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Writing the Equation of a Line

Recall that you can draw a line on the coordinate plane if you know the right information. That is, if you know either:
1. The coordinates of two points that lie on the line or
2. The coordinate of one point that lies on the line and the slope of the line.
Let's look at examples that demonstrate each case.

EXAMPLE 1:EXAMPLE 2:
Given:
The points (-4, 0) and (4, -2) lie on a line.
Given:

The point (-1, 3) lies on a line with a slope of 3.
Mark these two points on the plane. Mark the point (-1, 3) on the plane. Since the slope is 3, rise up 3 units, run across 1 unit, and mark the point (0, 6). This point will also lie on the line.
Draw the line passing through these two points. Draw the line passing through these two points.

Given a point on a line and the slope of the line, you can write the equation of a line using the point-slope equation:
y - y1 = m(x - x1)
where
  • (x1, y1) are the coordinates of a point on the line, and
  • m is the slope of the line.
    • Let's now write the equations of the lines drawn Examples 1 and 2.

    EXAMPLE 1:EXAMPLE 2:
    You can compute the slope of this line using the slope equation:
    m = y2 - y1
    x2 - x1
    where (x1, x2) and (y1, y2) represent the coordinates of two points lying on the line.
    Recall that the two points were (-4, 0) and (4, -2). This gives:
    m = -2 - 0
    4 - (-4)    		 = -2
      8    		 = -1
      4
    		For this example, you are given the slope of the line, m = 3, so no computation is necessary.
    Check with the above graph to verify that the slope is -1/ 4.
    You now have enough information to fill in the point-slope formula: Enter this information into the point-slope formula. Recall that the point given was (-1, 3).
    y - (-2) = -(1/4)(x - 4)
    y + 2
    		 = -(1/4)x + 1
    y
    		 = -(1/4)x - 1
    y - 3 = 3(x - (-1))
    y - 3 = 3(x + 1)
    y - 3 = 3x + 3
    y
    		 = 3x + 6
    The equation of the line is y = -(1/4)x - 1. The equation of the line is y = 3x + 6

    In both examples, we arrived at an equation of the form y = mx + b.
    Note that if you let x = 0, you find that the line intercepts the y-axis at b. Also note that m represents the slope of the line.
    Since it is easy to pick out the y-intercept and slope in the this equation, it is often referred to as the slope-intercept equation.
    Note that if you know the y-intercept and slope of a line, you can write the equation of the line by entering the appropriate values for m and b in the slope-intercept equation.
    Any linear equation can be rewritten into a general equation of the form ax + by + c = 0. Remember that the general form is not unique; you can always multiply the equation through by any nonzero real number and the resulting equation is still in general form.
    Let's write the linear equations from the examples in general form.

    EXAMPLE 1:EXAMPLE 2:
    y = -(1/4)x + 1 becomes
    -(1/4)x - y + 1 = 0
    or, multiplying both sides by -4 gives,
    x + 4y - 4 = 0
    y = 3x + 6 becomes
    3x - y + 6 = 0

    Go to this page, where you will find an action figure and some questions which will test your skill at writing linear equations.

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