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How Do You Write an Equation of a Line in Point-Slope Form and Standard Form If You Have Two Points?

Write the equation of the line passing through the points (0,-2) and (3,4) in 1) Point-Slope form and 2) Standard form

Summary

  1. 'm' represents slope
  2. (x1, y1) and (x2, y2) are the two points used to calculate the slope
  3. Plug in (0,-2) and (3,4) for (x1, y1) and (x2, y2) to find the slope
  4. Plug (3,4) in for (x1, y1) and 2 in for 'm'

Notes

    1. Point-slope form is one way of writing the equation for a line
    2. It's especially useful if you know the slope of the line and one point on it
    1. Point-slope form is one way of writing the equation for a line
    2. It's especially useful if you know the slope of the line and one point on it
    1. We need to know the slope before we can write the equation in point-slope form
    2. But since we have two points on the line, we can calculate the slope
    1. Since we were given two points on the line, we can use the formula for slope to calculate the slope between them
    1. (x1, y1) and (x2, y2) are the two points that we will use to calculate the slope
    2. In our case, we will use the points we were given: (0,-2) and (3,4)
    1. The points we were given were (0,2) and (3,4)
    2. These points correspond to (x1, y1) and (x2, y2) in the slope formula
    1. We have y2 = 4, y1 = -2, x2 = 3, and x1 = 0
    1. On the top, 4-(-2)=6
    2. On the bottom, 3-0=3
    3. This gives us 6/3, which equals 2
    4. This is the value of our slope
    1. We actually have two points, (0,-2) and (3,4)
    2. And we calculated our slope to be 2
    1. Point-slope form of a line is (y-y1) = m(x-x1), where:
    2. 'm' is the slope
    3. 'x1' and 'y1' are the x- and y- coordinates of a point on the line
    1. Point-slope form of a line is (y-y1) = m(x-x1), where:
    2. 'm' is the slope
    3. 'x1' and 'y1' are the x- and y- coordinates of a point on the line
    1. We were given two points: (0,-2) and (3,4)
    2. Since we know both of these points are on our line, we can use either one to write our equation
    1. Our equation will look slightly different if we use different points
    2. But you'll find that if you convert it to another form or if you graph it, the line will be the same
    1. This time we'll use (3,4)
    2. But if you follow the same steps using (0,-2), you'll end up with an equation for the same line
    1. Remember, (x1, y1) represents the point on our line
    2. So we'll have x1=3 and y1=4
    1. Here we plugged in 4 for 'y1', 3 for 'x1', and 2 (our slope) for 'm'
    1. Our line in point-slope form is y-4=2(x-3)
    1. Standard form of an equation is Ax+By=C, where A, B, and C are numbers and A is positive
    1. Standard form of an equation is Ax+By=C, where A, B, and C are numbers and A is positive
    1. Remember, our line in point-slope form was y-4=2(x-3)
    2. We're working with the same line here, just trying to put it in a different form
    1. Remember, our line in point-slope form was y-4=2(x-3)
    2. We can use algebra to rearrange the equation so it's in the form Ax+By=C
    1. Remember, our line in point-slope form was y-4=2(x-3)
    1. This is the equation that we'll start with, since we already know it goes through the two points we were given
    1. We haven't done anything to the left hand side yet
    2. When we distribute the 2 on the right we get:
    3. When we distribute the 2 on the right we get 2•x and 2•(-3)
    4. Simplifying this gives us 2x-6
    1. On the left we have y-4-y
    2. The y's cancel and we're left with -4
    3. On the right we also need to subtract a y, so we have 2x-y-6
    1. On the left we have -4+6, which is equal to 2
    2. On the right we have 2x-y-6+6
    3. The 6's cancel, leaving us with 2x-y
    1. Equality is preserved even if we flip things over the equals sign
    2. So 2=2x-y is the same thing as 2x-y=2
    1. Our equation is in the form Ax+By=C
    2. Here, A=2, B=-1, and C=2