Slope Help

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).

Slope

Fig. 11-23.

Slope

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

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:

Slope

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:

Slope

Hence, the slope of the line is Slope .

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:

Slope

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

Slope

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

Finding the Slope by Solving 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:

Slope

Hence, the slope is Slope , 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. Slope

2. Slope

3. Slope

4. Slope

5. Slope

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