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The Slope and Equation of a Line Help (page 3)

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By — McGraw-Hill Professional
Updated on Oct 4, 2011

Determining if Two Lines are Parallel or Perpendicular

We can decide whether two lines are parallel or perpendicular or neither by putting them in the form y = mx + b and comparing their slopes.

Examples

Determine whether the lines are parallel or perpendicular or neither.

  • 4 x − 3 y = −15 and 4 x − 3 y = 6

    The Slope and Equation of a Line Examples

    The lines have the same slope, so they are parallel.

  • 3 x − 5 y = 20 and 5 x − 3 y = −15

    The Slope and Equation of a Line Examples

    The slopes are reciprocals of each other but not negative reciprocals, so they are not perpendicular. They are not parallel, either.

  • x − y = 2 and x + y = −8

    x − y = 2        x + y = −8

          y = x − 2        y = −x − 8

    The slope of the first line is 1 and the second is −1. Because 1 and −1 are negative reciprocals, these lines are perpendicular.

  • y = 10 and x = 3

    The line y = 10 is horizontal, and the line x = 3 is vertical. They are perpendicular.

Finding the Equation of Parallel and Perpendicular Lines

Sometimes we need to find an equation of a line when we know only a point on the line and an equation of another line that is either parallel or perpendicular to it. We need to find the slope of the line whose equation we have and use this to find the equation of the line we are looking for.

Examples

  • Find an equation of the line containing the point (−4, 5) that is parallel to the line y = 2 x + 1.

    The slope of y = 2 x + 1 is 2. This is the same as the line we want, so we will let x = −4, y = 5, and m = 2 in y = mx + b. We get 5 = 2(−4) + b , so b = 13. The equation of the line we want is y = 2x + 13.

  • Find an equation of the line with x -intercept 4 that is perpendicular to x − 3y = 12.

    The x -intercept is 4 means that the point (4, 0) is on the line. The slope of the line we want will be the negative reciprocal of the slope of the line x − 3 y = 12. We will find the slope of x − 3 y = 12 by solving for y .

    The Slope and Equation of a Line Examples

    The slope we want is −3, which is the negative reciprocal of The Slope and Equation of a Line Examples . When we let x = 4, y = 0, and m = −3 in y = mx + b , we have 0 = −3(4) + b , which gives us b = 12. The line is y = −3 x + 12.

  • Find an equation of the line containing the point (3, −8), perpendicular to the line y = 9.

    The line y = 9 is horizontal, so the line we want is vertical. The vertical line passing through (3, −8) is x = 3. 

Practice problems for this concept can be found at Slope Practice Problems.

More practice problems for this concept can be found at: The Slope and Equation of a Line Practice Test.

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