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Slope and intercept meaning in context

Practice determining which feature of a linear model (the slope, the x-intercept, or the y-intercept) is useful for answering a given question in context.

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  • hopper cool style avatar for user Noah
    is slope run over rise or rise over run
    (12 votes)
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  • hopper happy style avatar for user Ian Wu ✅
    **********************************Hard Question**********************************
    What if John loaned Flynn 28 dollars? And Flynn's salary per week is 3 dollars. And every week the loan increases by 0.078 percent until Flynn paid every single dollar. How long will it take for Flynn to pay John back? And how much will it cost him at last?
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    P.S. Please comment the answer down below :)
    (1 vote)
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    • female robot grace style avatar for user Isaac Kriegerventh
      By pure algebraic manipulation I can't find a solution because the equation is of the form:

      D = D_0*(1+r)
      D_0*(1+r) = -3t + 28
      Where D_0 = previous value of D.
      r is the weekly interest rate.
      t is the time in weeks.
      This could be solved by iterations if I understood you correctly, Flynn is only allowed to pay 3$ per week and AFTER he has paid, the weekly interest applies. First example of the iteration is:

      D = 28
      D_0 = 28 - 3 = 25
      D = D_0*(1+7.8/100) = 25*(1.078) = 26.95
      D_0 = 26.95-3=23.95
      D = 23.95*1.078=25.82
      ... So on and so forth.

      Then we can measure the time in weeks spent up to 0$ or below even to know long will it take to pay back. The total amount spent is greater than 28$:
      P = 3*t
      A little less actually, because the last week Flynn is going to pay <= 3$ (very small probability of paying exactly 3$), the last equation is good aproximmation regardless.
      The answer for a 7.8% (that is what I think you meant) is 15 weeks and 44.93$ spent int total.

      If it is 0.078% (way too low weekly interest rate), 10 weeks and 28.09$ total spent.

      Compared to the easy version which is only: 28/3 = 9.33 weeks or 10 weeks realistically.

      A simple program in c++ or python works for the iterative approach... now, I couldn't come up with an analytic one maybe someone could.

      Good problem!
      (10 votes)
  • starky seed style avatar for user Joseph Billingsley
    Who is Flynn and why should I care?
    (5 votes)
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  • male robot hal style avatar for user Ling Qiu
    Did Flynn have to pay interest? Just kidding :)
    (4 votes)
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  • blobby green style avatar for user cheese
    interesting....
    (3 votes)
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  • starky tree style avatar for user CrazyCanuck
    So I understand both lessons, but I can't seem to understand the first practice (Relating linear contexts to graph features), the second one is very clear to me (Using slope and intercepts in context). Could someone explain it so I can better understand it?
    Thanks in advance :)
    (2 votes)
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  • piceratops seedling style avatar for user Vega, David; 201117643
    I do not have any questions
    (0 votes)
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Video transcript

- [Instructor] Flynn's sister loaned him some money, and he paid her back over time. Flynn graphed the relationship between how much time had passed, in weeks, since the loan and how much money he still owed his sister. What feature of the graph represents how long it took Flynn to pay back the loan? Pause this video, see if you can figure that out. All right, now let's go through each of these choices. So the first one is the slope. So the slope tells us how much do we change in the vertical direction for any given change in the horizontal direction. So for example, if we start from here, we can see that if one week goes by, we go from one week to two weeks, we can see that our loan went down. It went from $15 to $10. So when we had plus one weeks in time, our loan went down by $5. And then that happens over the next week. And that rate of how quickly the loan is paid, that's what the slope tells us. And it would be constant over the course of the entire time period is what makes this a line, is that slope is always going to be constant. So the slope is useful for the rate at which the loan is being paid back, but it's not the clearest way to figure out how long it took Flynn to pay back the loan. So I would rule that one out. The x-intercept, that's where the graph intersects the horizontal axis, which is often referred to as the x-axis. Or another way to think about it, that tells us, that says, what is the x-value when our vertical value, our y-value, is equal to zero? And our y-value is the money owed. So it says, hey how much time has passed when we owe, when we don't owe any more money or when Flynn doesn't owe any more money to his sister? Well that's exactly what they're asking for. How long did it take Flynn to pay back the loan? And that's what that x-intercept tells us. It took him four weeks. After four weeks, he didn't owe any more money to his sister. So I like this choice, but let's just review the other ones. The y-intercept, well that's where the graph intersects the y-axis right over there. This tells us what's going on at time t equals zero. At time t equal zero or time equal zero, when the horizontal variable is equal to zero, we see that Flynn owes $20. So this one isn't valuable for figuring out when he, how long it takes to pay it back. This one is useful for figuring out how much did he owe initially. So we'll rule that one out. And we already picked one choice, so we can rule out none of the above.