Warm-Up Determine the coordinates of each point in the graph below. y -12 -10 -8 -6 -4 -2 2 4 6 8 10 x y A B C D

Linear Equations and Their Graphs pt. 1 Objectives: To determine whether an ordered pair is a solution of an equation

Solutions of Equations How many solutions does the equation 4x + 6 = 14 have? What are they? 4x = 8 4 4 x = 2 An equation such as y = 3x + 7 has many solutions, which we write as ordered pairs of numbers. (x,y)

Example 1 Determine whether is a solution of y = 3x -2. ( , ) 2 4 ( , ) 2 4 y = 3x - 2 (4) = 3 (2) - 2 4 = 6 - 2 4 = 4 (2,4) is a solution of y = 3x -2

Practice 1) Determine whether (2,3) is a solution of y = 2x + 3.

Example 2 Find three solutions of y = 2x + 11. x y y = 2x + 11 11 y = 2(-1) + 11 y = 13 y = 11 y = 9 1 13 -1 9 *Our three solutions are (0,11), (1,13), and (-1,9).

Linear Equations and Their Graphs: pt. 2 Objective: To graph equations in two variables Using 3 Points Using Intercepts Horizontal / Vertical

Linear Equations Equations whose graphs are lines are linear equations. Here are some examples: Linear Equations y = 3x + 7 6y = -2 9x – 15y = 7 Nonlinear Equations y = x2 - 4 x2 + y2 = 16 xy = 3

Example 1 Graph the equation 2x + 2y = 6 using 3 points 2x + 2y = 6 solve for y 2x + 2y = 6 -2x -2x x y 2y = 6 – 2x 3 2 2 1 2 y = 3 - x -2 5 y = 3 - (0) = 3 y = 3 - (1) = 2 y = 3 - (-2) = 5

Example 1 Graph the equation 2x + 2y = 6. x y 3 1 2 -2 5 2x + 2y = 6 8 4 x y 2 3 -8 -6 -4 -2 2 4 6 8 1 2 -2 -2 5 -4 -6 -8

Example 2 Graph the equation 3y – 6 = 9x. 3y – 6 = 9x solve for y +6 2 3 3 1 5 y = 3x + 2 -2 -4 y = 3(0) + 2 = 2 y = 3(1) + 2 = 5 y = 3(-2) + 2 = -4

Example 2 Graph the equation 3y – 6 = 9x. x y 2 1 5 -2 -4 3y – 6 = 9x 8 3y – 6 = 9x 6 4 x y 2 2 -8 -6 -4 -2 2 4 6 8 1 5 -2 -2 -4 -4 -6 -8

Practice Graph these linear equations using three points. 1) 6x – 2y = -2 2) -10x – 2y = 8

Graphing Using Intercepts The x-intercept of a line is the x-coordinate of the point where the line intercepts the x-axis. The line shown intercepts the x-axis at (2,0). 8 6 4 We say that the x-intercept is 2. 2 -8 -6 -4 -2 2 4 6 8 -2 -4 -6 -8

Graphing Using Intercepts The y-intercept of a line is the y-coordinate of the point where the line intercepts the y-axis. The line shown intercepts the y-axis at (0,-6). 8 6 We say that the y-intercept is -6. 4 2 -8 -6 -4 -2 2 4 6 8 -2 -4 -6 -8

Example 1 Graph 4x – 3y = 12 using intercepts. *To find the y-intercept, let x = 0. 4x – 3y = 12 x y 4(0) – 3y = 12 -4 0 – 3y = 12 -3y = 12 -3 -3 y = -4

Example 1 Graph 4x – 3y = 12 using intercepts. *To find the x-intercept, let y = 0. 4x – 3y = 12 x y 4x – 3(0) = 12 -4 4x - 0 = 12 3 4x = 12 4 4 x = 3

Example 1 Graph 4x – 3y = 12 using intercepts. x y -4 3 8 6 4 2 -8 -6 -4 -8 -6 -4 -2 2 4 6 8 -2 3 -4 -6 -8

Example 2 Graph 2x + 5y = 10 using intercepts. *To find the y-intercept, let x = 0. 2x + 5y = 10 x y 2(0) + 5y = 10 2 0 + 5y = 10 5y = 10 5 5 y = 2

Example 2 Graph 2x + 5y = 10 using intercepts. *To find the x-intercept, let y = 0. 2x + 5y = 10 x y 2x + 5(0) = 10 2 2x + 0 = 10 5 2x = 10 2 2 x = 5

Example 2 Graph 2x + 5y = 10 using intercepts. x y 2 5 8 6 4 2 -8 -6 2 -8 -6 -4 -2 2 4 6 8 -2 5 -4 -6 -8

Practice Graph using intercepts. 1) 5x + 7y = 35 2) 8x + 2y = 24

Warm-Up 10 minutes Graph these equations: -x + 2y = 4 2x + 3y = 8

Graphing Horizontal and Vertical Lines The standard form of a linear equation in two variables is Ax + By = C, where A,B, and C are constants and A and B are not both 0. 3x + 4y = 12 6x + 7y = 23

Example 1 Graph y = -2. write the equation in standard form Ax + By = C (0)x + (1)y = -2 -8 -6 -4 -2 2 4 6 8 for any value of x y = -2

Example 2 Graph x = 7. write the equation in standard form Ax + By = C (1)x + (0)y = 7 -8 -6 -4 -2 2 4 6 8 x = 7 for any value of y

Practice Graph these equations. 1) x = 5 2) y = -4 3) x = 0