**Slope**

An important concept associated with lines is called the *slope* of a line. The slope of a line is associated with the "steepness" of a line. The **slope** of a line is the ratio of the vertical change to the horizontal change of a line when going from left to right. The slope is loosely defined as the rise divided by the run (see Fig. 11-23 ).

The slope of a line can be found in several ways. A line going uphill from left to right has a slope that is positive. A line going downhill from left to right has a slope that is negative. The slope of a horizontal line is zero and the slope of a vertical line is *undefined* (see Fig. 11-24).

On the graph of a line the slope can be found by selecting two points, forming a right triangle, and then counting the number of units in the rise and run and dividing those two values (see Fig. 11-25).

**Slope Intercept Formula**

*A better method is to find the coordinates of two points on the line, say (x _{1} , y _{1} ) and (x _{2} , y _{2} ) and then use the formula:*

**Example 1**

Find the slope of a line passing through the points whose coordinates are (5, 3) and (8, 1).

**Solution 1**

Let x _{1} = 5, y _{1} = 3, and x _{2} = 8, y _{2} = 1, and then substitute in the slope formula:

Hence, the slope of the line is .

**Finding Slope When the Equation is Known**

The slope whose equation is known can be found by selecting two points on the line then using the slope formula.

**Example 2**

Find the slope of a line whose equation is 5x + 2y = 10.

**Solution 2**

Select two points on the line:

Let x _{1} = 4, y _{1} = -5 and x _{2} = -2, y _{2} = 10. Now substitute in the slope formula:

The slope of the line 5x + 2y = 10 is .

**Finding the Slope by Solving ***y* in Terms of *x*

*y*in Terms of

*x*

Another way to find the slope of a line is to solve the equation for y in terms of x. The coefficient of x will be the slope.

**Example 3**

Find the slope of a line whose equation is 5x + 2y = 10.

**Solution 3**

Solve the equation for y, as shown:

Hence, the slope is , which is the result found in the previous example.

**Math Note:** An equation in the form of y in terms of x is called the slope–intercept form and in general is written as y = mx + b, where m is the slope and b is the y intercept.

**Slope Practice Problems**

**Practice**

1. Find the slope of the line containing the two points (-3, 4) and (6, 8).

2. Find the slope of the line containing the two points (9, 3) and (-4, -2).

3. Find the slope of the line whose equation is 5x - 3y = 8.

4. Find the slope of the line whose equation is -2x + 4y = 9.

5. Find the slope of the line whose equation is x + 7y = 10.

**Answers**

1.

2.

3.

4.

5.

Practice problems for these concepts can be found at: Graphing Practice Test.

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