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Constructing linear equations from context

Given a written description of a linear relationship in a some context, write an equation that represents the linear relationship described.

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  • starky ultimate style avatar for user Dabba
    I've redone this segment at least 10 times and everyday I redo it I don't get any closer to figuring it out, the word problems are so confusing. I understand what I'm supposed to do but I can't seem to determine what value goes to what.
    (30 votes)
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    • aqualine ultimate style avatar for user famousguy786
      I also had to do it many times before mastering it. My advice to you for this exercise is that you should first be proficient with all the forms of linear equations. Once you've done that. read the question and think-which form should be used here?
      For example, if only the slope and y-intercept is given you'll have to use the slope-intercept form.
      Also, this exercise demands the ability to convert verbal information into mathematical form. For example, the rate with which Mr.Mole digs is the slope of the line and the initial depth of his burrow will be the y-intercept.
      Finally, don't give up! All these times you've attempted the exercise, you've been developing your brain. You can learn anything!
      (14 votes)
  • winston default style avatar for user Michael Upleger
    How does he get 125 m/hours??
    (6 votes)
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  • male robot donald style avatar for user Marshal Guo
    kinda confusing ngl, all of the questions i do don't have a starting point like 1200m how do i complete it then?
    (8 votes)
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  • blobby green style avatar for user sebastian otero
    I'm stuck on this :/
    Addison painted her room. She had 50 square meters to paint, and she painted at a constant rate. After 2 hours of painting, she had 35 square meters left.
    Let y represent the area (in square meters) left to paint after x hours.
    (5 votes)
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  • aqualine ultimate style avatar for user LeannaDelgado7
    This problem is not even close to the ones that I encounter:

    A young sumo wrestler decided to go on a special high-protein diet to gain weight rapidly. After 11 months, he weighed 140 kilograms. He gained weight at a rate of 5.5 kilograms per month.

    Like how does this relate to the video at all?
    (4 votes)
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    • duskpin sapling style avatar for user proxima
      The sumo wrestler in that question is gaining weight at a constant ratio. Linear equations have slopes that do not change throughout the function, just like his rate of gaining weight. Like @hiroto.honda said, you find the y-intercept and slope to construct a linear equation. The slope is already given (rate at which he is gaining weight). So all you need to do is find the y-intercept and plug it into the form y=mx+b as b.
      Hope this helps
      (1 vote)
  • piceratops ultimate style avatar for user Evan Chen
    What is the formula for this?
    (1 vote)
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    • female robot grace style avatar for user loumast17
      there are two things you need to know. you are trying to find the function y = mx + b. you want m and b.

      m is the slope and b is the y intercept.

      For this problem the function is looking at elevation per time. Specifically meters per hour. if you have two points you can find the slope with the formula (y2 - y1)/(x2 - x1) where two points are (x1, y1) and (x2, y2). It doesn't matter which you make point and 2.

      The two points it gives are time = 0 at 1200 meters and time = 4 hours 1700 meters, so the points are (0, 1200) and (4, 1700) so let's go ahead and find the slope.

      (y2 - y1)/(x2 - x1) where (x1, y1) = (0, 1200) and (x2, y2) = (4, 1700)
      (1700 - 1200)/(4 - 0)
      500/4
      125

      so the slope m = 125

      now the y intercept. the y intercept is when x = 0. But hey, we know the point where x = 0, so we know the y intercept. b = 1200

      so now we can fill in y = mx + b
      y = 125x + 1200
      (6 votes)
  • blobby green style avatar for user kc07384
    How do you find the Y-intercept if they give you the rate of change and one point?
    (3 votes)
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    • leaf blue style avatar for user NatureNotes
      You can use the slope-intercept form, y = mx + b.

      I'll use an example to help explain: The rate of change of a line is 3, it passes through the point (1, 5). Find the y-intercept of the line.

      b = y-intercept, so we have to find b.

      The rate of change = slope = m
      Therefore sub m = 3:
      y = 3x + b

      Then sub the point (1, 5):
      5 = 3(1) + b
      2 = b

      So the y-intercept is (0, 2).
      (2 votes)
  • aqualine ultimate style avatar for user I like Speghetti and Math
    If an axis is given a variable other than x_ or _y, for example, t_, which stands for _time in Physics, will it be called the t-intercept?
    (4 votes)
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  • starky tree style avatar for user D -_-
    Mr. Mole left his burrow that lies 7 meters below the ground and started digging his way deeper into the ground, descending at a constant rate. After 6 minutes, he was 16 meters below the ground.

    Mr. Mole's burrow lies 7 meters below the ground. This corresponds to the point (0,−7), which is also the y-intercept.

    After 6 minutes, Mr. Mole was 16 meters below the ground, which corresponds to the point (6,-16)

    Let's use the slope formula with the points (0,-7) and (6,-16)

    m= −16−(−7)/6-0 = -9/6 = -1.5

    This means that Mr. Mole is descending at a rate of 1.51 meters per minute.

    Now we know the slope of the line is −1.5 and the y-intercept is (0, -7), so we can write the equation of that line:

    y=-1.5x-7



    SO, how would I know how to use the slope formula thing, which number are supposed to go on top and go on the bottom?
    (2 votes)
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    • duskpin sapling style avatar for user proxima
      The slope formula is “rise/run”. As implied, the “rise” part is the change in y-values between 2 points and the “run” part is the change between the x-values. So, the change on y-values always goes on the top and the change in the x-values always goes on the bottom.

      Hope this helps :)
      (2 votes)
  • scuttlebug green style avatar for user borsh525331
    How can I feel less overwhelmed by looking at word problems? I always skip them.
    (2 votes)
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    • mr pink green style avatar for user David Severin
      The first thing to do generally is to define your variables if the word problem does not define them for you. Then, you need to learn your vocabulary to tell what operations, parentheses, and where the equal goes (= are usually verbs in word problems). You can find some practice if you look up Kuda pre-algebra or algebra and find the worksheets on word problems or other sites.
      (2 votes)

Video transcript

- [Instructor] Tara was hiking up a mountain. She started her hike at an elevation of 1,200 meters and ascended at a constant rate. After four hours, she reached an elevation of 1,700 meters. Let y represent Tara's elevation in meters after x hours. And they ask us, and this is from an exercise on Khan Academy, it says, complete the equation for the relationship between the elevation and the number of hours. And if you're on Khan Academy, you would type it in, but we can do it by hand. So pause this video and work it out on some paper and let's see if we get to the same place. All right, now let's do this together. So first of all, they tell us that she's ascending at a constant rate. So that's a pretty good indication that we could describe her elevation based on the number of hours she travels with a linear equation. And we could even figure out that constant rate. It says that she goes from 1,200 meters to 1,700 meters in four hours. So we could say her rate is going to be her change in elevation over a change in time. So her change in elevation is 1,700 meters minus 1,200 meters and she does this over four hours. Over, her change in time is four hours. So her constant rate in the numerator here, 1,700 minus 1,200 is 500 meters. She's able to go up 500 meters in four hours. If we divide 500 by four, this is 125 meters per hour. And so we could use this now to think about what our equation would be. Our elevation y would be equal to, well, where is she starting? Well, it's starting at 1,200 meters. So she's starting at 1,200 meters. And then to that, we're going to add how much she climbs based on how many hours she's traveled. So it's going to be this rate, 125 meters per hour times the number of hours she has been hiking. So the number of hours is x times x. So this right over here is an equation for the relationship between the elevation and the number of hours. Another way you could have thought about it, you could have said, okay, this is going to be a linear equation because she's ascending at a constant rate. You could say the slope intercept form for a linear equation is y is equal to mx plus b, where b is your y-intercept. What is the value of y when x is equal to zero? And you'd say, all right, when x is equal to zero, she's at an elevation of 1,200. And then m is our slope. So that's the rate at which our elevation is increasing. And that's what we calculated right over here. Our slope is 125 meters per hour. So notice, these are equivalent. I just have, these two terms are swapped. So we could either write y is equal to 1,200 plus 125x or you could write it the other way around. You could write 125x plus 1,200. They are equivalent.