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Contents

- 1 The Expo
- 2 What Do We Know
- 3 What Do We Know
- 4 Cost Of A Nozzle
- 5 Cost Of A Nozzle
- 6 What Do We Want
- 7 What Do We Want
- 8 Word Equation
- 9 Word Equation
- 10 Cleaning Up
- 11 Cleaning Up
- 12 Enough Information
- 13 Enough Information
- 14 More Information
- 15 More Information
- 16 Enough Now
- 17 Enough Now
- 18 120 Nozzles Per Day
- 19 120 Nozzles Per Day
- 20 Ready Now
- 21 Ready Now
- 22 Features Of The Line
- 23 Features Of The Line
- 24 Find The Equation
- 25 Find The Equation
- 26 Grants Situation
- 27 Grants Situation
- 28 Point And Slope
- 29 Point And Slope
- 30 Solve For b
- 31 Solve For b
- 32 Substitute In
- 33 Substitute In
- 34 Clean Up
- 35 Clean Up
- 36 One Variable Per Box
- 37 One Variable Per Box
- 38 Distributing
- 39 Distributing
- 40 Undoing Distribution
- 41 Undoing Distribution
- 42 Factors
- 43 Factors
- 44 Common Factors
- 45 Common Factors
- 46 A Word Of Warning
- 47 A Word Of Warning
- 48 Try It Out
- 49 Try It Out
- 50 Another Tip
- 51 A Harder Example
- 52 A Harder Example
- 53 Challenge Question
- 54 Challenge Question
- 55 Back To The Line
- 56 Whats This Form Called
- 57 Whats This Form Called
- 58 Equation In Point Slope Form
- 59 Equation In Point Slope Form
- 60 Changing The Form
- 61 Changing The Form
- 62 Equation Through Two Points
- 63 Equation Through Two Points
- 64 Point Slope Form
- 65 Point Slope Form
- 66 Simplifying The Answer
- 67 Simplifying The Answer
- 68 General Form
- 69 General Form
- 70 Write In General Form
- 71 Write In General Form
- 72 A , B And C
- 73 A , B And C
- 74 More A B Cs
- 75 More A B Cs
- 76 Vertical Lines
- 77 Vertical Lines
- 78 Vertical Lines 2
- 79 Vertical Lines 2
- 80 General To Slope Intercept
- 81 General To Slope Intercept
- 82 Ad Agency
- 83 Ad Agency
- 84 Slope
- 85 Slope
- 86 Point Slope
- 87 Point Slope
- 88 Slope Intercept
- 89 Slope Intercept
- 90 Shotrtcut
- 91 Shotrtcut
- 92 Graphing
- 93 Graphing
- 94 Modify The Graph
- 95 Modify The Graph
- 96 Breaking Even
- 97 Breaking Even

People at the expo in the wonderful city, Udacity, love Grant's glasses wipers, but they're also really looking forward to his nozzles coming out. In order to ensure the success of his business, Grant wants to move into nozzle manufacturing really soon but nozzles, which if you remember, are the things that spray cleaning solution onto the glasses They're a little bit harder to make than glasses wipers. He's going to need to buy new machinery in order to start manufacturing them. On top of having to purchase this new machinery, Grant will also have to pay for the materials that are required to make a single nozzle. These cost about $2 per nozzle. As always, Grant's on a pretty tight budget and he'd like our help to figure out how much he should plan to spend on this new nozzle making machinery.

This is definitely going to be a big issue for us to help Grant tackle. So, I think we should break this problem-solving down. I also recognize that there are a lot of words on the screen right now, and that can definitely feel overwhelming. But if we just take things step-by-step, we can figure this out. Let's start by talking about what we actually know right now. What information does the problem give us? Read through this problem statement again, and then tell me, what do we already know. Do we know how much Grant should spend on nozzle machinery? Do we know how many nozzles he should make per month? Do we know how much materials for each nozzle cost? Or do we know how much Grant wants to spend total on nozzle production?

The problem tells us how much materials for each nozzle cost. We don't have any concrete information about any of the other three choices quite yet.

So how much does it cost to make each nozzle since we supposedly know that. Assume that the materials are the only thing that goes into the cost for making nozzles.

It costs $2 to make each nozzle. We can see that in the problem with this sentence. It says, the materials to make each nozzle cost $2.

Now that we've the information that we're given nailed down, what do we want to find out in the end? What is the final quantity that we're trying to figure out with this problem? There may be other things we need to figure out along the way to get to this final quantity, but right now, I'm only interested in what the problem is asking us for.

The last sentence of the problem tells us what we're looking for. It says that Grant wants to know how much money he should spend on the new nozzle making machinery. So this is asking how much he should spend on the new machinery, which is this first choice.

Now that we have our given information and the goal of our problem laid out explicitly, we can work toward creating and equation to help describe the situation. I'd like you to use these quantities down here to create a word equation in these blanks. I've made things a little bit easier for you and filled in operators already. So please, in each of these blanks, put the letter of the quantity down here that you think belongs in that position.

And the order of the letters that you should have is b equals a times c plus d. However, you could also have a and c switched here. Of course, because of the commutative property of multiplication. Doesn't matter whether we multiply the number of nozzles by the cost to make each nozzle or the cost to make each nozzle by the number of nozzles made. Either way you have it. This equation says that the total amount that Grant will spend on nozzle production depends both on how much he spends on the new machinery to allow for that production, and how much he spends in buying products to make each nozzle. Please notice what we're looking for the problem in the end is the quantity d over here. The cost of new machinery to Grant. And although it's not by itself an equation yet, you can use your algebra skills to pretty easily rearrange things so that d is by itself.

So here's the word equation we came up with in the last problem. Clearly though, considering I had to write this on two separate lines, there are a lot of words here and this is going to be kind of confusing to keep track of. So, let's use some different variables. I'd like to let x equal the number of nozzles made and I'd like to let y equal the total amount spent on nozzle production. So, with those substitutions, our equation will look a little bit cleaner. With these substitutions, it's pretty clear that this equation is written in slope intercept form, which we are now very familiar and comfortable with. So thinking about where the slope should show up in this equation and the information that we're given. What number do you think the slope is equal to in this equation?

The slope of our equation, because this is in slope-intercept form is just the coefficient in front of the x, so that's the cost to make each nozzle. We know that each nozzle costs $2 to make, so m is just 2. This is the thing that's going to determine how much the total amount of money spent or y will change as the number of nozzles or x changes. Now that we know that m is equal to 2, we can replace cost to make each nozzle with 2. Well, that looks even better. I am so excited about where we're going with this equation.

Since we know our equation is written in the form y equals mx plus b, cost of new machinery must be this b. In other words, cost of new machinery is the y-coordinate of the y-intercept of this line. I am just going to make that variable replacement right now, and just replace cost of machinery with b. So, this equation looks pretty good. But can we go any further? Think about what we are trying to find. We want to find out how much Grant should spend on nozzle machinery. Keeping in mind what it is we are trying to find out, do we have enough information to go further with this problem, and calculate that quantity? In other words, do we have enough information to come up with a number that is equal to the amount that Grant should spend on nozzle machinery? Please tell me yes or no.

And the answer is no. We have three things that we don't know in this equation. We don't have any numbers to replace y or x or b right now. We need to know more from Grant before we can move forward.

Conveniently, Grant just made the decision that he would like to keep the amount he spends on nozzles in the next month, at most $10,000. So we have something new to add to our what we know list. Assuming Grant maxes out this budget, he will spend $10,000 total on making nozzles. Thinking about what our variables stand for, which one of them could be replaced with 10,000? Do you think it belongs in the spot of y, or x, or b?

Since 10,000 is the amount that Grant wants to spend total on nozzles and y equals the total amount of money spent on nozzles, 10,000 is a value for y.

Knowing that we can replace y with 10,000 in the equation and thinking about what quantity we're looking for in the end, which of course is b. Do we now have enough information to come up with a number for b?

No. We still do not have enough info. To find out b. Even if we replace y with isolate one of them and have a pure number for what it's equal to. I guess we need to know more.

Fortunately, for us, again, Grant tells us more information. He tells us that he's going to make 120 nozzles per day for the next month. Taking this information into account, can you figure out a value for x based on this data? Remember, that x stands for the total number of nozzles made and I'd like to know what x will equal after 1 month.

Grant's going to make 120 nozzles every day for 30 days. So that means the total number of nozzles he'll make, is just 120 times 30. That is equal to 3,600.

Now that we've had even more information added to our pot of knowledge, can we now figure out what b is? Do we have enough information to solve our problem?

And the answer is, yes. We have a value for y and the value x that goes with that value for y. So b is the only unknown left in our equation.

What do we know about our line at this point? I'm not really interested in things that we can figure out. There are a lot of things that we can figure out. But just things that we know explicitly right now. Do we know its y intercept, its x intercept? Its slope? Or 1 point that lies on it? You can pick as many of these answers as you think are correct.

We figure out before that the slope of r minus 2 so we definitely know that. And we have a y and x coordinate or an x and y coordinate peer that go together and they make the point 3600, 10000. So we know one point that lies on our line as well.

Now that we have a point and a slope for our line, we can calculate b and come up with a final equation in slope-intercept form. Where x and y are the only variables. Please find that equation for me and put it in this box.

We can solve for b by simply substituting in the values x and y that we know from our point. If we do that and a bit simple algebra, we can[UNKNOWN] the value for b of 2800. Inserting this into our equation, gives us a final equation in[UNKNOWN] form of y equals 2x plus 2800.

What a beautiful equation we have now. But what does this mean for Grant's situation? Does this tell us that 2800 nozzles will be made in the first month? That he'll spend $2800 total on glasses in the first month? That every month he'll spend $2800 more on nozzles or that he can spend $2800 on new machinery to make his nozzles?

B or the y intercept, represented the amount that Grant could spend on new machinery and stay within the budget that he'd set for the first month. So, this last choice is correct.

As much as I love the new equation we just came up with, it took us a really long time and a bunch of steps to get it in this nice form. It would be really great if we could come up with a shortcut for how to get to the same end result without having to do quite as much work. The information that we used to develop our equation in the last problem was a point and a slope. Let's consider a general situation like this. Let's say that we have a line, and we know that the slope of the line is m. We also know that this line goes through the point, s, t, where s is our x coordinate, and t is our y coordinate. I'd like to start off by having you substitute these values in to the proper places of an equation that's in slope-intercept form. So we want to start off with y equals mx plus b, but then substitute in these quantities to the proper spots.

We should have t equals ms plus b.

Starting with this equation, what do you get if you solve for b? Write an expression that b is equal to in this box.

We just need to subtract ms from both sides to end up with t minus ms equals b. Or, because we can just switch things to either side of the equal sign if we know that they are equal, b equals t minus ms.

So now we have an expression for the y intercept of a line with slope m that goes through the point s, t. What happens if we plug this expression for b into the general equation for slope intercept form? Write the equation that, that would give us in this box.

We start off with y equals mx plus b, and then in place of b, substitute in t minus ms. Great. That's our new equation.

So now we have an equation in slope intercept form that takes into account both that our slope is m and that we have the point s,t on our line. This is really cool, but I don't think this looks very pretty right now. And you know me and how I like things in our equations to look pretty. So, let's see if we can clean it up a little bit. In an effort to make this look a little bit neater, what happens if we move t from the right hand side of the equation to the left hand side of the equation? Of course, we need to obey the rules of Algebra. But if I said I only wanted the term with t on the left, what equation would we have now?

To move t from the right side and have it on the left side instead, we just need to subtract it from both sides, so we end up y minus t equals mx minus ms.

We're definitely getting there. We have everything related to y on the left side of the equation, since t is our y coordinate, and everything related to x as well as our slope on the right side. However, I see one thing that we can do to change this a little bit that I think will help us out. I provide a new form for you that I'd like us to write this equation. So see if you can figure out which variable, that's on the right side right now, belongs in which box. Please note that I only want one variable in each box. And of course, I want this equation to be equivalent to this equation.

This is a little bit tricky. It's not exactly like anything we've done before, but here's the answer. We have y minus t equals m times the quantity x minus s. We'll talk more about why this works in just a second. What we have to notice about the right hand side of this equation, the second to last one we came up with, is that each term has a factor of m. We call this a common factor, since the two terms share it. So that means each of these terms can be divided by m, but it wouldn't end up with a remainder for either one. If we redistribute the m by multiplying it by either terms inside the parenthesis here, we would end up with same equation as we had just before.

Let's think more about this concept in a general case. If we have three numbers, a, b and c, and we have a times b plus c, what expression would you get if you wanted to get rid of the parentheses here?

Distributing multiplication by a to both terms inside the parentheses gives us a times b or ab plus a times c or ac. So now, we can see that both terms here have a common factor of a. This is basically the opposite of the step that we took in modifying our equation earlier. We're getting rid of our parentheses here instead of adding them in.

What if, instead we have xy plus xz? Can you write this in a form so that x is not distributed?

We get x times the entire quantity y plus z. Both terms in the first expression have a factor of x so we can sort of pull an x out of each term, write that as a factor that's multiplying two terms inside parenthesis and, of course, these two terms which are the terms left over from each of the original terms still have the same relationship as these two do. These are added together.

Its been pretty easy to see what the factors of these terms with only variables in them are, but what if we have a number instead? An integer factor of a number goes into that number, an integer number of times without leaving a remainder. So, this is a number that we can divide evenly into another number. So, for example, since 8 is equal to 4 times 2, 4 and 2 are both factors of 8. So, for example, what are all the factors of 6? Please fill in those four numbers here.

The factors of 6 are 1, 2, 3 and 6. We just need to think about all the combinations of 2 integers that can be multiplied together to equal 6. 1 times 6 equals 6 and 2 times 3 equals 6. So these the only 4 factors of 6.

As you saw earlier with mx minus ms equals m times x minus s, we can rewrite expressions by looking for what factor different terms in that expression have in common. So, let's try this with a few more expressions. Let's start off with expression inside parentheses. For this problem and for the several problems after this one, I'd like you to restrict the factors that you pull out of the original expression to the integers or variables. And make sure that what you have in the end, in the boxes and parentheses, doesn't have any common factors between the terms, aside from 1, of course, since everything is divisible by 1. Write the common factor that these two terms have in this box, and that the expression that needs to be multiplied by that factor in this box.

times 2x minus 3. Since 3 times 2x is 6x, and 3 times negative 3 is negative 9.

What about 6x plus 3? What belongs in these boxes then?

a little bit tricky. One mistake that is really easy to make, is to forget about this second term inside the parenthesis, the 1. But if we leave it out, then we only have one term inside of the parenthesis. 3 times 2x is just equal to 6x, and this 3 would have nowhere to come from. Checking to make sure that this answer when you simplify it, has the same number of terms as the first answer, is a pretty fundamental importance.

Let's try out 12x plus 18.

The answer is 6 times 2x plus 3.

Let's look at another case of a pretty common mistake people could make on this problem. Let's say that you looked at this and though oh, 12 and 18 are both even numbers that means they are divisible by 2. And, if we factor out a 2 from both terms here, we're left with 6x plus 9. But remember I said that I didn't want to see any common factors between terms. Inside the parentheses aside from 1. And 6x and 9 are both divisible by 3. That's not super hard to fix, though. If I pull out a 3 from here, then my coefficient, or rather my term multiplying the things inside parentheses, is 2 times 3.

How about 12xy plus 18x squared?

This time we have to worry about both variables and coefficients. If we look at the coefficient in each terms, they have a common factor of 6, so we know that we have to pull that out of our expression, but there's also an x in both the full common factor that 12xy and 18x squared share is 6x. I mean, you divide each term by that, you get 2y and 3x.

Now for a challenge question, how would you deal with 3x times x minus 1 plus 4 times x minus 1? What common factor is there between this term and this term and then what's left over when you pull that out. You may need to expand our definition of what can go in this box for this particular question.

The answer is x minus 1 times 3x plus 4. Recognizing that x minus 1 is a factor that both of these terms had, definitely takes a little bit of outside the box thinking. It is, however, in each term,multiplied by the other factors in the term so it count a factor. Just to get your brain jogging for later on in the course, think about what would happen if you multiply all this out. You did this before of course, but just think about this answer and the connection with, what this looks like when you distribute the terms.

Hopefully now the way that we modified this equation for our line earlier makes a little bit more sense. Remember that we had y minus t equals m times x minus s. But what does this really mean? Let's think back to what we called different parts of our line, or different characteristics of our line, when we created this equation. Well, this t was our y coordinate of the point that we knew. So this left hand side of the equation really says y minus the y coordinate and on the right side, m was our slope and s was our x coordinate. So we've just derived another way to work out the equation of the line. That's pretty impressive. This is really incredible that you guys did this yourself. Now, any time that we know the coordinates of a point on a line s and t, or the x coordinate and the y coordinate. And we know the slope of the line, we can automatically write that line's equation. This is a super, super powerful tool. However, I'm lazy, and writing it out this way is really time consuming. But writing it in the original way doesn't really show us what s and t actually stand for. It's a little bit Less clear what the meaning of these letters is. Let's see what we can do about this. Instead of calling our point here s, t I think we should call it x1, y1. That way whenever we use these terms in the equation we know what kind of values they are. We know whatever is in the place of x1 is going to be an x coordinate and whatever is in the place of y1 is going to be a y coordinate. So that updates this equation. Remember that y and x here are general variables, but x1 and y1 will be the fixed values that we pick based on a concrete point that we know. These values will vary. As we move up and down the line. These values will stay fixed.

Thinking about what information we used to develop this equation, what do you think that we should call this form of linear equation? Should we call it slope intercept form, standard form, point-slope form, general form or point-intercept form?

Since we started off knowing the coordinates of a point that lie on this line and the slope of the line, this is called point-slope form.

Now that we have this new formula, let's try using it. Let's say that we have a line with a slope of 5 that we know goes to the point 2,3. What would be an equation for this line in point-slope form?

Let's talk about the information we get from the problem. We see that the line has a slope of 5, so that means m equals 5. The fact that it goes to the point form equation, and y1 equals 3. Now we just need to insert these into the equation. We get y minus 3 equals 5 times x minus 2.

Great. Now that we have this equation in point-slope form, let's see if we can simplify and get it into slope-intercept form instead.

First things first, we need to distribute this multiplication by five to both the x and the negative two. Since in slope intercept form, y is all by itself on the left side of the equation, we're going to need to move this negative three term out of the left side. So we add three to both sides, since right now it's being subtracted. And we end up with a final equation of y equals five x minus seven. So, this equation. In our earlier equation should be equivalent or rather these should be equations that graph exactly the same line. To make sure that this equation fits the information that we used to create the equation end points slope form, lets see if the point we started are with lies on this line. We can already tell that the slope of this line is 5. So that gets rid of one of our criteria. Lets check each side of this equation done with the point 2, 3. One the left hand side we have y, and substituting in our value for y, we have expression on the right hand side of the equation is 5x minus 7. Let's see what that is equivalent to when we plug in our point. So you substitute 2 in for x. And for this side of the equation we also end up with 3. Since our left hand side and our right hand side match each other, we plug in our information. We're good to go. We have confirmed that this equation fits our information.

So, I hope you can tell, that both point-slope form and slope-intercept form, are super useful for us, depending on what information we start out with in our problem. What if we know that a line goes through 2 points, 7, 4 and 3, 9? Let's see if we can come up with an equation for this. Instead of doing this all in one step though, let's break it down. First things first, let's find the slope. Please fill in this box for the value of m.

And we get m equals negative 5 4ths.

If you feel comfortable carrying out all the steps at once right now for figuring out the equation of this line and then putting it in slope intercept form, please go right ahead. Then you can check your steps by walking through what I'm doing afterward. Of course, there are also multiple approaches to this problem. I'm just going to show you the one that I think is the most efficient. Now that we found the slope, I would recommend using the point slope form substituting in our value of m that we just found and the coordinates of one of these points. What equation end-points slope form do you get? Remember, you can choose either one of these points.

You could either have y minus 4 equals negative 5 4ths times x minus 7. Or if you used the other point, you could have y minus 9 equals negative 5 4ths times x minus 3.

Finally, let's put this equation into slope-intercept form. Please put your answer in this box.

I just chose one of our two possible equations in point-slope form to see what we end up within slope-intercept form. We should get the same answer regardless of which of these versions of the point-slope form equation we used. Either way, in the end, we get y equals negative 5 over 4x plus 51 over 4. Remembering, of course, that 4 is equal to 16 over 4.

So we have this equation written in slope intercept form now, which is great. But it's kind of yucky looking. At least, if you're me, and fractions aren't necessarily your favorite thing. There is however, yet another form of a linear equation that we can use to rewrite this, and maybe make it look a little bit more pleasing. And that is called the general form of, of linear equation. We can rate a general form of the general form proven to your equations as Ax plus By equals C where A, B and C are all constants. In addition to this, we have one more little requirement. One thing that's not allowed is for A and B to both be equal to 0. If that was the case of course, this term would be 0. And this term would be 0 as well. So we would just end up with the equation 0 plus 0 equals 0 which maybe true but isn't a linear equation because there are no variables in it. What the general form does then, is basically shift everything on the variable to one set of the equation and shift all the constant terms to the other side of the equation and then of course combine whatever constant you have into one number. So let's play around with this. Since the coefficient in form of y doesn't have to be 1, it can be some other constant in general form. Let's get rid of these fraction. What I'd like you to do then is to multiply both sides of these equation by some number so that there are no more fractional coefficients or constant terms. Try however, to make this number you multiply both sides by the smallest number that you could use to get rid of the fractions.

Since the denominator of our fractions on the right side is 4, to get rid of the fractions, we just need to multiply both sides by 4. Remember that since we're multiplying this whole side by 4, we need to multiply each term by 4. The 4 distributes. And we get 4y equals negative 5x plus 51. Well, I like that a lot better.

Now, I'd like you to write this equation in general form.

In general form we just have all the variable terms on one side, and then a constant term on the other side. The only change we need to make to this equation then is to get the term with the x onto the left side of the equation. We just need to add 5 x to both sides, and we get 5 x plus 4 y equals 51.

Let's look a little bit more at the general form for a linear equation. I'm going to do a little thought experiment concerning a, b, and c. Now what would you have if both a and b were equal to 0, but c was not equal to 0. Would you have an equation for a vertical line? An equation for a horizontal line? Would you have something that's not an equation at all? Or would you have a conditional equation?

It actually wouldn't be an equation at all. If we substitute 0 into the spots for A and B, in the equation we would get 0x and 0y equals C and if the left hand side is just equal to 0 and that would require C to be 0 as well, but the questions specifies that c is specifically not 0. This can not work. This is not an equation.

But what if A is zero and B is not, then what do we end up with?

We would get a horizontal line. If A is equal to 0, then this entire first term is equal to 0, so our equation would just be by equals C. If we solve for y or isolate y, we have the equation y equals C over B. So, y equals a constant, which we know is the form of a horizontal line.

We talked earlier about how a vertical line cannot be written in slope intercept form. Is it because the slope of the line is zero, the slope is not defined or the y intercept is zero?

For a vertical line, the slope is not defined. Remember that if we want to find slope, it would take the difference between two y coordinates along this line, But then, because all of the x coordinates are the same, we would have to divide by 0, which we know is a major no, no. Or, at least, not allowed, if we want to define this slope.

So we can't write the equation of a vertical line in slope intercept form. But what about the other two forms for linear equations that we know of? Can we write it in point slope form? Or can we write it in general form? Please pick yes or no for each of these questions.

We cannot write the equation of a vertical line in point slope form. Here, the slope is going to be an issue again. We still can't plug in a single numerical value for the slope in this form of the equation, so we can't use it. But yes, we can write the equation for a vertical line in general form. This is one of the major powerful aspects of the general form of linear equations. Looking back at general form, or the general form of the general form, all we would need to do would be to set B equal to 0. That would make this entire term with a y in it equal to 0. And we would just have Ax equal C, which very easily could be become an equation for a vertical line like many we've seen before.

Sometimes we want to convert equations from being in the general form to being in slope-intercept form, since it's easier to read off the slope and the intercept when it's written this way. Just real quick, try to do this with the equation 12x plus 5y equals 7.

Right off the bat, let's remember what slope-intercept form is. You've probably got it but it never hurts to write it down one more time. We want to get this equation in the form y with an coefficient of 1, or course, equals mx plus b. The first thing we need to do then is to get our term with y to be the only term on the left side of the equation. So we need to subtract 12x from both sides. Then, to get the coefficient of y to be 1, you just divide both sides by 5. Then, if you're picky like me and you want the equation to be exactly in this form, we just rearrange the order of our two terms on the right-hand side. We're allowed to do this because of commutativity and we end up with y equals negative is negative 12 over 5. And the y coordinate of the y intercept is 7 over 5.

Back at the glasses expo, Grant has noticed what awesome flashy displays some of the other vendors have. He just got his little glasses wipers and his little sign. Investing in some advertising seems like a really great idea for his business. Unfortunately, Grant runs into a woman who works for an ad agency and she guarantees with her agency's help, Grant can make $5,000 in just 2 weeks. Of course, there is a price to pay. That will cost Grant $2,000 to work with them. If we assume that Grant earns money at a constant rate or in other words that the graph of profit versus time is a linear graph and a linear equation. How long will it take Grant to break even? That's what we want to figure out. Let's say that x is equal to the number of days since the answer released, and let's say that y is equal to Grant's profit in dollars. Thinking about the information that the problem gave us, what two points we know lie on our line? Note also, that making $5,000 means making a profit of $5,000.

One point that we'll have is 14,5000. Since, after 2 weeks or 14 days, Grant will have a profit of $5,000. The other point is a little bit trickier. We know the ad agency wants Grant to pay them $2,000. So that means, at the start of his ad campaign, or after zero days have passed, he'll have a net loss of $2,000. So, the point 0, negative 2000. If this is a little bit tricky for you, not a big deal at all. I definitely see why this could be hard to figure out.

Now that we know two points, what can we say the slope of our line is?

Our slope is 500.

Now that we have a slope and 2 points to choose from, please write an equation in point-slope form for this data. Don't simplify just yet. And remember, it doesn't matter which point you use.

We have two possibilities. We could either have y plus 2000 equals 500x or we could have y minus 5000 equals 500 times x minus 14.

Now please choose one of these equations to start with and rearrange it so that it's written in slope-intercept form instead.

It's probably easiest to start with this equation on the left. Since all we need to do is subtract 2,000 from both sides. This gives us a final equation of y equals 500x minus 2000.

Thinking back to what information we started off with, or what information we had after we calculated the slope for this problem. Did using point slope form to get us eventually to the slope intercept form of our equation, give us a short cut? Or did it actually give us extra work? We know that, in some cases, it helps us out. But was that the case this time?

So actually, using point-slope form this time gave us extra work. If you look at which 2 points we had, one of these this first one, 0 negative 2000, is in fact the y interceptor of our graph. Since for slope intercept form we just need to know the slope and the y quadrant of the y intercept, we could have avoided all that work we did in the middle, using the point slope form. Instead we could have just written 500 in the place of m and the spot of b, negative 2000. The moral of the story is that point slope form is super, super useful in certain situations. But we need to think critically about which forms of equations we should use. Thinking not only about what kind of information we have but also the actual contents and meaning of that information.

Now that we have our equation, it would be great if we could graph it. Here are four different lines, and which one do you think represents this equation?

And the answer is line D. Note that the scale on the x-axis is that two days pass for every square we move over. Now on the y-axis, every grid line indicates another $1000 in profit. We can see that on our orange line, if we move over two days, we have to move up a $1000, and 1000 over 2 is 500. We also can see that the y intercept is at 0, negative 2000, which is what we're looking for.

We know that this line represents this equation. So this line shows all of the pairs of x and y coordinates that satisfy this equation, make you see that the line is infinitely long, both in the negative and the positive directions. So thinking about what the situation we're dealing with and trying to represent with this equation actually is. Especially keeping in mind what x and y both stand for in the story with Grant, maybe you think that our graph is perfect just the way it is. You shouldn't change it. Maybe you think we should start at x equals 0 and continue upward, or maybe you think we should start at y equals 0 and continue upward. Take your pick.

I think that we need to start at x equals 0. Remember that x stands for the number of days that have passed since Grant started using his advertising agency. We can't really have negative days, or at least in this situation that doesn't make sense. So I think that the graph should start here and then continue upward just as we had it. So this would just be omitting the values where we would have a negative number of days having passed.

The last thing that we want to figure out is when Grant is going to break even from this investment in advertising. Remember that, the moment of break even is when profit is equal to zero. So please tell me how many days it's going to take for this to happen.

If we're looking for profit equals 0, that means we're looking for the point on our line where y is equal to 0. That's just along the x axis. This is the x intercept of our graph and the coordinates of this point are 4,0. So that means that Grant will break even after just 4 days of using this ad agency. This investment is going to pay for itself in only four days. That seems like a pretty smart business decision to me.