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Determine equations of parallel lines

Parallel lines in the coordinate plane have equal slopes when their slopes are defined, so an equation through a given point is formed by preserving the reference line’s slope and determining the appropriate intercept; distinct lines with the same slope do not intersect. This includes recognizing horizontal lines as having slope zero and vertical lines as parallel when both have equations x=cx=c, while distinguishing parallel lines from coincident equations and expressing results in slope-intercept or point-slope form.

Detailed Explanation: Determine equations of parallel lines

To find an equation of a line parallel to a given line:

  1. Identify the slope of the given line.
  2. Use the same slope for the new line.
  3. Substitute the given point into the equation to find the intercept, or use point-slope form.
  4. Check that the new line is distinct from the original line.

Example: Find the equation of the line parallel to

y=−2x+5y=-2x+5

that passes through (3,4)(3,4).

The given line is already in slope-intercept form, y=mx+by=mx+b, so its slope is

m=−2.m=-2.

A parallel line must have the same slope. Using point-slope form,

y−y1=m(x−x1),y-y_1=m(x-x_1),

with (x1,y1)=(3,4)(x_1,y_1)=(3,4) and m=−2m=-2:

y−4=−2(x−3).y-4=-2(x-3).

Simplify:

y−4=−2x+6y-4=-2x+6 y=−2x+10.y=-2x+10.

Therefore, the equation is

y=−2x+10.\boxed{y=-2x+10}.

Both lines have slope −2-2, so they are parallel. They are not the same line because their yy-intercepts, 55 and 1010, are different.

For special cases, horizontal parallel lines have the same equation form y=cy=c and slope 00. Vertical parallel lines have the form x=cx=c; for example, x=2x=2 and x=7x=7 are parallel because both are vertical and have different xx-values.

Learn by doing: Determine equations of parallel lines

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    Slope - Find Parallel - Standard Form to Slope Y Intercept Form

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