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Interpreting points in context of graphs of systems

Given a graph of a system of equations and a context, practice interpreting what various points represent in that system.

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  • aqualine ultimate style avatar for user ndemisse401557
    what is this i dont get it
    (8 votes)
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  • blobby green style avatar for user jbrown97
    I think I am slowly understanding what this means, however, I feel like I need more practice with it all of it. Also, wherever the variable is on the graph, the number underneath or nearby represents that.
    (1 vote)
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    • blobby green style avatar for user Bilal Mohammed Ali
      Simply, Lauren has a sum of money. So she wants to spend this "sum of money" on things like "something." Now the data given about Lauren's situation through Sal's example — we are going to interpret them informally like:

      Lauren's sum of money → $ = $220
      something → beans = 80 kg

      Just remember that we have a couple of constraints that intersect at a given point on the graph. They can be given a description at any point on the graph by Khan Academy even when either of conditions: have no intersection, have intersection.
      Now, we as mathematicians trying to understand Lauren's problem to solve it, we found out that the problem is a bit complicated, but we are not afraid since mathematics supports us with its tools like variables and constants. That; however, is, it is true that Lauren wants to buy a 80 kg of beans, and, it is false that the beans out there in the market found in a single sort of mass. It is, again, true that the constituents here are two different ones in sort, but Lauren does not matter the equality of one sort, say x, over the other one sort, say y. What does matter for Lauren when buying the beans is the the couple of constituents, which are dark beans + light beans, has both to be equal to 80 kg in total amount. Again, she does not matter about the amount of dark beans to be equal or less to or more than the amount of light beans, and, it can apply to light beans in reverse too. We are going to see how this no-matter-of-equality relationship relates when we expand on the analysis of data given to Lauren's problem.

      If we go back to the context of the problem, we'd see clearly that the price of the dark beans ($3) is higher in one degree than the price of light beans ($2). So it's obvious to learn that Lauren is going to get more light beans than dark beans in kilograms! Again, we as mathematicians have to clear concise to the problem we are stated, which means Lauren's is neither interested in a relation of knowing which constituent x is less than or equal to constituent y, nor are we interested in a relation of knowing which constituent y is more than or equal to constituent x. We mathematicians use less words! We say that Lauren is not interested in a relation of an inequality equation to x and y, like for instance, x ≤ y or x ≥ y, but we are interested in beyond! What's interesting for Lauren's and we is we have constraints: a: 3(60)+ 2(20) = 220, b: 20 + 60 = 80. This what means to have constraints. Remember that the constraints given we need them to be met like 0 = 0. If not, for example, a: 3(10) + 2(96) = 220, b: 10 + 96 = 80, then we are going to use descriptions in language instead to show such a relation of a fact deviation.

      Now let's go deep into Lauren's problem. The constraints given are a and b. a represents the sum of money in $, b represents the amount of beans in kg. So far so good. Remember that both constraints have to intercept at the same point of (x, y) on the graph to have the a condition 3x + 2y = 220 to be exactly like a: 3(60) + 2(20) = 220 :: 220 = 220, and the b condition x + y = 80 :: 60 + 20 = 80 to be exactly like 80 = 80. *If both constraints don't meet at the same point, this means that *each constraint would have a point of (x, y) on the graph, like: a: (x2, y2), and b: (x2, y2).*

      Now let's pick each point on the graph and see how it looks like:

      Point *C
      shows at a point of *(10, 95)* for the a constraint, and the b constraint down right below C shows at point of (10, 70), we get:

      point C of (10, 95)*

      a: 3x + 2y = 220
      3(10) + 2(95) = 220
      220 = 220

      b: x + y = 80
      10 + 95 = 80
      105 = 80

      This is if *a
      and b were to intersect at (10, 95) whereas only a does, and, even though they do intersection at that point the system of equation would dissolve from the side of constraint b.*



      point down right below C (10, 70)

      a: 3x + 2y = 220
      3(10) + 2(70) = 220
      30 + 140 = 220
      170 = 220

      b: x + y = 80
      10 + 70 = 80
      80 = 80

      This is if *a
      and b were to intersect at (10, 70) whereas only b does, and, even though they do intersection at that point the system of our equation would dissolve from the side of constraint a.


      So what we can see is a and b as two constraints fail to intersect at point C of (10, 95)* to satisfy the system of our equation. In description, we would say about point C that, *Lauren spends the intended amount of money and buys more than the intended amount of beans.* The point *down right below C at (10, 70)* also fails like the one above it that C, which we would describe it like "Lauren spends less than the intended amount of money and buys the same intended amount of beans.*

      Let's continue with other the points of *D, F, and E
      and their descriptions: (Notice that I am using the same descriptive language by Khan Academy)


      point D of (20, 60)

      a: 3x + 2y = 220
      3(20) + 2(60) = 220
      60 + 120 = 220
      180 = 220

      b: x + y = 80
      20 + 60 = 80
      80 = 80

      Description
      Lauren spends less than the intended amount of money and buys the same intended amount of beans.*


      point F of (30, 30)

      a: 3x + 2y = 220
      3(30) + 2(30) = 220
      90 + 60 = 220
      150 = 220

      b: x + y = 80
      30 + 30 = 80
      60 = 80

      *Description

      Lauren spends less than the intended amount of money and buys less than the intended amount of beans.*


      point E of (60, 20)

      a: 3x + 2y = 220
      3(60) + 2(20) = 220
      180 + 40 = 220
      220 = 220

      b: x + y = 80
      60 + 20 = 80
      80 = 80

      *Description

      Lauren spends the same intended amount of money and buys the same intended amount of beans.*

      Finally, We can see how *the point E of (60, 20),* that by which both constraints of *a
      and b are intersected.
      (10 votes)
  • blobby green style avatar for user janjiss
    What doesn't quite make sense to me is the fact that the graph represents light (y) and dark (y), however the green graph represents price, which is not represented in this matrix. Why does this work?
    (3 votes)
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  • blobby green style avatar for user sam
    Answered my own and someone elses question so figured it would be worth posting.

    How can the xy axis represent both cost and kilograms at the same time!? Well, they dont at first glance.... the graph for the video was chosen for kilograms. To represent cost, remember cost per kilogram was 3x for dark beans and 2y for light beans. Cost makes sense in terms of what something costs... per kilogram
    (3 votes)
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  • blobby green style avatar for user love.thomas
    Why is this hella boring
    (2 votes)
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  • blobby green style avatar for user Patrick Pendola
    how do you know dark beans should be x? if dark beans were y, it doesn't work.
    (1 vote)
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  • sneak peak purple style avatar for user Vera_Lamplighter
    please help I do not understand anything
    (1 vote)
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  • blobby green style avatar for user Elana
    I am slowly understanding what this however, I feel like I need more practice with it. Also wherever the variable is on the graph, the number underneath represents that.
    (1 vote)
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  • purple pi purple style avatar for user louisaandgreta
    0.18(350)+0.04(150)=0.138(500)mL


    How do you get 0.138? This is from one of the practice exercise
    (0 votes)
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  • blobby green style avatar for user Lynette
    so practically, without the graphing included when solving these equations, it is just simply algebraic equations??
    (0 votes)
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Video transcript

- [Instructor] We're told that Lauren uses a blend of dark roast beans and light roast beans to make coffee at her cafe. She needs 80 kilograms of beans in total for her next order. Dark roast beans cost $3 per kilogram, light roast beans cost $2 per kilogram, and she wants to spend $220 in total. And they tell us here's a graph that shows a system of equations for this scenario where x is the number of kilograms of dark roast beans she buys and y is the number of kilograms of light roast beans she buys. All right, let me scroll down so we can take a look at this. And so sure enough, so this blue line, and I'll write it out in blue, this x is the number of kilograms of dark roast beans, y is the number of kilograms of light roast beans, and she wants to buy a total of 80 kilograms. That's what they told us up here. We can go back to look at that. She needs, I'll underline this in blue. She needs 80 kilograms of beans. So that constraint that the sum of the kilograms of dark and light is equal to 80, that's represented by this equation. And if we were to graph it, that is this blue line right over here. And then this other constraint, three x, well, the dark roast beans cost $3 per kilogram, so three x is how much she spends on dark roast. Two y is how much she spends on light roast 'cause it's $2 per kilogram. And 220 is the amount that she spends in total. And they tell us that up here. Dark roast beans cost $3 per kilogram, light roast beans cost $2 per kilogram, and she wants to spend $220. So this equation is another way of expressing what I just underlined up here in green. And the green line shows all of the x y combinations that would match these constraints. And so now let's do something interesting. They've labeled some points here, point C, D, F, and E. And we're gonna think about what do each of these points represent? So for example, this point C that is on the green line, but it sits above the blue line, what does this tell us? What does this point C represent? Pause this video and think about it. Well, if we're on the green line, that means that the amount that she spends on dark roast plus the amount that she spends on light roast is adding up to exactly $220. So she's definitely spending $220 at C, but how many total kilograms is she using? Well, the fact that for this given x, we're sitting above the line, that means that she's not using exactly 80 kilograms. And we can see that over here. She's using, looks like 10 kilograms of dark, and it looks like something like 95 kilograms of light. If you were to add those two points together, it looks like she's using something closer to 105 kilograms. So point C is a situation where she is spending exactly $220, but she's using more than 80 kilograms 'cause it's not meeting this second constraint. It's sitting above that line. Now let's think about point D. What does that represent? Pause the video and try to figure that out. Well, because we sit on the blue line, that means that we are meeting this constraint that the kilograms of dark and light combined is equal to 80 kilograms. So she's using exactly 80 kilograms here, but what about her spending? Well, because this point lies below the green line, that tells us that we are spending less than $220. And we could even try it out. Three times 20 plus two times 60 is what? 60 plus 120 is $180. And so this is a point where we're meeting this constraint, but we're not meeting this constraint. We're underspending right over here. Now, what about point F? Well, point F sits below both of these lines. So pause your video and think about what that means. Well, if we're sitting below both of these lines, that means that neither are we spending $220, nor are we using 80 kilograms. And you can see that if you actually look at the numbers. You don't have to do this, but this is just to make you feel good about it. It looks like she is using 30 kilograms of dark and 30 kilograms of light, so in total she is using, so this is a situation where she's using 60 kilograms in total, not 80. And so that's why we're not sitting on this blue line. And if you look at how much she's spending, she has 30 kilograms of each, so three times 30 plus two times 30, that's going to be 90 plus 60. That's also less than 220, and so that's why we see this point is below these lines. And then last but not least, what does point E represent? Well, point E sits on both of these lines, so that means that it meets both of these constraints. This is a situation where she is spending exactly $220, and the total number of kilograms she's using of dark and light is exactly 80. And so if we wanted to say, hey, what combination of dark and light would she need in order to meet both constraints, E represents that, the intersection of these two lines.