Chapter 3 Linear Equations and Inequalities in two variables.
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1 Chapter 3 Linear Equations and Inequalities in two variables. 3.1 Paired Data and Graphing Ordered Pairs 3.2 Graphing linear equations in two variables. 3.3 Graphing using intercepts 3.4 The slope of a line 3.5 Slope-intercept Form of a line 3.6 Point-Slope Form of a line 3.7 Linear Inequalities in Two variables. 1
2 Bar Graph A blood-alcohol level of 0.08% or higher makes driving illegal in the United States. The bar graph shows how man drinks a person of a certain weight would need to consume in 1 hr to achieve a bloodalcohol level of 0.08%. Copright 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesle 2
3 Eercise: Answer the following using the bar graph on the previous page. 1) Approimatel how man drinks would a 180-pound person have consumed if he or she had a blood-alcohol level of 0.08%? 2) What can be concluded about the weight of someone who can consume 2 drinks in an hour without reaching a blood-alcohol level of 0.08%? 3
4 3.1 Paired Data and Graphing an Ordered Pair (,) Qualitative Bar Graph 4
5 3.1 Paired Data and Graphing an Ordered Pair (,) Qualitative Bar Graph 5
6 3.1 Paired Data and Graphing an Ordered Pair (,) Quantitative Bar Graph 6
7 3.1 Paired Data and Graphing an Ordered Pair (,) Quantitative Bar Graph 7
8 3.1 Paired Data and Graphing an Ordered Pair (,) Scatterplot 8
9 3.1 Paired Data and Graphing an Ordered Pair (,) Scatterplot 9
10 3.1 Paired Data and Graphing an Ordered Pair (,) Scatterplot 10
11 Line Graph Value of Anmols Car E: According to the graph, how much did the value of the car go down from the third to the fourth ear Anmol owned it? What percent does this represent? Number of ears since Anmol purchased his Car 11
12 3.1 Paired Data and Graphing an Ordered Pair (,) Where the aes cross is called the origin. We will plot points (or ordered pairs): (, ) which consists of two numbers called coordinates. 12
13 Plot the following points. Identif the quadrant each point lies. 5) (-4, -3) 13
14 Plot the following points. Is there a trend? X Y
15 You leave home and drive a car at 60mph. Find our distance () after each 30 minutes () for the net 2.5 hours. Is there a trend here? X Y 15
16 Jud takes a job at Gigi's boutique. Her job pas $9.00 per hour plus $50 per week in commission. The graph shows how much Jud earns for working from 0 to 40 hours in a week. a) List three ordered pairs and write what the mean in sentence form. b) How much will she earn for working 40 hours? c) If her check for one week is $230, how man hours did she work? d) She works 35 hours one week, but her pacheck before deductions are subtracted out is for $365. Is this correct? Eplain. 16
17 3.2 Graphing Linear Equations in 2-variables The Standard Form of a Linear Equation of 2-varibles is given b: A + B = C, where A, B, & C are constants. Eample: - 3 = 1. What are the values for A, B, and C? 17
18 3.2 Graphing Linear Equations in 2-variables Eercise: How man solutions does the equation - 3 =1 have? Definition: The solution to a linear equation in two variables is an infinite set of ordered pairs (,) that make the equation TRUE. Eercise: Which pairs (,) are solutions of - 3 = 1? 1) (4,1) 2) (1,4) 3) (-2,-1) 18
19 3.2 Graphing Linear Equations in 2-variables Eercise: Given the equation = 6, complete the following so the WILL be solutions to the equation. a) (0,?) b) (?,1) c) (3,?) 19
20 3.2 Graphing Linear Equations in 2-variables Eercise: Given the equation, = 2-1, complete the following table: X Y
21 3.2 Graphing Linear Equations in 2-variables Eercise: Given the equation, = 2-1, complete the following table: X Y
22 3.2 Graphing Linear Equations in 2-variables Eercise: Graph the solution set for + = 5. Some solutions are: (0,5), (2,3), (3,2), and (5, 0). Verif this
23 3.2 Graphing Linear Equations in 2-variables How to graph a Linear Equation in two variables: Step 1: Find an three ordered pairs that make the equation TRUE. You can create a table for this. Step 2: Plot those three points. If all three do not line up, choose a forth point to find our mistake. Step 3: Draw a line through the points with arrows on the ends. Eample: Graph the solution to =
24 3.2 Graphing Linear Equations in 2-variables Eample: Graph the solution to 3-2 = 6 24
25 3.2 Graphing Linear Equations in 2-variables Eample: Graph the solution to 25
26 3.2 Graphing Linear Equations in 2-variables Eample: Graph the solution to 26
27 3.2 Graphing Linear Equations in 2-variables Eample: Graph the solution to 27
28 3.3 Graphing with Intercepts (0,b) is the -intercept. It is the point ON the -ais. (a,0) is the -intercept. It is the point ON the -ais. 28
29 3.3 Graphing Using Intercepts Eercise: Find the following on the given graph. 1) -intercept 2) -intercept
30 3.3 Graphing Using Intercepts Eercise: Find the following on the given graph. 1) -intercept 2) -intercept
31 3.3 Graphing Using Intercepts Eercise: Find the following on the given graph. 1) -intercept 2) -intercept
32 3.3 Graphing Using Intercepts Eercise: Find the following on the given graph. 1) -intercept 2) -intercept
33 3.3 Graphing with Intercepts Eample: Find the - and - intercepts for 3-2 = 6 and then use them to draw the graph
34 3.3 Graphing with Intercepts Eample: Use the intercepts to draw the graph of
35 3.3 Graphing with Intercepts Propert: The -intercept of the line = a + b is (0,b). The line will cross the -ais at the point (0,b). Note: This is not Standard Form! Eercise: Find the -intercept of the following: a) c) b) d) 35
36 3.3 Graphing with Intercepts Propert: The -intercept of the line = a + b is (0,b). The line will cross the -ais at the point (0,b). Eercise: Find the -intercept of the following: e) g) f) h) 36
37 Just for fun. Eercise: Find the following on the given graph. 1) -intercepts 2) -intercept
38 3.4 The Slope of a Line Slope = m = Graphical interpretation: The slope of the line measures the rate of the line (how fast it is increasing or decreasing). 38
39 The slope will be the same anwhere on the line. It is the rate in which the coordinates change from point to point (rise) as the coordinate changes (run) over those same points. 39
40 Definition Slope If points ( 1, 1 ) and ( 2, 2 ) are an two different points on the line, then the slope of the line is calculated b the formula: 40
41 Eample: Find the slope of the line if ( 1, 1 ) = (1,2) and ( 2, 2 ) = (3,5)
42 Eample: Find the slope of the line if ( 1, 1 ) = (-2,1) and ( 2, 2 ) = (5,-4)
43 Eample: Find the slope of the line if ( 1, 1 ) = (3,-1) and ( 2, 2 ) = (3,4)
44 Eample: Find the slope of the line if ( 1, 1 ) = (4,2) and ( 2, 2 ) = (-3,2)
45 3.4 The Slope of a Line 45
46 Eample: Find the slope of the line
47 Eample: Graph the line with slope m = (0,-4). passing through the point
48 Eample: Graph the line with slope m = (1,1). passing through the point
49 Slope is a rate of change. A rate is a ratio that indicates how two quantities change with respect to each other. Change in values Change in values Eercise: On March 4, Nichole rented a mini-van with a full tank of gas and 10,324 mi on the odometer. On March 9, she returned the mini-van with 10,609 mi on the odometer. If the rental agenc charged Nichole $126 for the rental and needed 15 gal of gas to fill up the gas tank, find the following rates: 1) The cars' average rate of gas consumption, in miles per gallon. 2) The average cost of the rental, in dollars per da. 3) The cars' rate of travel, in miles per da. 49
50 50
51 Eercise: The chart below show the in a beach lot for ) How much would it cost to park for 4 hours, 6 hours? in dollars, for parking ) How long can ou park if ou had $3 to spend? $12? 3) What is the rate of increase in dollars for parking? 4) How much would it cost to park for 10 hours? 5) What is the -intercept and interpret this point? 51
52 Eercise: Wanda s Hair Salon has a graph displaing data from a recent da of work. 1) What is this graph describing? 2) Interpret the point (3,6) 3) What rate (ratio) can be determined from the graph? 4) What is slope of the line? 52
53 3.5 Slope-Intercept form Eercise: Graph the line that has slope m = and -intercept (0,1)
54 3.5 Slope-Intercept form Eercise: Graph the line that has equation b: a) Fill in the chart: b) Connect our two points c) Find the slope of the line d) What do ou notice about the equation and the features of the graph? 54
55 3.5 Slope-Intercept form Definition: Slope-intercept form of a line. The equation of the line with slope m and -intercept (0,b) is given b: Eercise: State the slope, m, and the -intercept for each of the equations below: a) c) b) d) 55
56 3.5 Slope-Intercept form How to graph with slope and intercept 1. Plot the intercept (0,b) 2. Use the slope = to collect more points. Eercise: Graph, using slope and intercept: a) c) b) d)
57 Eercise: 3.5 Slope-Intercept form 1) Find the slope and -intercept for 3-2 = 6. Then, use them to graph all solutions. 2) Rewrite = 12 in slope-intercept form. Then, use them to graph all solutions
58 3.5 Slope-Intercept form Finding the equation of a line. 1. Figure out the slope, m, from the information given. 2. Find a solution (,) and use it to solve for b, the -coordinate of the -intercept. 3. Write the equation as = m + b Eercise: Be a detective! Find m and b. 1) Find the equation of the line with slope -2 that passes through (-4,3). 2) Find the equation of the line that goes through (2,5) and is parallel to the line 3+4=12 3) Find the equation of the line that goes through the two points (0,4) and (2,3) 58
59 3.5 Slope-Intercept form Finding the equation of a line. 1. Figure out the slope, m, from the information given. 2. Find a solution (,) and use it to solve for b, the -coordinate of the -intercept. 3. Write the equation as = m + b Eercise: Be a detective! Find m and b. 4) Find the equation of the line perpendicular to 4-8 = 32 which goes through the point (2,5). 59
60 Mied Eercises 1. Graph 6-12 = 36 using - and -intercepts 2. Graph = 6-5 using slope-intercept. 3. Graph the line through the points (0,-3) and (5,2). a) What is the slope of that line? b) What is the -intercept of that line? c) Find the equation of that line in slope-intercept form. 4. Without graphing the line that goes through the points (-2, 4) and (6, -3). a) What is the slope of that line? b) What is the -intercept of that line? c) Find the equation of the line in slope-intercept form 60
61 3.6 Point-Slope form Finding the equation of a line. 1. Figure out the slope, m, from the information given. 2. Find a point ( 1, 1 ) on the line (or ordered pair solution to the equation) 3. Write the equation as - 1 = m( - 1 ) Eercise: Be a detective! Find m and ANY point ( 1, 1 ) 1) Find the equation of the line with slope 2 that goes through the point (5,2) then graph the line
62 3.6 Point-Slope form Eercise: Be a detective! Find m and ANY point ( 1, 1 ) 2) Find the equation of the line with slope -1/3 that goes through the point (-4,6) and graph. 3) Find the equation of the line parallel to = that goes through the point (-2, -4) and graph. 4) Find the equation of the line that goes through points (4, 7) and (-5, 0) and graph. 5) Find the equation of the line perpendicular to and has -intercept (0,4) and graph 6) Find the equation of the line that goes through (-4, 12) and (5,14). Write our answer in slope-intercept form and then in standard form. Graph the line. 7) Find the equation of the line with -intercept 4 and -intercept 7. Write our answer in slopeintercept form and then in standard form. 62
63 3.7 Solving Linear Inequalities Review: Solutions to Linear Equations. 1) Graph + 3 = -2 ( - 5) ) Graph 3) Graph 3-5 =
64 3.7 Solving Linear Inequalities Definition: A Linear Inequalit has the form: where a, b, and c are constants. The inequalit smbol < can be replaced with an of the following: Definition: The solution set for an inequalit in two variables is the set of all pairs (,) that make the inequalit true. Eercise: Determine if the given (,) is part of the solution set for the inequalit. 64
65 3.7 Solving Linear Inequalities Preview: 1) What is the equation of the line on the graph? ) Choose a point above or below the line. Test that point in the inequalit
66 3.7 Solving Linear Inequalities Definition: The solution set is the set of all points that make the inequalit true. Since there are infinitel man solutions, we need to graph the solution set. To graph the solution to an inequalit in two variables. 1) Graph the line (equation with an = sign) 2) Use a test point in the inequalit to check for a true statement. Shade in the true region. Eample. Graph the solution to 3-5 > 15 1) Graph the line 3-5 = 15 (dotted, wh?) 2) Select a test point in region 1 or 2 to test in the inequalit: 3-5 >15 then shade in the true region
67 3.7 Solving Linear Inequalities Definition: The solution set is the set of all points that make the inequalit true. Since there are infinitel man solutions, we need to graph the solution set. To graph the solution to an inequalit in two variables. 1) Graph the line (equation with an = sign) 2) Use a test point in the inequalit to check for a true statement. Shade in the true region. Eamples. 1) Graph the solution to 2) Graph the solution to
68 4.1 Solving Linear Sstems b Graphing Definition: is a set of two or more equations, in two or more variables, for which a common Definition: solution to a sstem are all values (,) 68
69 69
70 2. Solve the sstem algebraicall (gives EXACT solutions) = MathB70 70
71 Eercise: 1) Which line does the the following equation go with? 2) Which line does the the following equation go with? 3) From the graphs, what is the solution to the following sstem? 71
72 Eercise: Solve b graphing. 72
73 Eercise: Solve b graphing. 73
74 Eercise: 1) Find the slope-intercept form of the following lines. 2) Solve the sstem b graphing. 74
75 Eercise: 1) Find the slope-intercept form of the following lines. 2) Solve the sstem b graphing. 75
76 Eercise: 76
77 Question: When solving a linear sstem of two equations in two variables, what are the possible outcomes? 77
78 Application: T-shirt Villa sold 52 shirts, one kind at $8.25 and another at $11.50 each. In all, $ was taken in for the shirts. The above is modeled b the following sstem: 1) What does represent? 2) What does represent? 78
79 Application Continued... 3) Verif that (41,11) is a solution to the sstem ) Interpret the meaning of the solution (41, 11) in terms of the meaning to the application
80 80
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