Section 1 2 Graphing Linear Equations Definition of

Section 1. 2 Graphing Linear Equations

Definition of Solution, Satisfy, and Solution Set Consider the equation So, when the ordered pair Section 1. 2 . Let’s find y when , which cab be represented by Lehmann, Intermediate Algebra, 3 ed Slide 2

Definition of Solution, Satisfy, and Solution Set Definition 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. Section 1. 2 Lehmann, Intermediate Algebra, 3 ed Slide 3

Definition of Solution, Satisfy, and Solution Set The equation becomes a true statement when we substitute 3 for xcoordinate and 1 for y-coordinate. Section 1. 2 Lehmann, Intermediate Algebra, 3 ed Slide 4

Definition of Solution, Satisfy, and Solution Set Definition • 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. Section 1. 2 Lehmann, Intermediate Algebra, 3 ed Slide 5

Graphing an Equation Definition of Solution, Satisfy, and Solution Set Example Find five solutions to the equation and plot them in the coordinate system (on the right). Section 1. 2 Lehmann, Intermediate Algebra, 3 ed Slide 6

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

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

Graphing an Equation Definition of Solution, Satisfy, and Solution Set • Every point on the line is a solution to the equation Section 1. 2 Lehmann, Intermediate Algebra, 3 ed Slide 9

Graphing an Equation Definition of Solution, Satisfy, and Solution Set • The point lies on the line • Should satisfy the equations • Whereas is not on the line • Thus should not satisfy the equation Section 1. 2 Lehmann, Intermediate Algebra, 3 ed Slide 10

Graphing an Equation Definition of Solution, Satisfy, and Solution Set • The Section 1. 2 is not a solution to the equation Lehmann, Intermediate Algebra, 3 ed Slide 11

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

Definition: Graph Definition of Solution, Satisfy, and Solution Set Definition 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 ). Section 1. 2 Lehmann, Intermediate Algebra, 3 ed . . Slide 13

Graphs of Linear Equations of the form If an equation can be put into the form where m and b are constants, then the graph of the equation is a line. Example What is m and b for the equations Section 1. 2 Lehmann, Intermediate Algebra, 3 ed Slide 14

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

Graphing Linear Equations Graphs of Linear Equations Solution First we solve for y Section 1. 2 Lehmann, Intermediate Algebra, 3 ed Slide 16

Graphing Linear Equations Graphs of Linear Equations Solution Continued • . 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 Section 1. 2 Lehmann, Intermediate Algebra, 3 ed Slide 17

Graphing Linear Equations Graphs of Linear Equations Solution Continued Table of solutions x Section 1. 2 y Lehmann, Intermediate Algebra, 3 ed Slide 18

Graphing Linear Equations Graphs of Linear Equations Graphing Calculator • Enter for y 1. • Use Zstandard followed by Zsquare. • The graph is correct assuming that y was isolated correctly. Section 1. 2 Lehmann, Intermediate Algebra, 3 ed Slide 19

Using the Distributive Law to Help Graph a Linear Equation Graphs of Linear Equations Example Sketch the graph of Solution • Use the distributive property on the left-hand side. • Collect like terms. • Isolate y. Section 1. 2 Lehmann, Intermediate Algebra, 3 ed Slide 20

Using the Distributive Law to Help Graph a Linear Equation Graphs of Linear Equations Solution Continued Section 1. 2 Lehmann, Intermediate Algebra, 3 ed Slide 21

Using the Distributive Law to Help Graph a Linear Equation Graphs of Linear Equations Solution Continued Table of solutions x Section 1. 2 y Lehmann, Intermediate Algebra, 3 ed Slide 22

Graphing an Equation That Contains Fractions Graphs of Linear Equations Example Sketch a graph of Solution • Avoid fraction values for y • Use even values for x Section 1. 2 Lehmann, Intermediate Algebra, 3 ed Slide 23

Graphing an Equation That Contains Fractions Graphs of Linear Equations Solution Continued Table of solutions x 0 y 2 4 Section 1. 2 Lehmann, Intermediate Algebra, 3 ed Slide 24

Graphing an Equation That Contains Fractions Graphs of Linear Equations Graphing Calculator Use Zdecimal to verify the solution. Section 1. 2 Lehmann, Intermediate Algebra, 3 ed Slide 25

Property Finding Intercepts of a Graph 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 Directions • 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 Section 1. 2 Lehmann, Intermediate Algebra, 3 ed Slide 26

Using Intercepts to Sketch a Graph Finding Intercepts of a Graph Example Use intercepts to sketch a graph of Solution x-intercept: Set y = 0. Section 1. 2 Lehmann, Intermediate Algebra, 3 ed Slide 27

Using Intercepts to Sketch a Graph Finding Intercepts of a Graph Solution Continued y-intercept: Set x = 0. So, the x-intercept is and y-intercept is Section 1. 2 Lehmann, Intermediate Algebra, 3 ed Slide 28

Using Intercepts to Sketch a Graph Finding Intercepts of a Graphing Calculator Use ZStandard followed by ZSquare. Use TRACE to verify the y-intercept. Use “zero” to verify the x-intercept. Section 1. 2 Lehmann, Intermediate Algebra, 3 ed Slide 29

Graphing a Vertical Line Finding Intercepts of a Graph Example Graph the equation of Solution x y 3 5 3 3 3 1 3 -3 Section 1. 2 Notice that the values of x must be 3, but y can have any value. Some solutions are listed to the left. Lehmann, Intermediate Algebra, 3 ed Slide 30

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

Graphing a Horizontal Line Vertical and Horizontal Lines Graphing Calculator Use ZStandard to verify the graph. Section 1. 2 Lehmann, Intermediate Algebra, 3 ed Slide 32

Vertical and Horizontal Line Property Vertical and Horizontal Lines Property 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 Section 1. 2 Lehmann, Intermediate Algebra, 3 ed Slide 33

Vertical and Horizontal Line Property Vertical and Horizontal Lines Property 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. Section 1. 2 Lehmann, Intermediate Algebra, 3 ed Slide 34