Chapter 3 Section 1 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Reading Graphs; Linear Equations in Two Variables Interpret graphs. Write a solution as an ordered pair. Decide whether a given ordered pair is a solution of a given equation. Complete ordered pairs for a given equation. Complete a table of values. Plot ordered pairs

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1 Objective 1 Interpret graphs. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Interpret graphs. Recall that a bar graph is used to show comparisons. It consists of a series of bars (or simulations of bars) arranged either vertically or horizontally. In a bar graph, values from two categories are paired with each other. Slide A line graph is used to show changes or trends in data over time. To form a line graph, we connect a series of points representing data with line segments.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Compare sales of motor scooters in 1999 and Solution: Sales were about 25 thousand in 1999 and about 50 thousand in 2001, so sales doubled over those years. Interpreting a Bar Graph Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Estimate the number of households that purchased a real tree in About how much did the number of households purchasing real trees increase from 2002 to 2004? EXAMPLE 2 Solution: about 27 million Interpreting a Line Graph Slide Solution: about 5 million

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Interpret Graphs. (cont’d) A linear equation in two variables is an equation that can be written in the form where A, B, and C are real numbers and A and B are not both 0. Some examples of linear equations in two variables in this form, called standard form, are and Slide Other linear equations in two variables, such as and are not written in standard form, but could be. We discuss the forms of linear equations in more detail in Section 3.4. Linear equations in two variables

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2 Objective 2 Write a solution as an ordered pair. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley A solution of a linear equation in two variables requires two numbers, one for each variable. For example, a true statement results when we replace x with 2 and y with 13 in the equation since Write a solution as an ordered pair. Slide The pair of numbers x = 2 and y = 13 gives a solution of the equation The phrase “x = 2 and y = 13” is abbreviated x-value y-value The x-value is always given first. A pair of numbers such as (2,13) is called an ordered pair, since the order in which the numbers are written is important. The ordered pairs (2,13) and (13,2) are not the same. The second pair indicates that x = 13 and y = 2. For ordered pairs to be equal, their x-coordinates must be equal and their y-coordinates must be equal.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3 Objective 3 Slide Decide whether a given ordered pair is a solution of a given equation.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Decide whether a given ordered pair is a solution of a given equation. We substitute the x- and y-values of an ordered pair into a linear equation in two variables to see whether the ordered pair is a solution. An ordered pair that is a solution of an equation is said to satisify the equation. Slide When listing ordered pairs, be sure to always list the x -value first. Infinite numbers of ordered pairs can satisfy a linear equation in two variables.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Decide whether each ordered pair is a solution of the equation EXAMPLE 3 Solution: Deciding Whether Ordered Pairs Are Solutions of an Equation Slide No Yes

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 4 Objective 4 Slide Complete ordered pairs for a given equation.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Solution: Completing Ordered Pairs Slide Complete each ordered pair for the equation

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5 Objective 5 Complete a table of values. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Complete the table of values for the equation Then write the results as ordered pairs. Solution: Completing Tables of Value Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 6 Objective 6 Plot ordered pairs. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Plot ordered pairs. Slide Every linear in two variables equation has an infinite number of ordered pairs as solutions. Each choice of a number for one variable leads to a particular real number for the other variable. To graph these solutions, represented as ordered pairs (x,y), we need two number lines, one for each variable. The two number lines are drawn as shown below. The horizontal number line is called the x-axis and the vertical line is called the y-axis. Together, these axes form a rectangular coordinate system, also called the Cartesian coordinate system.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The coordinate system is divided into four regions, called quadrants. These quadrants are numbered counterclockwise, starting with the one in the top right quadrant. Slide Plot ordered pairs. (cont’d) Points on the axes themselves are not in any quadrant. The point at which the x-axis and y-axis meet is called the origin, labeled 0 on the previous diagram. This is the point corresponding to (0, 0). The x-axis and y-axis determine a plane— a flat surface illustrated by a sheet of paper. By referring to the two axes, we can associate every point in the plane with an ordered pair. The numbers in the ordered pair are called the coordinates of the point. In a plane, both numbers in the ordered pair are needed to locate a point. The ordered pair is a name for the point.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley For example, locate the point associated with the ordered pair (2,3) by starting at the origin. Slide Plot ordered pairs. (cont’d) Since the x-coordinate is 2, go 2 units to the right along the x-axis. Since the y-coordinate is 3, turn and go up 3 units on a line parallel to the y-axis. The point (2,3) is plotted in the figure to the right. From now on the point with x-coordinate 2 and y-coordinate 3will be referred to as point (2,3).

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Plot the given points in a coordinate system: EXAMPLE 6 Plotting Ordered Pairs Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Completing Ordered Pairs to Estimate the Number of Twin Births Slide Complete the table of ordered pairs for the equation,, where x = year and y = number of twin births in thousands. Round answers to the nearest whole number. Interpret the results for Solution: There were about 125 thousand twin births in the U.S. in 2002.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The ordered pairs of twin births in the U.S. for 1998, 2000, and 2003 are graphed to the right. This graph of ordered pairs of data is called a scatter diagram. Notice how the how the axes are labeled: x represents the year, and y represents the number of twin births in thousands. Plot ordered pairs. (cont’d) Slide A scatter diagram enables us to tell whether two quantities are related to each other. These plotted points could be connected to form a straight line, so the variables x (years) and y (number of births have a linear relationship.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Think of ordered pairs as representing an input value x and an output value y. If we input x into the equation, the output is y. We encounter many examples of this type of relationship every day. The cost to fill a tank with gasoline depends on how many gallons are needed; the number of gallons is the input, and the cost is the output The distance traveled depends on the traveling time; input a time and the output is a distance. The growth of a plant depends on the amount of sun it gets; the input is the amount of sun, and the output is growth. Plot ordered pairs. (cont’d) Slide