1. Chapter 1
Linear Equations and Linear Functions

2. 1.4 Meaning of Slope for Equations, Graphs, and Tables

3. Example: Finding the Slope of a Line
Find the slope of the line y = 2x + 1.

4. Solution
We use x = 0, 1, 2, 3 in the table to list the solutions, and sketch the graph of the equation.
If the run is 1, the rise is 2. So the slope is

5. Finding the Slope from a Linear Equation of the Form y = mx + b
For a linear equation of the form y = mx + b, m is the slope of the line.

6. Example: Identifying Parallel or Perpendicular Lines
Are the lines and 12y – 10x = 5 parallel,
perpendicular, or neither?

7. Solution For the line the slope is
For 12y – 10x = 5, the slope is not –10. To find the slope, we must solve the equation for y:

8. Solution For the slope is the same as the
slope of the line Therefore, the two
lines are parallel. We use a graphing calculator to draw the lines on the same coordinate system.

9. Vertical Change Property
For a line y = mx + b, if the run is 1, then the rise is the slope m.

10. Finding the y-intercept from a Linear Equation of the Form y = mx + b
For a linear equation of the form y = mx + b, the y-intercept is (0, b).

11. Slope-intercept form Definition
If an equation is of the form y = mx + b, we say it is in slope-intercept form.

12. Example: Using Slope to Graph a Linear Equation
Sketch the graph of y = 3x – 1.

13. Solution Note that the y-intercept is (0, –1) and that the
slope is To graph,
1. Plot the y-intercept, (0, –1).
2. From (0, –1), look 1 unit to the right and 3 units up to plot a second point. (see the next slide)
3. Sketch the line that contains these two points. (see the next slide)

14. Solution

15. Using Slope to Graph a Linear Equation of the Form y = mx + b
To sketch the graph of a linear equation of the form y = mx + b.
1. Plot the y-intercept (0, b).
2. Use to plot a second point.
3. Sketch the line that passes through the two plotted points.

16. Using Slope to Graph a Linear Equation
Warning
Before we can use the y-intercept and the slope to graph a linear equation, we must solve for y to put the equation into the form y = mx + b.

17. Example: Working with a General Linear Equation
1. Determine the slope and the y-intercept of the graph of ax + by = c, where a, b, and c are constants and b is nonzero.
2. Find the slope and the y-intercept of the graph of 3x + 7y = 5.

18. Solution 1.
The slope is and the y-intercept is

19. Solution
2. We substitute 3 for a, 7 for b, and 5 for c in our results from Problem 1 to find that the slope is
and the y-intercept is

20. Slope Addition Property
For a linear equation of the form y = mx + b, if the value of the independent variable increases by 1, then the value of the dependent variable changes by the slope m.

21. Example: Identifying Possible Linear Equations
Four sets of points are shown on the next slide. For each set, decide whether there is a line that passes through every point. If so, find the slope of that line. If not, decide whether there is a line that comes close to every point.

22. Example: Identifying Possible Linear Equations

23. Solution
1. For set 1, when the value of x increases by 1, the value of y changes by –3. So, a line with slope –3 passes through every point.
2. For set 2, when the value of x increases by 1, the value of y changes by 5. Therefore, a line with slope 5 passes through every point.

24. Solution
3. For set 3, when the value of x increases by 1, the y value does not change. Further, a line does not come close to every point, because the value of y changes by such different amounts each time the value of x increases by 1.
4. For set 4, when the value of x increases by 1, the value of y changes by 0. Therefore, a line with slope 0 (the horizontal line y = 8) passes through every point.