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4. Using the line equation

Vertical lines have an undefined slope, so how can we solve for the intersection point? Instead of using slope-intercept form, we need to use the line equation, or implicit form. Click here to see why dividing by zero is bad. Click here to try the interactive program shown in the video.

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  • leaf red style avatar for user Blaze
    I have heard the term implicit before; what does it mean?
    (1 vote)
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    • blobby green style avatar for user Matt
      An explicit equation shows a clear relation between x and y, such as y=3x+7. This way it is very clear what the relationship between x and y is. If x=2, then it is easy to compute y=3(2)+7=13. An implicit equation, on the other hand, represents an implied relationship between x and y, such as 5x+2y=7. It is not as obvious that y=-4 when x=3.

      In everyday use, I might say I prefer not working with numbers, which might be an implicit way of saying I don't like math.
      (5 votes)
  • blobby green style avatar for user moot
    How do you do this (ray/line intercept) with a line drawn in 3D space? It seems like the slope intercept form would not be usefull in this case?
    (2 votes)
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  • leaf green style avatar for user Dezső Finda
    At A(2.973,2) B(2.973,-1) => AB: 3x-8.92=0 (vertical line case)
    I just can't figure it out, how c has become -8.92 from the Δyx-Δxy+Δxi=0 <=> ax+by+c=0 identity. I know it is true, but this conversion not takes me there.
    If c=Δx*i, the constant would be 0 * i = 0! There is no change in x, this way only 3x=0 remains. What did I miss out?
    (1 vote)
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    • leafers sapling style avatar for user Peter Collingridge
      It's a good question, which was glossed over. The issue is that you need to find out what iΔx is, but Δx is 0 and i is undefined since there is no intercept for a vertical line. And yet we can still do it.

      Remember one way to write the equation of a line is:
      y = (Δy/Δx)x + i

      If we multiply by Δx (so we can avoid dividing by 0), then we get
      yΔx = xΔy + iΔx

      So iΔx = yΔx - xΔy.

      For the case A = (2.973, 2) and B = (2.973, -1), Δx = 0 and Δy = 3, so we have:
      iΔx = 0y - 3x = -3x - notice we are keeping the left hand side as iΔx since this is what we want to calculate.
      Then we just plug in a value for x that we know, maybe Ax, but it doesn't matter since all x on the line will be the same.

      Ax = 2.972, so iΔx = -3(2.972) = -8.916 (-8.92 when rounded).
      So now xΔy - yΔx + iΔx = 0 becomes 3x - 0y + (-8.92) = 0.
      Hence 3x - 8.92 = 0

      This is why we multiply by Δx so we can get numbers we can deal with rather 1/0 multiplied by 0.
      (2 votes)
  • mr pink red style avatar for user GvR37.P.L.Boon
    how do you find the y-intercept given 2 points without having to divide by zero if there is no change in x?
    (1 vote)
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  • aqualine ultimate style avatar for user mercedes/ash venne
    i thot i was just going to animate
    (1 vote)
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  • blobby green style avatar for user anonymoussrkh91
    At time ,how did the y intercept became 11 ? please solve this !
    (1 vote)
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  • mr pink green style avatar for user jami.colmore
    i have a question y do we have to do this
    (0 votes)
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Video transcript

- There's a detail that we need to attend to: there's a slight problem with the slope intercept form of the sine geometry. The problem is that if AB is vertical, the slope isn't defined. To see that, look at the slope intercept form in general: y equals m x plus i, where m is the slope and i is the y-intercept. The slope m is the change in y divided by the change in x, meaning that if AB is a vertical line, there is no change in x. So, computing the slope would mean dividing by zero, which is bad. (bell) But we can eliminate this problem by multiplying through by the change in x. So, multiplying through by the change in x, we get change in x times y equals change in y times x plus i times change in x. It's common to move everything to one side and re-write this as change in y times x minus change in x times y plus i times change of x equals zero. Call this term, change in y, a value: a; this term, negative change in x, a value: b; and this term, i times change of x, a value: c; meaning we can write an equation for the line as a x plus b y plus c equals zero. An equation like this for a line goes by several names. It is sometimes called the line equation. It's also called the implicit form for the line. Let's do an example for this specific line, AB. Change in y is negative three. Change in x is one, and i is 11. So, negative three x minus y plus 11 equals zero. That line equation is shown here. Notice that as I move A and B around, the line equation updates accordingly. The line equation can be used with the parametric form of array to compute intersection points, this time, for any type of line, even vertical ones. Use the next exercise to practice computing intersection points using line equations.