficients, and constants. From here, solving is pretty straightforward. You can take a second to look at the steps that I did to come up with the same answer as we did using the first method, 36 over 5. Both of these are completely fine ways of solving this problem. It's just a matter of what you prefer doing. Which kind of math you enjoy doing more.
Back in the world of glasses wipers, Grant just found out that the situation with manufacturing cost is not quite as simple as he thought. He not only has to pay for materials to build his wipers, which remember, we figured out that one wiper set cost $6.50 to buy. He not only has to pay for materials to build his wipers, which you'll recall, we figured out that materials for one wiper set costs $6.50, he also has to pay for the factory space that he's using. This is just a fixed amount, per month, of $1200. Taking this information into account, which one of these expressions do you think should replace the question mark to create an equation for the money that Grant spends each month?
Even if Grant makes 0 glasses wipers, we know that he's going to have to pay $1,200 for rent each month. So, his expenses are going to start at $1,200 and they continue to increase incrementally for each wiper set he produces. So, we need to include both of these pieces of information in our equation. That means, that this answer choice, 6.5 times number of glasses wiper sets made, can't be right because it doesn't account for the monthly rent. The first choice here is actually not right either because this is saying that for every glasses wiper set that Grant makes, he'll pay $1,200. That would be pretty unfortunate for him, that would be a very, very expensive wiper business. And we also know that it costs $6.50 not $1,200 to make each wiper set. That leaves us with two more choices, both of which include 1,200 and 6.5 times the number of glasses wipers made. The difference between these is the sign here. If we pick this third answer choice right here, we'll be seeing that Grant's monthly expenses decrease because of this negative sign right here. Decrease for every glasses wiper sets he makes. This would mean that he would actually be making money off of paying for materials, which doesn't really make sense. That means this answer choice isn't right either. The last one is correct. Grant will spend a fixed amount of $1,200 and then, on top of that, he will spend $6.50 for every wiper set he made.
Now let's use some numbers and variables. It turns out that after one month of just producing wipers, Grant had spent $5,685. I'd like you to think about the last equation that we came up with in the previous quiz, and write a new version of it, including all this information. I'd also like you to call the number of wiper sets made in a month, x. Please write your new equation for the situation in this box. And leave out any dollar signs that you might want to include for now.
At the end of the quiz before this one, we said that a correct equation to describe this situation is that each month would be money Grant spends each month equals 6.5 times number of wiper sets made, plus 1200. All we need to do now is replace number of wiper sets made with x, and money Grant spends each month with $5685. So again, as we've done many times before, this is just a simple case of substitution. And we come up with an equation of 5,685 equals
Now we have an equation that we know exactly how to handle. What does x need to equal to make this equation true? In other words, how many glasses wipers did Grant make last month?
The first step we need to take to move toward getting x alone, is to subtract left side, we end up with 690. And on the right side, we have just x. So, that's our answer. Connecting this new piece of information back to our story, the solution to this equation tells us that Grant manufactured 690 glasses wiper sets in the first month. If you think back to where we were at the very beginning of this lesson, you've made a ton of progress in learning about how to solve equations. If there's anything that you don't feel 100% comfortable with right now, go back and re-watch the videos covering it. Awesome job.
If we divide this entire left side of the equation by 3, then we just get rid of this coefficient in front of the parentheses and we end up with x plus 2. Over here, things get a little bit more complicated. Since we're dividing the whole side by 3, we have to divide each term by 3. 12 over 3 is 4 and negative 2x divided by 3 gives us minus 2 3rds x.