2. CHAPTER 6 Introduction to Graphing and Statistics Slide 2Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 6.1Tables and Pictographs 6.2Bar Graphs and Line Graphs 6.3Ordered Pairs and Equations in Two Variables 6.4Graphing Linear Equations 6.5Means, Medians, and Modes 6.6Predictions and Probability

3. OBJECTIVES 6.4 Graphing Linear Equations Slide 3Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. aFind solutions of equations in two variables. bGraph linear equations in two variables. cGraph equations for horizontal or vertical lines.

4. 6.4 Graphing Linear Equations a Find solutions of equations in two variables. Slide 4Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. To solve an equation with one variable, like 3x + 2 = 8, we isolate the variable, x, on one side of the equation. To solve an equation with two variables, we will first replace one variable with some number choice and then solve the resulting equation.

5. EXAMPLE 6.4 Graphing Linear Equations a Find solutions of equations in two variables. 1 Slide 5Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. If x is 5, then the solution will be an ordered pair (5, y). We find y by substituting 5 for x in x + y = 7:

6. EXAMPLE 6.4 Graphing Linear Equations a Find solutions of equations in two variables. 3 Slide 6Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. We are free to use any number as a replacement for either x or y. To find one solution, we select 1 as a replacement for x. We then solve for y:

7. EXAMPLE 6.4 Graphing Linear Equations a Find solutions of equations in two variables. 3 Slide 7Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

8. EXAMPLE 6.4 Graphing Linear Equations a Find solutions of equations in two variables. 3 Slide 8Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. To find a second solution, we choose to replace y with 0 and solve for x:

9. EXAMPLE 6.4 Graphing Linear Equations a Find solutions of equations in two variables. 3 Slide 9Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. To find a third solution, we can replace x with 0 and solve for y:

10. Title 6.4 Graphing Linear Equations Slide 10Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. To find a solution of an equation with two variables: 1. Choose a replacement for one variable. 2. Solve for the other variable. 3. Write the solution as an ordered pair.

11. 6.4 Graphing Linear Equations b Graph linear equations in two variables. Slide 11Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. All equations that can be written Ax + By = C are said to be linear because the solutions of each equation, when graphed, form a straight line. An equation Ax + By = C is called the standard form of a linear equation. When the line representing the solutions is drawn, we say that we have graphed the equation. Since solutions of Ax + By = C are written in the form (x, y), we label the horizontal axis as the x-axis and the vertical axis as the y-axis.

12. EXAMPLE 6.4 Graphing Linear Equations b Graph linear equations in two variables. 4 Slide 12Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. We first need to calculate several solutions of 2x – y = 5. In Example 3, we found that (1, –3), (2.5, 0), and (0, –5) are solutions of the equation. Next, we plot the points. As expected, the points describe a straight line. We draw the line and label it with the equation.

13. EXAMPLE 6.4 Graphing Linear Equations b Graph linear equations in two variables. 4 Slide 13Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

14. 6.4 Graphing Linear Equations b Graph linear equations in two variables. Slide 14Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Note that two points are enough to determine a line, but we generally calculate and graph at least three ordered pairs before drawing each line. If the points do not all line up, we know that a mistake has been made.

15. 6.4 Graphing Linear Equations b Graph linear equations in two variables. Slide 15Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Linear equations are not always written in standard form. To find solutions of equations like y = 2x, we usually choose values for x and then calculate y.

16. EXAMPLE 6.4 Graphing Linear Equations b Graph linear equations in two variables. 5 Slide 16Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. First, we find some ordered pairs that are solutions. To find three ordered pairs, we can choose any three values for x and then calculate the corresponding values for y.

17. EXAMPLE 6.4 Graphing Linear Equations b Graph linear equations in two variables. 5 Slide 17Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Next, we plot these points. We draw the line, or graph, with a ruler and label it y = 2x.

18. 6.4 Graphing Linear Equations b Graph linear equations in two variables. Slide 18Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. For linear equations, tables can be formed using any numbers for x.

19. 6.4 Graphing Linear Equations c Graph equations for horizontal or vertical lines. Slide 19Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Any equation in the form Ax + By = C is linear, provided A and B are not both zero. If A is 0 and B is nonzero, there is no x-term and the graph is a horizontal line. If B is 0 and A is nonzero, there is no y-term and the graph is a vertical line.

20. EXAMPLE 6.4 Graphing Linear Equations c Graph equations for horizontal or vertical lines. 9 Slide 20Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

21. EXAMPLE 6.4 Graphing Linear Equations c Graph equations for horizontal or vertical lines. 9 Slide 21Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. When we plot (–2, 3), (0, 3), and (4, 3) and connect the points, we obtain a horizontal line. Any ordered pair of the form (x, 3) is a solution, so the line is 3 units above the x-axis.

22. EXAMPLE 6.4 Graphing Linear Equations c Graph equations for horizontal or vertical lines. 10 Slide 22Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

23. EXAMPLE 6.4 Graphing Linear Equations c Graph equations for horizontal or vertical lines. 10 Slide 23Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. When we plot (–4, 5), (–4, 1), and (–4, 3) and connect them, we obtain a vertical line. Any ordered pair of the form (–4, y) is a solution, so the line is 4 units left of the y-axis.

24. 6.4 Graphing Linear Equations HORIZONTAL AND VERTICAL LINES Slide 24Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.