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Math 40 Prealgebra Section 8.2 – Introduction to Graphing Linear Equations 8.2 Introduction to Graphing Linear Equations So far we have dealt with linear equations in one variable such as 3x 4 11 . It has only one solution, x 5 , since 3 5 4 11 . This is always the case for linear equations in one variable. In this section, we consider linear equations in two variables such as y x 2 . A solution to this type of equation is an ordered pair x, y that satisfies the equation. For example 4, 6 is a solution since replacing x with 4 and y with 6 gives us a true statement: y x2 ? 6 42 66 TRUE 1 1 , 5 are some solutions to 2 2 name a few. We can verify this by substituting the x-coordinate and y-coordinate of each ordered pair into y x 2 and obtaining a true statement. Can we find more ordered pairs that satisfy y x 2 ? Yes! y x2 7, 5 , 2, 0 , 3 y x2 y x2 ? ? 5 7 2 0 22 5 5 00 TRUE T RUE 1 ? 1 5 3 2 2 2 1 1 5 5 TRUE 2 2 It is important to note that one can find infinitely many ordered pairs to satisfy a linear equation in two variables. We will discuss this in more detail throughout this section. In the meanwhile, let’s practice our understanding of what it means to be a solution to a linear equation in two variables. Example 1: Determine if the following ordered pairs are solutions of the equation y 2 x 8 . a) Solution: 8, 11 b) 1, 6 c) 2, 14 d) 3, 13 a) To determine if 8, 11 is a solution of y 2 x 8 , replace x with 8 and y with 11 : y 2 x 8 ? 11 2 8 8 ? 11 16 8 ? 11 8 FALSE Since the resulting statement is false, 8, 11 is not a solution of y 2 x 8 . 1 2015 Campeau Math 40 Prealgebra Section 8.2 – Introduction to Graphing Linear Equations b) To determine if 1, 6 is a solution of y 2 x 8 , replace x with 1 and y with 6 : y 2 x 8 ? 6 2 1 8 ? 6 2 8 ? 6 6 TRUE Since the resulting statement is true, 1, 6 is a solution of y 2 x 8 . c) To determine if 2, 14 is a solution of y 2 x 8 , replace x with 2 and y with 14 : y 2 x 8 ? 14 2 2 8 ? 14 4 8 ? 14 12 FALSE Since the resulting statement is false, 2, 14 is not a solution of y 2 x 8 . d) To determine if 3, 13 is a solution of y 2 x 8 , replace x with 3 and y with 13 : y 2 x 8 ? 13 2 3 8 ? 13 6 8 ? 13 14 FALSE Since the resulting statement is false, 3, 13 is not a solution of y 2 x 8 . You Try It 1: Determine if the following ordered pairs are solutions of the equation y 3x 4 . a) 6, 20 b) 4, 16 c) 4, 8 d) 6, 14 2 2015 Campeau Math 40 Prealgebra Section 8.2 – Introduction to Graphing Linear Equations Example 2: Find the missing coordinate so that the ordered pair will satisfy the given linear equation. a) y 5 x 7; Solution: 6, ? b) y 4 x 9; ?, 1 1 c) 2 x 6 y 12; ?, 3 a) In 6, ? the x-coordinate is 6 and the y-coordinate is unknown. Replace x with 6 in y 5x 7 and solve for y: y 5 6 7 y 30 7 The ordered pair 6, 23 satisfies y 5x 7 . y 23 b) In ?, 1 the x-coordinate is unknown and the y-coordinate is 1 . Replace y with 1 in y 4x 9 and solve for x: y 4 x 9 1 4 x 9 10 4 x 10 x 4 5 x 2 1 2 x 2 1 The ordered pair 2 , 1 satisfies y 4x 9 . 2 1 1 c) In ?, the x-coordinate is unknown and the y-coordinate is . 3 3 Replace y with 1 in 2 x 6 y 12 and solve for x: 3 2 x 6 y 12 1 2 x 6 12 3 2 x 2 12 2 x 14 1 The ordered pair 7, satisfies 2 x 6 y 12 . 3 x7 3 2015 Campeau Math 40 Prealgebra Section 8.2 – Introduction to Graphing Linear Equations You Try It 2: Find the missing coordinate so that the ordered pair will satisfy the given linear equation. 1 a) y 8 x 5; , ? 4 b) y x 3; ?, 1 c) 3 x 4 y 12; 2, ? Now that we have had some practice with what it means to be a solution (and what it means to not be a solution) of a linear equation in two variables, we focus our attention on finding all solutions of a linear equation in two variables. Let’s consider the equation y 3x 4 . Since this equation is solved for y, y depends on the value of x that is substituted into the equation. We can quickly calculate ordered pairs that are solutions of y 3x 4 by choosing many values for x. To make it easier on ourselves, we choose integer values of x and calculate the corresponding y-value for each. x 4 : y 3 4 4 12 4 8 y 3x 4 x 3 : y 3 3 4 9 4 5 x y x 2 : y 3 2 4 6 4 2 4 8 x 1 : y 3 1 4 3 4 1 3 5 x 0 : y 3 0 4 0 4 4 2 2 x 1 : y 3 1 4 3 4 7 1 1 0 4 1 7 2 10 x 2 : y 3 2 4 6 4 10 (x, y) 4, 8 3, 5 2, 2 1, 1 0, 4 1, 7 2, 10 We organize the results in a table so we can easily refer to it while graphing the ordered pairs. 4 2015 Campeau Math 40 Prealgebra Section 8.2 – Introduction to Graphing Linear Equations We plot the ordered pairs below: 10 y 8 6 4 2 x 10 8 6 4 2 0 2 4 6 8 10 2 4 6 8 10 This is only a plot of seven solutions of y 3x 4 . We can find more solutions by choosing non-integer values of x and calculating the corresponding y-value for each. To make it easier on ourselves, we will choose xvalues in between the integer values we found earlier. x 4.5 : y 3 4.5 4 13.5 4 9.5 y 3x 4 x 3.5 : y 3 3.5 4 10.5 4 6.5 x y x 2.5 : y 3 2.5 4 7.5 4 3.5 4.5 9.5 x 1.5 : y 3 1.5 4 4.5 4 0.5 3.5 6.5 x 0.5 : y 3 0.5 4 1.5 4 2.5 2.5 3.5 x 0.5 : y 3 0.5 4 1.5 4 5.5 1.5 0.5 x 1.5 : y 3 1.5 4 4.5 4 8.5 0.5 2.5 0.5 5.5 1.5 8.5 (x, y) 4.5, 9.5 3.5, 6.5 2.5, 3.5 1.5, 0.5 0.5, 2.5 0.5, 5.5 1.5, 8.5 Again, we organize the results in a table so we can easily refer to it while graphing the ordered pairs. 5 2015 Campeau Math 40 Prealgebra Section 8.2 – Introduction to Graphing Linear Equations Plotting the six ordered pairs we just found, along with the seven ordered pairs we found earlier, gives us the following: 10 y 8 6 4 2 x 10 8 6 4 2 0 2 4 6 8 10 2 4 6 8 10 We can continue to choose more and more x-values and add more and more points to the graph. Eventually we will have so many points plotted that the graph of the solutions resembles a straight line. 10 y 8 6 4 2 x 10 8 6 4 2 0 2 4 6 8 10 2 4 6 8 10 By drawing a line through the ordered pairs we plotted, we are representing the infinitely many ordered pairs that satisfy the equation y 3x 4 . The line represents the graph of the linear equation y 3x 4 . Note: In order to get an accurate representation of the graph of a linear equation, it is very important that you use a straight edge to connect the points you plot. A ruler, student id card, or sheet of paper folded in half will serve us nicely for the problems in this section. 6 2015 Campeau Math 40 Prealgebra Section 8.2 – Introduction to Graphing Linear Equations You may be wondering if you need to choose as many x-values as we did in our graph of y 3x 4 . Luckily for us, it has been determined that the minimum number of ordered pairs needed to graph a linear equation is two ordered pairs. However, we recommend to find at least three ordered pairs. Finding at least three ordered pairs is can be helpful; if you happen to make a minor error in your calculations, you will catch it right away when you see that the ordered pairs don’t fall in a straight line. You also may be wondering if you are allowed to choose any x-values you want. Pay attention to the directions of the problem. Sometimes, you are given a table of specific x-values for which you must calculate the corresponding y-values and sometimes you are asked to construct your own table of values. Example 3: Graph the linear equation by completing the table of values. y 3x 1 x Solution: (x, y) y 2 Substitute x 2 into y 3x 1 and solve for y. y 3 2 1 6 1 7 0 Record the result in the table. Plot the ordered pair. 3 Substitute x 0 into y 3x 1 and solve for y. y 3 0 1 0 1 1 Record the result in the table. Plot the ordered pair. Substitute x 3 into y 3x 1 and solve for y. y 3 3 1 9 1 8 Record the result in the table. Plot the ordered pair. 10 y 3x 1 y 2 7 2, 7 0 1 0, 1 3 8 6 (x, y) x y 8 4 2 x 10 8 6 3, 8 4 2 0 2 4 6 8 10 2 4 6 Use a straight edge to connect the plotted points. 8 10 Make sure to extend the line through both edges of the axes! 7 2015 Campeau Math 40 Prealgebra Section 8.2 – Introduction to Graphing Linear Equations You Try It 3: Graph the linear equation by completing the table of values. 10 y 2x 4 x y (x, y) y 3 8 6 0 4 6 2 x 10 8 6 4 2 0 2 4 6 8 10 2 4 6 8 10 Example 4: Graph y x 7 by constructing a table of values. Solution: Here we have the freedom to choose any values of x we desire. We recommend a minimum of three x-values. The following is a possible solution. It is important to note that choosing different x-values will give us different ordered pairs, however, the final graph will look the same. Substitute x 5 into y x 7 and solve for y: y 5 7 5 7 2 Substitute x 0 into y x 7 and solve for y: y 0 7 0 7 7 Substitute x 3 into y x 7 and solve for y: y 3 7 3 7 10 10 y x 7 x y 8 (x, y) 5 2 5, 2 0 7 0, 7 3 10 3, 10 y 6 4 2 x 10 8 6 4 2 0 2 4 6 8 10 2 4 Use a straight edge to connect the plotted points. 6 8 Make sure to extend the line. 10 8 2015 Campeau Math 40 Prealgebra Section 8.2 – Introduction to Graphing Linear Equations You Try It 4: Graph y x 5 by constructing a table of values. 10 y 8 6 4 2 x 10 8 6 4 2 0 2 4 6 8 10 2 4 6 8 10 3 Example 5: Graph y x 2 by constructing a table of values. 4 Solution: Here we have the freedom to choose any values of x we desire. We recommend a minimum of three x-values. Notice that the coefficient of x is a fraction. If we want to ensure we only obtain integer values of y in our solution, we choose values of x that are multiples of the denominator. It is still ok if you decide to choose values of x that are not multiples of the denominator, however, you will obtain fractional or decimal values of y and you will have to be more careful when plotting the ordered pairs you obtain. The following is a possible solution. 3 3 Substitute x 4 into y x 2 and solve for y: y 4 2 3 2 5 4 4 3 3 Substitute x 4 into y x 2 and solve for y: y 4 2 3 2 1 4 4 3 Substitute x 8 into y x 2 and solve for y: 4 y 3 8 2 6 2 4 4 9 2015 Campeau Math 40 Prealgebra Section 8.2 – Introduction to Graphing Linear Equations x 4 4 8 y 5 1 4, 1 4 8 6 (x, y) 4, 5 y 10 3 y x2 4 4 2 x 10 8 6 4 2 0 2 4 6 8 10 2 8, 4 4 6 8 Use a straight edge to connect the plotted points. 10 Make sure to extend the line. You Try It 5: Graph y 3 x 4 by constructing a table of values. 2 10 y 8 6 4 2 x 10 8 6 4 2 0 2 4 6 8 10 2 4 6 8 10 10 2015 Campeau