1. Representing Equations
Graphing in the Coordinate Plane

2. What You’ll learn To graph points on a plane
Graphing Points
What You’ll learn
To graph points on a plane

3. Rene Descartes develops Coordinate System
Coordinate Plane is a grid formed by a horizontal number line called the X axis and a vertical number line called the Y axis
Ordered Pair (x,y) gives the coordinates of the location of a point
X-coordinate is the first number of horizontal units from the origin
Y-coordinate is the second number of vertical units for the origin
-1,1
X axis
Y axis
origin
Ordered pair

4. The X and Y axes divide the coordinate plane into 4 QuadrantsII I
III IV

5. Writing Coordinates
The smiley face is 2 units to the right of the y axis, so the x coordinate is 2
The smiley face is 2 units above the x axis, so the y coordinate is 2
The ordered pair for the location of the fly is (2,2)

6. Graph point A (3,-5); B (-3,-2); C (-2, 4)
Text page ; Pr 10-1

7. Ordered Pair
(3, 5)
Domain of a relation is the set of first coordinates (or x values)
Range is the set of second coordinates (y values)

8. Relations and Functions
Relation: a set of ordered pairs
Function: a special type of relation that pairs each domain with exactly on range

9. Representations of Relations
Table
Graph
Mapping Diagram

10. Identifying Functions
State the domain and range, then whether the relation is a function
Alg. book p. 173 # 1-14

11. Linear Equations
A function is linear if it can be written in standard form
Ax + By = C
When an equation is written in standard form:
x and y both have an exponent of 1
x and yare not multiplied together
and y do not appear in denominators, exponents, or radicals

12. Linear or Nonlinear?
y = x + 3
3xy + x = 1
y – 2 = 3x
x² + y = - 1

13. Graphing Linear Equations using a table
What You’ll Learn:
To find ordered pairs that are solutions of linear equations
To graph linear equations

14. Determine whether each ordered pair is a solution of y = x + 5
(40,45)
y = x + 5
45 =
45 = 45
(21,27)
y = x + 5
27 =
27 = 26
Substitute for x and y in the equation

15. Determine whether each ordered pair is a solution of y = 3x - 1
(4, 11)
(7, 12)
(17, 23)

16. Graphing Linear Equations
Step 1
Make table
Step 2
Insert x values
Step 3
Find y values
Step 4
Graph the ordered pairs and draw a line through the points X
Equation = y y
(X, Y)

17. Graph: y = x + 1 x X + 1 Y (x,y) -4 -4 + 1 -3 -4,-3 -2 -2 + 1 -1 -2,-1
0 + 1 1
0,1
1 + 1 2
1,2 3
3 + 1 4
3,4
Solutions of the equation
Graph points, draw line
Choose values for x

18. Graph points from table

19. Practice Graph y – 2 = ¼ x Graph y = 5/6x + 3 Graph 2x + 3y = -6
Alg I bk p. 235 #’s 30-49

20. Graphing Linear Equations using Intercepts
What You’ll Learn:
To find X and Y Intercept
To graph linear equations

21. EX. 1 Graphing using the Intercepts
Graph 2x + 3y = 6 using the x and y intercepts.
Step 1: Find the intercepts
Find the y intercept letting x = 0
2x + 3y = 6
2(0) + 3y = 6
3y = 6
y = 2

22. Graphing using the Intercepts
Graph 2x + 3y = 6 using the x and y intercepts.
Step 1: Find the intercepts
Find the x intercept letting y = 0
2x + 3y = 6
2x+ 3(0) = 6
2x + 0 = 6
x = 3

23. Step 2 Graph the equation
Plot points (0,2) & (3,0)

24. Practice On graph paper, graph the following using x and y intercepts:
4x + 2y = 8
3y + x = 9
3x – 4y = 24
Alg I bk p. 240 #’s 1-21

25. 3-3 Finding the Slope of a line
What You’ll Learn:
To find and use the slope of a line
Investigation text page 533

26. Slope of a line
Slope is a ratio that describes the steepness of a line
Slope = rise
run
Rise – compares the vertical change a line
Run – the horizontal change of a line

27. Finding Slope Slope = rise / run -6 6/-6 1/-1 or -1 +6
Pr 10-3; page 538 practice quiz, quiz 10-1 to 10-3

28. Finding Slope
Negative Slope
Positive slope

29. Slopes of Horizontal and Vertical Lines
m = 0
m = undefined

30. Which Slope is Steeper?

31. Which Slope is Steeper?

32. Find The Slopes Slope Formula (m) m = y2 – y1 m = (-3) – 3 1 – (-2)
x2 - x1
m = (-3) – 3
1 – (-2)
m = = or -2
-2,3
1,-3

33. Find the Slope of the following Lines that contain the following Points
(-3, 7) ; (3, 3)
(0, -2) ; (4,0)
(-5,-3) ; (-2,-3)
Verify you answers by graphing each pair of coordinates
Alg 1 bk Text. P ; p257 #1-10

34. Review: For y = -1/3x + 3 ; graph and find the slope of the line
Find 3 solutions to graph
Use X/Y chart
Graph order pairs and connect points
Find slope
m = y2 – y1
x2 – x1
Verify by counting on graph

35. Using the y - Intercept
Remember Slope is the ratio of vertical change and horizontal change (Rise/Run)
The y - intercept is the y-coordinate of the point where a line crosses the y-axis

36. Below is the graph of y = -2/3 x +6
Find the y intercept of the line
Find the Slope of the line
What do you notice about the Equation?

37. Slope Intercept Form
An equation written in the form y = mx + b is in slope intercept form.
The graph is a line with slope “m” and y- intercept “b”

38. Identify Slope (m) and y-intercept (b)
Find the slope of each equation:
Find the y-intercept of each equation:
y = 6x
y = -x – 4
y = 2x – 1
y = - x
y = - 4x + 2 5
y = 6x
y = -x – 4
y = 2x – 1
y = - x
y = - 4x + 2 5

39. Identify Slope (m) and y-intercept (b)
Find the slope of each equation
Find the y intercept of each equations
y + ½ x = -6
-1/4x = y + 3
y – x = -1
y + ½ x = -6
-1/4x = y + 3
y – x = -1
Text p. 139 # 1-6; 25-27

40. Using slope intercept to graph a line
Step 1: Find the slope and y-intercept
Step 2: Graph the y-intercept
Step 3: use slope to graph next 2 points
Step 4: Draw line through the points

41. Graph y =1/3x + 4

42. Graph the following Equations
y = ½ x – 3
y + 2 = 3/4x
Text p. 140 #7-12 odd; 21-23

43. Writing an Equation for a Line
Write and equation for a line with a slope of ½ and y - intercept of -6
Remember you formula
y = mx + b
Insert your given/found values
y = ½ x - 6

44. Writing Equations for a Line
a. m = 4, b = -2
b. m = - 3/2 , b = 5
c. m = ¼ , b = 0

45. Write equation from graph
Steps:
y = mx + b
Step 1:
Find the y - intercept
Step 2:
Find the slope
Count or use formula
Step 3:
Write equation in slope intercept form
m = 3/1
b = -1
Alg. I bk p. 272 #’s 1-12;14-22

46. 3-4 Key Concepts
Write an equation
Graph y = -2x + 4

47. Practice

48. Point Slope Form
Objective: To graph a line and write a linear equation using point slop form

49. What you should know Find the slope of a line containing points
(0,2) and (3,4)
Write the following in slope-intercept form, find m and b
y – 5 = 3(x +2)

50. Point -Slope Form of a Linear Equations
y – y1 = m ( x – x1)
Point ( x, y) Slope: Rise/Run

51. Example 1: Writing an Equation in Point - Slope Form
Remember: y – y1 = m ( x – x1)
m = 5/2 ; (-3,0)
m = ; (4,2)
m = ; (-2,-3)

52. Example 2a: Using Point-Slope Form to Graph
Graph: y – 1 = 3 (x – 1)

53. Example 2b: Using Point-Slope Form to Graph
Graph: y + 2 = - 1/2(x – 3)
Rewrite to match the formula

54. Homework
Text page 279 #’s 1-6

55. Example 3a: Writing an Equation in Slope Intercept Form
Remember: y – y1 = m ( x – x1 )
y = m ( x – b)
Containing: m = -4 ; (-1, - 2)
Step 1: Write equation in point slope form
Step 2: Transform equation in slope intercept form (get y by itself)

56. Example 3b: Writing an Equation in Slope Intercept Form
Containing: point (1, -4) and (3, 2)
Step 1: Find the slope
Step 2: Write in point slope
Step 3: Transform into slope intercept
Do with both points

57. Example 3c: Writing an Equation in Slope Intercept Form
Remember: y – y1 = m ( x – x1 )
y = m ( x – b)
Containing: X intercept = -2 ; Y Intercept = 4
Step 1: Write ordered pairs of intercepts
Step 2: Find Slope
Step 3: Write in point slope form
Step 4: Transform equation in slope intercept form

58. Example 4: Using Two Points to find Intercepts
Containing: (4, 8) ; (- 1 , - 12)
Step 1: Find Slope
Step 2: Write in point slope form
Step 3: Transform into slope intercept form
Find X & Y Intercepts

59. Homework
Text page 279 #’s 7-15

60. Parallel / Perpendicular
Parallel Lines – Two lines that do not intersect
Perpendicular – Two lines the do intersect to form right angles

61. Theorem Two lines are parallel if their slopes are equal
Two lines are perpendicular if the product of their slopes are equal -1

62. 3x – y = 4 y = 3x + 1 What are these two lines? Parallel/Perpendicular
Put each in y int. form
3x – y = 4 >> y = 3x – 4
y = 3x +1
Identify slopes
3/1 & 3/1
Identify relationship
Lines are Parallel because the slopes are the same

63. Identify which lines are Parallel and Perpendicular
Ln 1: y = 2x – 6
Ln 2: 4x – 2y = 1
Ln 3: x – y + 2 = 0
Ln 4: 2y = 10 – x
Page

64. Writing Equations to Parallel Lines
Write and equation in slope intercept form for the line that passes through (4,5) and is parallel to y = 5x + 10.
Step 1: Find the slope of the line
Step 2: Write the equation in point slope form
Step 3: Transform the equation to slope intercept form

65. Write and equation in slope intercept form for the line
Passes through point (4, 10) and is parallel y = 3x + 8
Passes through point (2, -1) and is parallel y = ½ x

66. Writing Equations to Perpendicular Lines
Write and equation in slope intercept form for the line that passes through (4,5) and is Perpendicular to y = 5x + 10.
Step 1: Find the slope of the line (opposite reciprocal)
Step 2: Write the equation in point slope form
Step 3: Transform the equation to slope intercept form

67. Write and equation in slope intercept form for the line
Passes through point (4, 10) and is perpendicular to y = 3x + 8
Passes through point (2, -1) and is perpendicular to y = ½ x

68. Homework
Text p #’s 22-26; 34,35,37,38

69. Solving Linear Systems by Graphing
Two or more linear equations form a system of linear equations.
A solution of the system is any ordered pair that is a solutions of each equation
Review problems from p. 325 on board as warm up activity

70. Solving Systems by Graphing
Ex. 1) Tell whether the ordered pair is a solution of a given system
(4,1) ; x + 2y = 6 & x – y = 3
substitute x & y values in each equation
(-1,2) ; 2x + 5y = 8 & 3x – 2y = 5

71. Finding a Solution to Systems of Linear Equations
(4, 2)
y = ½ x
y = 2x - 6

72. Ex.1)Solve the system of equations y= x - 3 and y = - x – 1

73. Ex 2).Solve the system of equations x + y= 0 and y = ½ x + 1

74. Homework
Page 332
#’s 1-7
Review
#’s 9-15

75. Solving Systems by Substitution
What You will Learn:
To solve systems on equations with 2 variables by substitution

76. Solving Systems by Substitution
Ex.1a solve for each variable by using substitution
y = 2x
y = x + 5
Step 1: pick a variable to isolate (get by itself)
Step 2: Substitute the equation of isolated variable in 2nd equation
Step 3: Solve for the variable
Step 4: Substitute the value of the variable to solve for the other variable in the original equation
Step 5: Write each variable as ordered pair (x, y)

77. Solving Systems by Substitution
Ex.1b solve for each variable by using substitution
2x + y = 5
y = x - 4
Step 1: pick a variable to isolate (get by itself)
Step 2: Substitute the equation of isolated variable in 2nd equation
Step 3: Solve for the variable
Step 4: Substitute the value of the variable to solve for the other variable in the original equation
Step 5: Write each variable as ordered pair (x, y)

78. Solving Systems by Substitution
Ex.1c solve for each variable by using substitution
x + 4y = 6
x + y = 3
Step 1: pick a variable to isolate (get by itself)
Step 2: Substitute the equation of isolated variable in 2nd equation
Step 3: Solve for the variable
Step 4: Substitute the value of the variable to solve for the other variable in the original equation
Step 5: Write each variable as ordered pair (x, y)

79. Homework
Text page 340 #’s 1-3
Review

80. Solving Systems by Substitution
Ex.2 solve for each variable by using substitution and distributive property
4y – 5x = 9
x - 4y = 11
Step 1: pick a variable to isolate (get by itself)
Step 2: Substitute the equation of isolated variable in 2nd equation
Step 3: Solve for the variable using distributive property
Step 4: Substitute the value of the variable to solve for the other variable in the original equation
Step 5: Write each variable as ordered pair (x, y)

81. Homework
Text page 340 #’s 4-6
Review
#’s 9, 10 , 13, 15, 19, 22

82. Solving Systems by Elimination
Ex.1 Elimination using addition
x – 2y = - 19
5x + 2y = 1
Step 1: Align like term, pick a variable to eliminate
Step 2: add equations to eliminate variable
Step 3: Solve for the variable
Step 4: Substitute the value of the variable to solve for the other variable in the original equation
Step 5: Write each variable as ordered pair (x, y)

83. Solving Systems by Elimination
Ex.2 Elimination using subtraction
3x + 4y = 18
-2x + 4y = 8
Step 1: Align like term, pick a variable to eliminate
Step 2: subtract equations to eliminate variable(add opposites of entire equation)
Step 3: Solve for the variable
Step 4: Substitute the value of the variable to solve for the other variable in the original equation
Step 5: Write each variable as ordered pair (x, y)

84. Solving Systems by Elimination
Ex.3 Elimination using multiplication (common multiple)
2x + y = 3
-x + 3y = -12
Step 1: Align like term, pick a variable to eliminate
Step 2: Multiply equation to make common multiple to eliminate
Step 3: Solve for the variable
Step 4: Substitute the value of the variable to solve for the other variable in the original equation
Step 5: Write each variable as ordered pair (x, y)

85. Homework
Text page 347
#’s 1-9
Review
#’s 11-19

86. Solving Special Systems
Consistent System
A system with at least one solution or when two lines intersect at least one point
Independent system
Has exactly 1 solution
Dependent system
Has infinite solutions
Inconsistent System
When 2 lines do not intersect, therefore there is no ordered pair to satisfy both equations (parallel lines)

87. Ex.1 Systems with no solutions
Show that the systems have no solution
y = x – 1
-x + y = 2
Step 1: Write the equations to line up the like terms
Step 2: Add terms
Step 3: identify relationship

88. Ex.2 Systems with infinite solutions
Show that the systems have no solution
y = 2x +1
2x – y = -1
Step 1: Write the equations to line up the like terms
Step 2: Add terms
Step 3: identify relationship

89. Homework
Text page 335 #’s 8-10
Review
Text page 336 #’s 20-22

90. Practice