1. Graphing Linear Equations Section 1.2

2. Lehmann, Intermediate Algebra, 3ed Section 1.2 Consider the equation. Let’s find y when So, when, which cab be represented by the ordered pair Slide 2 Definition of Solution, Satisfy, and Solution Set

3. Lehmann, Intermediate Algebra, 3ed Section 1.2 For an ordered pair, we write the value of the independent variable in the first (left) position and the value of the dependent variable in the second (right) position. The numbers a and b are called coordinates. For the x-coordinate is 3 and the y-coordinate is 1. Slide 3 Definition of Solution, Satisfy, and Solution Set Definition Definition of Solution, Satisfy, and Solution Set

4. Lehmann, Intermediate Algebra, 3ed Section 1.2 The equation becomes a true statement when we substitute 3 for x- coordinate and 1 for y-coordinate. Slide 4 Definition of Solution, Satisfy, and Solution Set

5. Lehmann, Intermediate Algebra, 3ed Section 1.2 An ordered pair is a solution of an equation in terms of x and y if the equation becomes a true statement when a is substituted for x and b is substituted for y. We say satisfies the equation. The solution set of the equation is the set of all solution of the equation. Slide 5 Definition of Solution, Satisfy, and Solution Set Definition Definition of Solution, Satisfy, and Solution Set

6. Lehmann, Intermediate Algebra, 3ed Section 1.2 Find five solutions to the equation and plot them in the coordinate system (on the right). Slide 6 Graphing an Equation Example Definition of Solution, Satisfy, and Solution Set

7. Lehmann, Intermediate Algebra, 3ed Section 1.2 We begin be arbitrarily choosing the values 0, 1, and 2 to substitute for x: The ordered pairs and are also solutions. Slide 7 Graphing an Equation Solution Definition of Solution, Satisfy, and Solution Set

8. Lehmann, Intermediate Algebra, 3ed Section 1.2 Create a table of solutions xy -2 5 -1 3 0 1 1-1 2-3 Slide 8 Graphing an Equation Plot the solutions Points form a linear line. Solution Continued Definition of Solution, Satisfy, and Solution Set

9. Lehmann, Intermediate Algebra, 3ed Section 1.2 Every point on the line is a solution to the equation Slide 9 Graphing an Equation Definition of Solution, Satisfy, and Solution Set

10. Lehmann, Intermediate Algebra, 3ed Section 1.2 The point lies on the line Should satisfy the equations Whereas is not on the line Thus should not satisfy the equation Slide 10 Graphing an Equation Definition of Solution, Satisfy, and Solution Set

11. Lehmann, Intermediate Algebra, 3ed Section 1.2Slide 11 Graphing an Equation The is not a solution to the equation Definition of Solution, Satisfy, and Solution Set

12. Lehmann, Intermediate Algebra, 3ed Section 1.2 Use ZDecimal on a graphing calculator. To enter press (–) 2 X,T,,n + 1. The key – is used for subtraction, and the key. (–) is used for negative numbers as well as taking the opposite. Slide 12 Graphing an Equation Calculator Definition of Solution, Satisfy, and Solution Set

13. Lehmann, Intermediate Algebra, 3ed Section 1.2 The graph of an equation in two variables is the set of points that correspond to all solutions of the equation. In the last example we found that the equation. is a line. Notice that the equation. is of the form (where and ). Slide 13 Definition: Graph Definition Definition of Solution, Satisfy, and Solution Set

14. Lehmann, Intermediate Algebra, 3ed Equations of the form Section 1.2 If an equation can be put into the form where m and b are constants, then the graph of the equation is a line. Slide 14 Graphs of Linear Equations What is m and b for the equations Example Graphs of Linear Equations

15. Lehmann, Intermediate Algebra, 3ed is of the form : and is of the form because we write the equation as (so and ). Sketch the graph of the equation Graphing Linear Equations Definition Example Graphs of Linear Equations

16. Lehmann, Intermediate Algebra, 3ed Section 1.2 First we solve for y Slide 16 Graphing Linear Equations Solution Graphs of Linear Equations

17. Lehmann, Intermediate Algebra, 3ed Section 1.2. is of the form The graph of the equation is a line Find 2 points of the line Plot the two points Sketch the line Find a third point Verify that the third point lies on the line Slide 17 Graphing Linear Equations Graphs of Linear Equations Solution Continued

18. Lehmann, Intermediate Algebra, 3ed Section 1.2Slide 18 Graphing Linear Equations Table of solutions x y Solution Continued Graphs of Linear Equations

19. Lehmann, Intermediate Algebra, 3ed Section 1.2 Enter for y 1. Use Zstandard followed by Zsquare. The graph is correct assuming that y was isolated correctly. Slide 19 Graphing Linear Equations Graphing Calculator Graphs of Linear Equations

20. Lehmann, Intermediate Algebra, 3ed Section 1.2 Sketch the graph of Use the distributive property on the left-hand side. Collect like terms. Isolate y. Slide 20 Using the Distributive Law to Help Graph a Linear Equation Example Solution Graphs of Linear Equations

21. Lehmann, Intermediate Algebra, 3ed Section 1.2Slide 21 Using the Distributive Law to Help Graph a Linear Equation Solution Continued Graphs of Linear Equations

22. Lehmann, Intermediate Algebra, 3ed Section 1.2Slide 22 Using the Distributive Law to Help Graph a Linear Equation Table of solutions x y Solution Continued Graphs of Linear Equations

23. Lehmann, Intermediate Algebra, 3ed Section 1.2 Sketch a graph of Avoid fraction values for y Use even values for x Slide 23 Graphing an Equation That Contains Fractions Example Solution Graphs of Linear Equations

24. Lehmann, Intermediate Algebra, 3ed Table of solutions x y 0 2 4 Section 1.2Slide 24 Graphing an Equation That Contains Fractions Solution Continued Graphs of Linear Equations

25. Lehmann, Intermediate Algebra, 3ed Section 1.2 Use Zdecimal to verify the solution. Slide 25 Graphing an Equation That Contains Fractions Graphing Calculator Graphs of Linear Equations

26. Lehmann, Intermediate Algebra, 3ed Section 1.2 Sometimes we find intercepts to graph a line. x-intercept is on the y-axis, so y = 0 y-intercepts in on the x-axis, so x = 0 Slide 26 Property For an equation containing the variables x and y x-intercept: Substitute y = 0 and solve for x y-intercept: Substitute x = 0 and solve for y Directions Finding Intercepts of a Graph

27. Lehmann, Intermediate Algebra, 3ed Section 1.2 Use intercepts to sketch a graph of Slide 27 Using Intercepts to Sketch a Graph x-intercept: Set y = 0. Example Solution Finding Intercepts of a Graph

28. Lehmann, Intermediate Algebra, 3ed Section 1.2 y-intercept: Set x = 0. So, the x-intercept is and y-intercept is Slide 28 Using Intercepts to Sketch a Graph Solution Continued Finding Intercepts of a Graph

29. Lehmann, Intermediate Algebra, 3ed Section 1.2 Use ZStandard followed by ZSquare. Use “zero” to verify the x-intercept. Slide 29 Using Intercepts to Sketch a Graph Use TRACE to verify the y-intercept. Graphing Calculator Finding Intercepts of a Graph

30. Lehmann, Intermediate Algebra, 3ed Section 1.2 Graph the equation of Slide 30 Graphing a Vertical Line xy 3 5 3 3 1 3-1 3-3 Notice that the values of x must be 3, but y can have any value. Some solutions are listed to the left. Example Solution Finding Intercepts of a Graph

31. Lehmann, Intermediate Algebra, 3ed Section 1.2 Graph the equation of Slide 31 Graphing a Horizontal Line x y –2 –5 –1 –5 0 –5 1 –5 2 –5 Notice that the values of y must be –5, but x can have any value. Some solutions are listed to the left. Example Solution Vertical and Horizontal Lines

32. Lehmann, Intermediate Algebra, 3ed Section 1.2 Use ZStandard to verify the graph. Slide 32 Graphing a Horizontal Line Graphing Calculator Vertical and Horizontal Lines

33. Lehmann, Intermediate Algebra, 3ed Section 1.2 If a and b are constants: An equation that can be put into the form. has a vertical line as its graph An equation that can be put into the form.has a horizontal line as its graph Slide 33 Vertical and Horizontal Line Property Property Vertical and Horizontal Lines

34. Lehmann, Intermediate Algebra, 3ed Section 1.2 In an equation can be put into either form where m, a, and b are constants, then the graph of the equation is a line. We call such an equation a linear equation in two variables. Slide 34 Vertical and Horizontal Line Property Property Vertical and Horizontal Lines