1. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.4, Slide 1 Chapter 1 Linear Equations and Linear Functions

2. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.4, Slide 2 1.4 Meaning of Slope for Equations, Graphs, and Tables

3. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.4, Slide 3 Example: Finding the Slope of a Line Find the slope of the line y = 2x + 1.

4. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.4, Slide 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. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.4, Slide 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. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.4, Slide 6 Example: Identifying Parallel or Perpendicular Lines Are the lines and 12y – 10x = 5 parallel, perpendicular, or neither?

7. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.4, Slide 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. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.4, Slide 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. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.4, Slide 9 Vertical Change Property For a line y = mx + b, if the run is 1, then the rise is the slope m.

10. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.4, Slide 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. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.4, Slide 11 Slope-intercept form Definition If an equation is of the form y = mx + b, we say it is in slope-intercept form.

12. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.4, Slide 12 Example: Using Slope to Graph a Linear Equation Sketch the graph of y = 3x – 1.

13. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.4, Slide 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. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.4, Slide 14 Solution

15. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.4, Slide 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. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.4, Slide 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. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.4, Slide 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. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.4, Slide 18 Solution 1. The slope is and the y-intercept is

19. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.4, Slide 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. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.4, Slide 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. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.4, Slide 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. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.4, Slide 22 Example: Identifying Possible Linear Equations

23. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.4, Slide 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. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 1.4, Slide 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.