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Graphing Linear Equations

Graphing Linear Equations

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Graphing Linear Equations

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  1. Section 1.2 Graphing Linear Equations

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

  3. Section 1.2 Slide 3 Definition of Solution, Satisfy, and Solution Set 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.

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

  5. Section 1.2 Slide 5 Definition of Solution, Satisfy, and Solution Set 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.

  6. Section 1.2 Slide 6 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).

  7. Section 1.2 Slide 7 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 and are also solutions.

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

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

  10. Section 1.2 Slide 10 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

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

  12. Section 1.2 Slide 12 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.

  13. Section 1.2 Slide 13 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 ).

  14. Section 1.2 Slide 14 Graphs of Linear Equations Graphs of Linear Equations 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

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

  16. Section 1.2 Slide 16 Graphing Linear Equations Graphs of Linear Equations Solution First we solve for y

  17. Section 1.2 Slide 17 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

  18. Section 1.2 Slide 18 Graphing Linear Equations Graphs of Linear Equations Solution Continued Table of solutions x y

  19. Section 1.2 Slide 19 Graphing Linear Equations Graphs of Linear Equations Graphing Calculator Enter for y1. Use Zstandard followed by Zsquare. The graph is correct assuming that y was isolated correctly.

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

  21. Section 1.2 Slide 21 Using the Distributive Law to Help Graph a Linear Equation Graphs of Linear Equations Solution Continued

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

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

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

  25. Section 1.2 Slide 25 Graphing an Equation That Contains Fractions Graphs of Linear Equations Graphing Calculator Use Zdecimal to verify the solution.

  26. Section 1.2 Slide 26 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

  27. Section 1.2 Slide 27 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.

  28. Section 1.2 Slide 28 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

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

  30. Section 1.2 Slide 30 Graphing a Vertical Line Finding Intercepts of a Graph Example Graph the equation of Solution x y 3 5 3 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.

  31. Section 1.2 Slide 31 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 Notice that the values of y must be –5, but x can have any value. Some solutions are listed to the left.

  32. Section 1.2 Slide 32 Graphing a Horizontal Line Vertical and Horizontal Lines Graphing Calculator Use ZStandard to verify the graph.

  33. Section 1.2 Slide 33 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

  34. Section 1.2 Slide 34 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.