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The Slope and Equation of a Line Help (page 2)

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

Finding the Equation of a Line Using Slope Formula

We can find an equation of a line if we know any two points on the line. First we need to use the slope formula to find m . Then we will pick one of the points to put into y = mx + b .

Examples

Find an equation of the line containing the given points.

  • (−2, 3) and (10, 15)

    The Slope and Equation of a Line Examples

    We will use x = −2 and y = 3 in y = mx + b to find b .

    3 = 1(−2) + b

    5 = b

    The equation is y = 1 x + 5, or simply y = x + 5.

  • The Slope and Equation of a Line Examples and (4, 3)

    The Slope and Equation of a Line Examples

    Using x = 4 and y = 3 in y = mx + b, we have

    The Slope and Equation of a Line Examples

  • (0, 1) and (12, 1)

    The y -values are the same, making this a horizontal line. The equation is y = 1.

Finding Slope from a Graph

If a graph is clear enough, we can find two points on the line or even its slope. If fact, if the slope and y -intercept are easy enough to see on the graph, we know right away what the equation is.

Examples

  •  

    The Slope and Equation of a Line Examples

    Fig. 1.8.

    The line in Figure 1.8 crosses the y -axis at 1, so b = 1. From this point, we can go right 2 and up 3 to reach the point (2, 4) on the line. “Right 2” means that the denominator of the slope is 2. “Up 3” means that the numerator of the slope is 3. The slope is The Slope and Equation of a Line Examples, so the equation of the line is The Slope and Equation of a Line Examples.

  •  

    The Slope and Equation of a Line Examples

    Fig. 1.9.

    The y-intercept is not easy to determine, but we do have two points. We can either find the slope by using the slope formula, or visually (as we did above). We can find the slope visually by asking how we can go from (−4, 3) to (2, −1): Down 4 (making the numerator of the slope −4) and right 6 (making the denominator 6). If we use the slope formula, we have

    The Slope and Equation of a Line Examples

    Using x = 2 and y = −1 in y = mx + b, we have The Slope and Equation of a Line Examples. From this, we have The Slope and Equation of a Line Examples . The equation is The Slope and Equation of a Line Examples.

  •  

    The Slope and Equation of a Line Examples

    Fig. 1.10.

    The line in Figure 1.10 is vertical, so it has the form x = a. All of the x-values are −2, so the equation is x = −2.

Solving for y to Find the Slope of an Equation

When an equation for a line is in the form Ax + By = C , we can find the slope by solving the equation for y . This will put the equation in the form y = mx + b.

Example

  • Find the slope of the line 6 x − 2 y = 3.

    6 x − 2 y = 3

    −2 y = −6x + 3

    The Slope and Equation of a Line Example

    The slope is The Slope and Equation of a Line Example .

Slopes of a Parallel and Perpendicular Lines

Slopes of Parallel Lines - Equal Slopes

Two lines are parallel if their slopes are equal (or if both lines are vertical).

The Slope and Equation of a Line Example

Fig. 1.11.

Slopes of Perpendicular Lines - Slopes with Negative Reciprocals

Two lines are perpendicular if their slopes are negative reciprocals of each other (or if one line is horizontal and the other is vertical). Two numbers are negative reciprocals of each other if one is positive and the other is negative and inverting one gets the other (if we ignore the sign).

Examples

  • The Slope and Equation of a Line Examples are negative reciprocals
  • The Slope and Equation of a Line Examples are negative reciprocals

    The Slope and Equation of a Line Examples

    Fig. 1.12.

  • The Slope and Equation of a Line Examples are negative reciprocals
  • 1 and − 1 are negative reciprocals
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