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Comparing linear functions: equation vs. graph

Sal is given the formula of a linear function and the graph of another, and is asked to determine which function increases faster. Created by Sal Khan.

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  • leaf green style avatar for user Rajeswari Murukan
    If ax + by + c = 0 is a linear equation in two variables. Is x + 0y + 7 = 0 a linear equation in 2 variables?
    (17 votes)
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    • orange juice squid orange style avatar for user Marvin Humphrey
      Yes, the second example is also a linear equation in two variables. Similarly, 0x + 0y + 0z = 0 is a linear equation in three variables, albeit one with an infinite number of solutions.

      Having a coefficient of 0 doesn't mean that the variable goes away. It still exists, it is simply irrelevant and can take on any value without affecting the validity of the equation.
      (9 votes)
  • leaf green style avatar for user Madina Shukurova
    At , how did Sal plot a "g" out of nowhere?
    (8 votes)
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  • purple pi purple style avatar for user Zachary Hage
    my teacher wants me to (sooner or later) describe a linear function. how should I do that?
    (4 votes)
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    • aqualine ultimate style avatar for user MathBunnyRabbit
      Zachary,
      Lets break up the word. Linear means something that makes a line (pretty simple,) and a function is a set of coordinates or points on a graph that have only one x value to every y value. So basically, a linear function does not curve, but goes in a strait line on a graph. If it curves, this is called a non-linear function.
      I hope this helped!
      (16 votes)
  • blobby green style avatar for user Theresa Lopez
    is y=1/x - 7 a linear function
    (3 votes)
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  • eggleston orange style avatar for user Aesha
    At , if both are increasing then don't the negatives cancel out and become positive?
    (5 votes)
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    • stelly blue style avatar for user Kim Seidel
      You are comparing the slopes, not multiplying the slopes.
      Both lines are negative, so both lines slant down from left to right. The slope of line F is decreasing faster because its slope is more negative than the slope of line G.
      Hope this helps.
      (3 votes)
  • leafers seed style avatar for user Keenan Benitez-Hay
    Graph the linear function with the equation y=1/2x+3
    (4 votes)
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  • blobby green style avatar for user Posts Only
    Why are both f and g both decreasing? Looking at the graph, I'm totally confused by this. In which direction is the line "moving"?
    (5 votes)
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  • mr pink red style avatar for user Diyana Zlatanowa
    I dont understand where is line f. How do we know that it is decreasing? And when calculating the slope of f ,why is just -7/3x considered ,and not +1 (f = -7/3x + 1)
    (4 votes)
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    • leaf green style avatar for user giveitago
      f is only shown as an equation. Its slope is -7/3. The negative means that it is a downward slope. [-7/3 is a downward slope. 7/3 would be an upwards slope]. The slope of the line, g is -2/1. When calculating the slope we want to figure out the numbers before x
      example: -7/3 x . The +1 is the point where the line crosses the y axis. It isn't used to calculate slope unless we are using it as one of two points to use the y2-y1 / x2-x1 process. We are not using this process because we already know the slope is -7/3.
      (2 votes)
  • female robot grace style avatar for user Anna
    -7/3 = -7/3 = 7/-3 so how do you know whether it is 3 right 7 down or 7 up 3 left? In other words both are change in y over change in x so how do you know what the slope is if you are only given -7/3 = 7/-3?
    (3 votes)
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    • leaf red style avatar for user Noble Mushtak
      3 right, 7 down is the same as 7 up, 3 left just like how -7/3=7/-3 so it doesn't actually matter. Usually, you'll see that the vertical direction (change in y) is first and right is always used for the horizontal direction (change in x), so in this case, 7 down, 3 right would be used.
      (1 vote)
  • leafers ultimate style avatar for user Andrew Escobedo
    Sal had an equation that showed the relationship between f and x, and we also had the graph of the equation that describes the relationship between x and g. What I'm having trouble with is understanding the relationship between f and g. Would this be something like g(f(x))?
    (2 votes)
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Video transcript

Two functions, f and g, are described below. Which of these statements about f and g is true? So they defined function f as kind of a traditional linear equation right over here. And this right over here is g. So this right over here is g of x. And that also looks like a linear function. We see it's a kind of a downward sloping line. So let's look at our choices and see which of these are true. f and g are both increasing, and f is increasing faster than g. Well, when I look at g-- Well, first of all, g is definitely decreasing. So we already know that that's false. And f is also decreasing. We see here it has a negative slope. Every time we move forward 3 in the x direction, we're going to move down 7 in the vertical direction. So neither of these are increasing so that's definitely not right. f and g are both increasing. Well, that's definitely not right. So we know that both f and g are decreasing. So this first choice says they're both decreasing, and g is decreasing faster than f. So let's see what the slope on g is. So the slope on g is every time we move 1 in the x direction, positive 1 in the x direction, we move down 2 in the y direction. So for g of x, if we were to write our change in y over our change in x-- which is our slope-- our change in y over change in x, when we move one in the x direction, positive 1 in the x direction, we move down 2 in the y direction. So our change in y over change in x is negative 2. So g has a slope of negative 2. f has a slope of negative 7/3. Negative 7/3 is the same thing as negative 2 and 1/3. So f's slope is more negative. So it is decreasing faster. So g is not decreasing faster than f. f is decreasing faster than g. So this is not right. And then we have this choice-- f and g are both decreasing, and f is decreasing faster than g. This is right, right over here. We have this last choice-- g is increasing but f is decreasing. We know that's not true. g is actually decreasing.