Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Graphs of Linear Equations 11 11.1 Graphs and Applications of Linear Equations 11.2 More with Graphing and Intercepts 11.3 Slope and Applications 11.4 Equations of Lines 11.5 Graphing Using the Slope and the y-Intercept 11.6 Parallel and Perpendicular Lines 11.7 Graphing Inequalities in Two Variables Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 2

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.1 Graphs and Applications of Linear Equations a Plot points associated with ordered pairs of numbers; determine the quadrant in which a point lies. b Find the coordinates of a point on a graph. c Determine whether an ordered pair is a solution of an equation with two variables. d Graph linear equations of the type y = mx + b and Ax + By = C, identifying the y-intercept. e Solve applied problems involving graphs of linear equations. d Use < or > for  to write a true statement in a situation like 6  10. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 3

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.1 Graphs and Applications of Linear Equations a Plot points associated with ordered pairs of numbers; determine the quadrant in which a point lies. On the number line, each point is the graph of a number. On a plane, each point is the graph of a number pair. To form the plane, we use two perpendicular number lines called axes. They cross at a point called the origin. The arrows show the positive directions. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 4

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.1 Graphs and Applications of Linear Equations a Plot points associated with ordered pairs of numbers; determine the quadrant in which a point lies. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 5

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.1 Graphs and Applications of Linear Equations a Plot points associated with ordered pairs of numbers; determine the quadrant in which a point lies. Consider the ordered pair (3, 4). The numbers in an ordered pair are called coordinates. In (3, 4), the first coordinate (the abscissa) is 3 and the second coordinate (the ordinate) is 4. To plot (3, 4) ,we start at the origin and move horizontally to the 3. Then we move up vertically 4 units and make a “dot.” Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 6

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.1 Graphs and Applications of Linear Equations a Plot points associated with ordered pairs of numbers; determine the quadrant in which a point lies. 1 Plot the point (–5, 2). The first number, –5 is negative. Starting at the origin, we move –5 units in the horizontal direction (5 units to the left). The second number, 2, is positive. We move 2 units in the vertical direction (up). Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 7

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.1 Graphs and Applications of Linear Equations a Plot points associated with ordered pairs of numbers; determine the quadrant in which a point lies. 1 Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 8

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.1 Graphs and Applications of Linear Equations a Plot points associated with ordered pairs of numbers; determine the quadrant in which a point lies. In region I (the first quadrant), both coordinates of any point are positive. In region II (the second quadrant), the first coordinate is negative and the second positive. In region III (the third quadrant), both coordinates are negative. In region IV (the fourth quadrant), the first coordinate is positive and the second is negative. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 9

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.1 Graphs and Applications of Linear Equations a Plot points associated with ordered pairs of numbers; determine the quadrant in which a point lies. 2 In which quadrant, if any, are the points (–4, 5), (5, –5), (2, 4), (–2, –5), and (–5, 0) located? Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 10

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.1 Graphs and Applications of Linear Equations a Plot points associated with ordered pairs of numbers; determine the quadrant in which a point lies. 2 Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 11

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.1 Graphs and Applications of Linear Equations a Plot points associated with ordered pairs of numbers; determine the quadrant in which a point lies. 2 The point (–4, 5) is in the second quadrant. The point (5, –5) is in the fourth quadrant. The point (2, 4) is in the first quadrant. The point (–2, –5) is in the third quadrant. The point (–5, 0) is on an axis and is not in any quadrant. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 12

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.1 Graphs and Applications of Linear Equations b Find the coordinates of a point on a graph. 3 Find the coordinates of points A, B, C, D, E, F, and G. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 13

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.1 Graphs and Applications of Linear Equations b Find the coordinates of a point on a graph. 3 Point A is 3 units to the left (horizontal direction) and 5 units up (vertical direction). Its coordinates are (–3, 5). Point D is 2 units to the right and 4 units down. Its coordinates are (2, –4). The coordinates of the other points are as follows: Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 14

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.1 Graphs and Applications of Linear Equations c Determine whether an ordered pair is a solution of an equation with two variables. When an equation contains two variables, the solutions of the equation are ordered pairs in which each number in the pair corresponds to a letter in the equation. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 15

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.1 Graphs and Applications of Linear Equations c Determine whether an ordered pair is a solution of an equation with two variables. 4 Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 16

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.1 Graphs and Applications of Linear Equations c Determine whether an ordered pair is a solution of an equation with two variables. 4 Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 17

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.1 Graphs and Applications of Linear Equations GRAPH OF AN EQUATION c The graph of an equation is a drawing that represents all of its solutions. Determine whether an ordered pair is a solution of an equation with two variables. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 18

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.1 Graphs and Applications of Linear Equations d Graph linear equations of the type y = mx + b and Ax + By = C, identifying the y-intercept. In general, any equation equivalent to one of the form y = mx + b or Ax + By = C where m, b, A, B, and C are constants (not variables) and A and B are not both 0, is linear. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 19

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.1 Graphs and Applications of Linear Equations To graph a linear equation: 1. Select a value for one variable and calculate the corresponding value of the other variable. Form an ordered pair using alphabetical order as indicated by the variables. 2. Repeat step (1) to obtain at least two other ordered pairs. Two points are essential to determine a straight line. A third point serves as a check. 3. Plot the ordered pairs and draw a straight line passing through the points. d Graph linear equations of the type y = mx + b and Ax + By = C, identifying the y-intercept. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 20

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.1 Graphs and Applications of Linear Equations d Graph linear equations of the type y = mx + b and Ax + By = C, identifying the y-intercept. 7 We select a value for x, compute y, and form an ordered pair. Then we repeat the process for other choices of x. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 21

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.1 Graphs and Applications of Linear Equations d Graph linear equations of the type y = mx + b and Ax + By = C, identifying the y-intercept. 7 Results are listed in the table. The points corresponding to each pair are then plotted. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 22

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.1 Graphs and Applications of Linear Equations y-INTERCEPT d Graph linear equations of the type y = mx + b and Ax + By = C, identifying the y-intercept. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 23

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.1 Graphs and Applications of Linear Equations d Graph linear equations of the type y = mx + b and Ax + By = C, identifying the y-intercept. 10 Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 24

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.1 Graphs and Applications of Linear Equations d Graph linear equations of the type y = mx + b and Ax + By = C, identifying the y-intercept. 10 Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 25

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.1 Graphs and Applications of Linear Equations d Graph linear equations of the type y = mx + b and Ax + By = C, identifying the y-intercept. 10 We find two other pairs using multiples of 4 for x to avoid fractions. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 26

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.1 Graphs and Applications of Linear Equations e Solve applied problems involving graphs of linear equations. 11 World Population The world population, in billions, is estimated and projected by where x is the number of years since 1980. That is, x = corresponds to 1980, y = corresponds to 1992, and so on. Source: U.S. Census Bureau Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 27

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.1 Graphs and Applications of Linear Equations e Solve applied problems involving graphs of linear equations. 11 World Population a) Determine the world population in 1980, in 2005, and in 2030. b) Graph the equation and then use the graph to estimate the world population in 2015. c) In what year would we estimate the world population to be 7.761 billion? Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 28

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.1 Graphs and Applications of Linear Equations e Solve applied problems involving graphs of linear equations. 11 World Population a) The years 1980, 2005, and 2030 correspond to x = 0, x = 25, and x = 50, respectively. We substitute 0, 25, and 50 for x and then calculate y: The world population in 1980, in 2005, and in 2030 is estimated to be 4.593 billion, 6.393 billion, and 8.193 billion, respectively. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 29

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.1 Graphs and Applications of Linear Equations e Solve applied problems involving graphs of linear equations. 11 World Population b) We have three ordered pairs from part (a). We plot these points and see that they line up. Thus our calculations are probably correct. Since we are considering only the year 2015 and the number of years since 1980 and since the population, in billions, for those years will be positive (y > 0), we need only the first quadrant for the graph. We use the three points we have plotted to draw a straight line. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 30

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.1 Graphs and Applications of Linear Equations e Solve applied problems involving graphs of linear equations. 11 World Population Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 31

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.1 Graphs and Applications of Linear Equations e Solve applied problems involving graphs of linear equations. 11 World Population To use the graph to estimate world population in 2015, we first note that this year corresponds to x = 35. We need to determine which y-value is paired with x = 35. We locate the point on the graph by moving up vertically from x = 35 and then find the value on the y-axis that corresponds to that point. It appears that the world population in 2015 will be about 7.1 billion. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 32

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.1 Graphs and Applications of Linear Equations e Solve applied problems involving graphs of linear equations. 11 World Population Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 33

Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 11.1 Graphs and Applications of Linear Equations e Solve applied problems involving graphs of linear equations. 11 World Population To find a more accurate value, we can simply substitute into the equation: The world population in 2015 is projected to be 7.113 billion. Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Slide 34