Equations of parallel and perpendicular lines from points

¶ … equation of a line, find equations for lines parallel or perpendicular to it going through specified points. Find the appropriate equations and points from the table below. Simplify your equations into slope-intercept form.

Write the equation of a line parallel to the given line but passing through the given point.

y = -1/2x + 1; (4,2)

Parallel lines -- lines that never meet, and remain a constant distance from each other -- have the same slope, defined the coefficient of x in the above slope-intercept form of the equation, commonly identified as "m." In this case, m = -1/2x and will remain so in the second equation. Using the values for the given point and putting the equation into point-slope form yields:

y -- 2 = -1/2x --

…and solving the equation for y yields the slope-intercept form of the equation:

y = -1/2x --

Write the equation of a line perpendicular to the given line but passing through the given point.

y = -3x -- 6; (-1,5)

Perpendicular lines -- lines that meet at a ninety degree angle -- have inverse and opposite slopes (coefficients of x or m values). In other words, for a line with slope m, a perpendicular line would have the slope -1/m. In this case, that means a perpendicular line will have a slope (m) of 1/3. Again plugging in the values from the given point and putting the equation in point-slope form yields:

y -- 5 = 1/3x -- (-1)

…and solving for y yields the slope-intercept form:

y = 1/3x + 6.

Discussion

Lines are defined by equations, and certain equations make it very easy to place and graph a line and to find similar lines. First, it is important to understand how graphs work. An x-axis (a horizontal line) and y-axis (a vertical line) are set to represent "0" for each variable, and they meet at the origin. The origin is defined as the point (0,0). This notation for a point is known as an ordered pair, and it provides an x-value (the first 0) and a y-value (the second 0). For another example, the ordered pair (1,3) has an x-value of 1 (it is one unit right -- in the positive direction -- along the horizontal x-axis) and a y-value of 3 (it is three units up -- the positive direction). Starting at the origin and using the order pair as a guide is a simple way to find any point.