**Introduction to The Slope of a Line in the Plane**

A line in the plane may rise gradually from left to right, or it may rise quite steeply from left to right (Fig. 1.13).

Likewise, it could fall gradually from left to right, or it could fall quite steeply from left to right (Fig. 1.14).

The number “slope” differentiates among these different rates of rise or fall.

Look at Fig. 1.15. We use the two points *P* = ( *p* _{1} , *p* _{2} ) and *Q* = ( *q* _{1} , *q* _{2} ) to calculate the slope. It is

It turns out that, no matter which two points we may choose on a given line, this calculation will always give the same answer for slope.

**Examples**

**Example 1**

Calculate the slope of the line in Fig. 1.16 .

**Solution 1**

We use the points *P* = (−1, 0) and *Q* = (1, 3) to calculate the slope of this line:

We could just as easily have used the points *P* = (−1, 0) and *R* = (3, 6) to calculate the slope:

If a line has slope *m*, then, for each unit of motion from left to right, the line rises *m* units. In the last example, the line rises 3/2 units for each unit of motion to the right. Or one could say that the line rises 3 units for each 2 units of motion to the right.

**Example 2**

Calculate the slope of the line in Fig. 1.17 .

**Solution 2**

We use the points *R* = (−2, 10) and *T* = (1, −5) to calculate the slope of this line:

We could just as easily have used the points *S* = (−1, 5) and *T* = (1, −5):

In this example, the line falls 5 units for each 1 unit of left-to-right motion. The negativity of the slope indicates that the line is falling.

The concept of slope is undefined for a vertical line. Such a line will have any two points with the same *x* -coordinate, and calculation of slope would result in division by 0.

**You Try It:** What is the slope of the line *y* = 2 *x* + 8?

**You Try It:** What is the slope of the line *y* = 5? What is the slope of the line *x* = 3?

Two lines are perpendicular precisely when their slopes are negative reciprocals. This makes sense: If one line has slope 5 and the other has slope −1/5 then we see that the first line rises 5 units for each unit of left-to-right motion while the second line falls 1 unit for each 5 units of left-to-right motion. So the lines must be perpendicular. See Fig. 1.18(a).

**You Try It:** Sketch the line that is perpendicular to *x* + 2 *y* = 7 and passes through (1, 4).

Note also that two lines are parallel precisely when they have the *same slope* . See Fig. 1.18(b).

Practice problems for this concept can be found at: Calculus Basics Practice Test.

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