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Writing the Equation of a Line: Point-Slope Form by Christopher Monahan

One of the advantages of using the point-slope form for a line is that two key pieces of information are given in the equation: the slope and a point (the point being the y-intercept). A disadvantage of using this formula is that when the slope or the coordinates of the points are especially easy to work with, the arithmetic for determining the slope and intercept gets sloppy. When we write an equation, the purpose of the variables x and y in the equation is to enable us to substitute values to determine whether a particular point is on the graph or to find a missing element of the ordered pair if one part is known. This can be used to write an equation for a line in what is called the point-slope form.

A line that passes through the known point (x1, y1) with slope m has the equation yy1 = m(xx1).

Example: Write an equation for the line that passes through the point (3, 5) and has slope 3/4.

Because the slope is 3/4, any point with coordinates (x, y) must satisfy the equation

. Do you realize this is all you need to do? You can check any point to determine whether it is on the line by substituting for x and y and seeing if the ratio on the left side of the equation is 3/4. This equation is usually modified by multiplying both sides of the equation by the denominator of the left-hand side so that the equation becomes y − 5 = 3/4 (x − 3).

Example: Write an equation for the line that passes through the point (-4, 9) with slope −5/7.

x1 = −4 and y1 = 9, so the equation is y − 9 = −5/7 (x − (-4)), which is simplified to y − 9 = −5/7 (x + 4).

Example: Determine the coordinates of the y-intercept and the x-intercept of the line with equation y − 9 = −5/7 (x + 4).

The y-intercept has an x-coordinate of 0. Substituting, y − 9 = −5/7 (0 + 4) becomes y − 9 = −20/7, so y = 43/7.

The x-intercept has a y-coordinate of 0. Substituting, 0 − 9 = −5/7 (x + 4) becomes −9 = −5/7(x + 4). Multiply by −7/5 to get 63/5 = x + 4, or x = 43/5.