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Graphing Linear Equations

Graphing Linear Equations. What is a linear equation?. Any equation that can be written as Ax+By=C where A and B are both not 0. Ax+By=C is called the general or standard form of the equation of a line. Examples of linear equations. 2x-3y=6 3x=1-2y. Examples of linear equations. y=-2x+3

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Graphing Linear Equations

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  1. Graphing Linear Equations

  2. What is a linear equation? • Any equation that can be written as Ax+By=C where A and B are both not 0. • Ax+By=C is called the general or standard form of the equation of a line.

  3. Examples of linear equations • 2x-3y=6 • 3x=1-2y

  4. Examples of linear equations • y=-2x+3 • (1/2)x=.75y-3

  5. Examples of non-linear equations

  6. Examples of non-linear equations

  7. Ways to Graph Linear Equations • Table of Values • x and y intercepts • Slope Intercept Form: y = mx + b

  8. x y Table of Values

  9. Table of Values • You can select any value for x OR y, then you can solve for the other variable. • If you choose a value for x, then input that value into the linear equation and solve for y. If you choose a y value, then solve for x.

  10. Table of Values • To create a line you need 2 coordinates. • To be safe find 3 coordinates. If one point is incorrect, you will know because the coordinates will not create a straight line.

  11. x y 0 1 -2 Table of Values • Example: Given: 2x + 3y = 6 • Solve:

  12. x y 0 1 -2 Table of Values • When x = 0 • 2(0) + 3y = 6 • 2 + 3y = 6 • -2 -2 • 3y = 6 • 3 3 • y = 2 2

  13. x y 2 3/2 0 1 -2 Table of Values • When y = 1 • 2x + 3(1) = 6 • 2x + 3 = 6 • - 3 - 3 • 2x = 3 • 2 2 • x = 3/2

  14. x y 2 3/2 0 1 10/3 -2 Table of Values • When x = -2 • 2(-2) + 3y = 6 • -4 + 3y = 6 • +4 +4 • 3y = 10 • 3 3 • Y = 10/3

  15. x y 2 3/2 0 1 10/3 -2 Table of Values List the points (0, 2) (3/2, 1) = (1.5, 1) (-2, 10/3) = (-2, 3 1/3)

  16. Table of Values Plot the points (0, 2) (1.5, 1) (-2, 3 1/3)

  17. Table of Values • Draw a line through the points • You have graphed the equation: 2x + 3y = 6

  18. Graph by x and y intercepts • The x-intercept of a line is the point (a,0) where the line intersects the x-axis. To find a, substitute 0 for the y and solve for x. • The y-intercept of a line is the point (0,b) where the line intersects the y-axis. To find b, substitute 0 for the x and solve for y.

  19. Graph by x and y intercepts • Example: • Graph 2x +3y = 6 by finding the x and y intercepts.

  20. Graph by x and y intercepts • Find the x-intercept by making y = 0. • 2x – 3(0) = 6 • 2x = 6 • 2 2 • x = 3 • x-intercept = (3, 0)

  21. Graph by x and y intercepts • Find the y-intercept by making x = 0. • 2(0) + 3y = 6 • 3y = 6 • 3 3 • y = 2 • y-intercept = (0, 2)

  22. Graph by x and y intercepts • Plot the: • x-intercept (3, 0) • y-intercept (0, 2)

  23. Graph by x and y intercepts • Draw a line through the intercepts and you have graphed: 2x + 3y = 6

  24. Slope intercept form:y = mx + b • The slope intercept form is y = mx + b where “m” is the slope and “b” is the y- intercept. • The x and y represent the coordinates which satisfy the linear equation.

  25. Slope intercept form:y = mx + b The slope is all of the following: • rate of change of a linear graph. • change in y over the change in x. • vertical change over the horizontal change. • rise over run.

  26. Slope intercept form:y = mx + b • Different Types of slopes • Positive • Negative • Slope of zero • Slope is undefined

  27. Slope intercept form:y = mx + b • Examples of: Positive Slopes

  28. Slope intercept form:y = mx + b • Examples of: Negative Slopes

  29. Slope intercept form:y = mx + b • Examples of: Slopes of zero

  30. Slope intercept form:y = mx + b • Examples of: Slopes is undefined

  31. Slope intercept form:y = mx + b • The slope formula is used to find the slope given two coordinates. • Slope formula:

  32. Slope intercept form:y = mx + b • Example: Find the slope given: (2, -3) and (-4, -1)

  33. Slope intercept form:y = mx + b • First label the coordinates:

  34. Slope intercept form:y = mx + b • Substitute into the slope formula and solve.

  35. Slope intercept form:y = mx + b • Slope can be determined by calling it rise/run. • The rise determines if you go up or down. • If the number is (+), then go up • If the number is (-), then go down.

  36. Slope intercept form:y = mx + b • The run determines if you go left or right. • If the number is (+), then go to the right. • If the number is (-), then go to the left.

  37. Slope intercept form:y = mx + b • In our example, we got a slope of: • (-1/4) • The rise is: -1 • The run is: 4 • This means from a point on the graph, I would go down 1 and to the right 4 to find another point on the graph.

  38. Slope intercept form:y = mx + b • To graph using the slope intercept form: • the equation must be in slope intercept form • you must determine your “m” and “b” • plot your “b” • use “m” to find another point • then draw a line through those coordinates

  39. Slope intercept form:y = mx + b • Example: • Graph: 2x +3y = 6 using slope intercept form

  40. Slope intercept form:y = mx + b • Step 1: Convert to slope intercept form • 2x + 3y = 6 • -2x -2x • 3y = -2x + 6 • 3 3 • y = (-2/3) x + 2

  41. Slope intercept form:y = mx + b • Step 2: Determine your “m” and “b” • y = (-2/3) x + 2 • m = (-2/3) • b = 2

  42. Slope intercept form:y = mx + b • Step 3: • Plot your • “b” (y-intercept) • b = 2

  43. Slope intercept form:y = mx + b • Step 4: Use “m” to find another point • m = (-2/3) • rise = -2 and the run = 3 • From the “b” (y-intercept) you would go down 2 • Then go to the right 3

  44. Slope intercept form:y = mx + b • After you use the “m” to find the other point, then you draw a line through the 2 points and you have graphed 2x+3y=6.

  45. Now that you are done: “Try these practice problems using the various methods.”

  46. y= -2x + 3

  47. 3y = -4x - 1

  48. x – 2y = 4

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