Linear Equations The Mathematics Readiness Project was funded jointly by the Eisenhower State Grant Program in Mathematics and Science and the California Academic Partnership Program. Overview for the Teacher Mathematics Readiness Projec t Linear Equations by Katrine Czajkowski Overview for the Teacher Consider the following question: Which of the following is a portion of the graph of y = –2x + 4? (a) (b) (d) (e) (c) The incorrect answer most commonly chosen is (e). Among the reasons students might miss this question are the following: 1. 2. 3. 4. 5. 6. 7. 8. Students might not realize that y = –2x + 4 describes the relationship between two variables. Students might not recognize that the equation is linear. Students might not recognize a line graphed on the Cartesian plane as the graphical representation of a linear equation. Students might reverse the slope and y-intercept terms. Students might not understand how to identify the y-intercept of a line. Students might not understand that slope is simply a way to express the ratio of the vertical change vs. the horizontal change. Students might not be able to connect the direction of the line drawn to the sign of its slope. Students might not distinguish between a slope of 2 and a slope of –2. Mathematics Readiness Project 1997 The misunderstandings described above can become particularly acute when students face a more complex question such as x+y=2 If , then y = x–y=6 (a) –8 (b) –4 (c) –2 (d) 2 (e) 8 The incorrect answer most commonly chosen is (b). This unit is intended to provide teachers with ways to supplement their curriculum in order to better prepare students to succeed in their study of algebra. Mathematics Readiness Project 1997 Page 2 Table of Contents Section 1: Characteristics of Linear Equations Section 2: Tables of Values for Graphing Section 3: Using Intercepts for Graphing Section 4: Using Slope-Intercept Form with Graphs Section 5: Families of Linear Equations Section 6: Graphing Linear vs. Non-Linear Equations Section 7: Creating, Graphing and Using Linear Equations Section 8: Simple System of Equations Section 9: What Went Wrong? Section 10: Exploring with a Graphing Calculator Mathematics Readiness Project 1997 Page 3 Section 1: Characteristics of Linear Equations Teacher Page For the purpose of this lesson, we will not discuss constant linear equations (such as x = 7 or y = –10), as they offer exceptions to the rules governing most linear equations. Explain to the students that linear equations have several basic characteristics: 1) 2) 3) The expression contains an equals (=) sign. The expression contains two variables (usually denoted by "x" and "y"). The expression can be manipulated using addition or subtraction so that it appears like either y = mx + b or cx + dy = r, where b, c, d, m and r are constant. Work through the following examples with the students: Consider the equation y = 2x + 3. Is this equation linear? Analysis 1) Does the expression contain an equals (=) sign? Yes. 2) Does the expression contain two variables? Yes. 3) Can the expression be manipulated using addition or subtraction so that it appears like either y = mx + b or cx + dy = r, where b, c, d, m and r are constant? Yes, it already looks like y = mx + b. Is the equation linear? Yes. Make a table of values where x always changes by the same amount. One possible response is x y –2 –1 –1 1 0 3 1 5 2 7 3 9 Notice that when the values for x change by the same amount, the values for y also always change by the same amount. Mathematics Readiness Project 1997 Page 4 Section 1: Characteristics of Linear Equations Teacher Page Consider the equation xy = 8. Is this equation linear? Analysis 1) Does the expression contain an equals (=) sign? Yes. 2) Does the expression contain two variables? Yes. 3) Can the expression be manipulated using addition or subtraction so that it appears like either y = mx + b or cx + dy = r, where b, c, d, m and r are constant? No. Is the equation linear? No. Make a table of values where x always changes by the same amount. One possible response is x y 1 6 2 3 3 2 4 1.5 Notice that the values for y do not change by the same amount. The problems in the student pages of Characteristics of Linear Equations reinforce these ideas. Students may work individually, in pairs or in groups of no more than four. Be sure to review their work with them. Answers to student page questions: I. The equation is linear. II. The equation is linear. III. The equations y = 3x + 3, 2y + 5x = 10, 4 = x – 2y and 2x – 3y = 12 are linear. Mathematics Readiness Project 1997 Page 5 Section 1: Characteristics of Linear Equations Student Page I. Given the equation y = 4 – x, complete the table below. • Under "yes/no," check whether or not the equation meets each characteristic in the first column. • In the final column, provide evidence to support your choice of either "yes" or "no." Be sure to complete the table of values box. Choose values for x that always change by the same amount. In the box at the bottom of the table, answer the question and provide a onesentence reason for your answer. Characteristics of linear equations Evidence to support your decision 1. Does the expression contain an equals (=) sign? 2. Does the expression contain two variables? 3. Can the expression be manipulated using addition or subtraction so that it appears like either y = mx + b or cx + dy = r, where b, c, d, m and r are constant? x y Is this equation linear? Once you have completed this table and reviewed it with your teacher, proceed to the next page and complete the chart based on the expression given. Mathematics Readiness Project 1997 Page 6 Section 1: Characteristics of Linear Equations Student Page II. Given the equation x – y = 8, complete the table. • Under "yes/no," check whether or not the equation meets each characteristic in the first column. • In the final column, provide evidence to support your choice of either "yes" or "no." Be sure to complete the table of values box. Choose values for x that always change by the same amount. In the box at the bottom of the table, answer the question and provide a onesentence reason for your answer. Characteristics of linear equations Evidence to support your decision 1. Does the expression contain an equals (=) sign? 2. Does the expression contain two variables? 3. Can the expression be manipulated using addition or subtraction so that it appears like either y = mx + b or cx + dy = r, where b, c, d, m and r are constant? x y Is this equation linear? Once you have completed this table and reviewed it with your teacher, proceed to the next page and complete the chart based on the expression given. Mathematics Readiness Project 1997 Page 7 Section 1: Characteristics of Linear Equations Student Page III. Now complete the table below. Check either "linear" or "not linear" for each equation. In the final column, provide evidence to support your decision. Expression Linear? Not linear? Evidence to support your decision y = 3x + 3 2y + 5x = 10 3xy = 12 4 = x – 2y x2 + 2 = y 2x – 3y = 12 Mathematics Readiness Project 1997 Page 8 Section 2: Tables of Values for Graphing Teacher Page Provide the students with a brief review of plotting points in the Cartesian plane. They then should be able to work through the student pages of Tables of Values for Graphing. Graph paper should be available for the students. Answers to student page questions: I. II. III. Mathematics Readiness Project 1997 Page 9 Section 2: Tables of Values for Graphing I. Student Page Consider the equation y = –2x + 2. Step 1: Make an (x,y) table using at least three of your favorite values for x. Pick at least one positive value for x, at least one negative value for x, and use zero as a value for x. x y Step 2: Plot the points on the Cartesian plane. Step 3: In the plot above, draw a line running through all the points. Mathematics Readiness Project 1997 Page 10 Section 2: Tables of Values for Graphing Student Page For each equation below, construct the (x,y) table of values and graph the equation on the Cartesian plane. Work with a partner and follow the steps for using a table of values to graph a linear equation. Show your work in the space provided. II. y = 3x – 2 x y III. 2x + y = 8 x y Mathematics Readiness Project 1997 Page 11 Section 3: Using Intercepts for Graphing Teacher Page Work through the graphing of 3x + 4y = 12 with the students. Do this by first identifying the x-intercept—(4,0)—and the yintercept—(0,3)—and then drawing the line that runs through these two points. Then work through the somewhat more complex 1 problem of graphing y = 2 x – 4 by using the x-intercepts and the y-intercepts. Have the students work through the Using x-Intercepts and y-Intercepts for Graphing student pages. Mathematics Readiness Project 1997 Page 12 Section 3: Using Intercepts for Graphing Teacher Page Answers to student page questions: I. II. III. IV. Mathematics Readiness Project 1997 Page 13 Section 3: Using Intercepts for Graphing Student Page The x-intercept is the point where the line crosses the x-axis. The y-intercept is the point where the line crosses the y-axis. The two intercepts can be used to quickly graph a linear equation. I. Follow the procedure below to graph the equation y = 2x + 4. Step 1: Find the y-intercept: In the equation substitute the value 0 for x and solve for y. Step 2: Fill in the missing number: (0, ) is the y-intercept of y = 2x + 4. Step 3: Find the x-intercept: In y = 2x + 4 substitute the value 0 for y and solve for x. Step 4: Fill in the missing number: ( ,0) is the x-intercept of y = 2x + 4. Step 5: Plot the x-intercept and the y-intercept on the Cartesian plane. Step 6: On the graph above, draw a line running through the two intercepts. Mathematics Readiness Project 1997 Page 14 Section 3: Using Intercepts for Graphing Student Page For each of the equations below, a) b) c) d) e) Identify the x-intercept of the graph by substituting zero for y. Plot the x-intercept on the Cartesian plane. Identify the y-intercept of the graph by substituting zero for x. Plot the y-intercept on the Cartesian plane. Draw a line running through the two intercepts. II. y = 3x – 1 x-intercept: ( , ) y-intercept: ( , ) Coordinates of third point: ( , ) Does this point lie on the line you drew after finding the xand y-intercepts? Why is the important? Mathematics Readiness Project 1997 last question Page 15 Section 3: Using Intercepts for Graphing Student Page III. x – 2y = 8 x-intercept: ( , ) y-intercept: ( , ) Coordinates of third point: ( , ) Does this point lie on the line you drew after finding the xand y-intercepts? Why is the important? Mathematics Readiness Project 1997 last question Page 16 Section 3: Using Intercepts for Graphing Student Page IV. x = 4 + y x-intercept: ( , ) y-intercept: ( , ) Coordinates of third point: ( , ) Does this point lie on the line you drew after finding the xand y-intercepts? Why is the important? Mathematics Readiness Project 1997 last question Page 17 Section 4: Using Slope-Intercept Form with Graphs Teacher Page In this section you should present to the students the ideas of slope and yintercept. Below we outline for you a possible presentation of these concepts. Demonstrate for the students that in a table of values for the equation y = 2x + 5, the y values increase by 2 as the x values increase by 1. Show also that for any two points on this line, the ratio of the vertical change divided by the horizontal change is 2. Define this number to be the slope of the line. Use the equation y = –3x + 1 to graphically illustrate the idea of negative slope (–3 in this case). Point out to the students that the line represented by the equation y = –3x + 1 crosses the y-axis at the point (0,1). This is called the y-intercept. Mathematics Readiness Project 1997 Page 18 Section 4: Using Slope-Intercept Form with Graphs Teacher Page Use the graph of the equation y = 2x + 5 to reinforce the idea of y-intercept. In this case, the y-intercept is at (0,5) Summarize for the students the idea that the line represented by an equation of the form y = mx + b has slope "m" and y-intercept (0,b). We call y = mx + b the slope-intercept form of the equation of the line. Have the students work through the Using Slope-Intercept Form with Graphs student pages. Graph paper should be available for the students. Mathematics Readiness Project 1997 Page 19 Section 4: Using Slope-Intercept Form with Graphs Teacher Page Answers to student page questions: I. y = –x + 5 II. y = 2x – 4 1 III. y = 2 x + 2 IV. y = –2x – 6 2 V. y = 3 x + 2 VI. y = 3x – 5 Mathematics Readiness Project 1997 Page 20 Section 4: Using Slope-Intercept Form with Graphs Student Page In problems I through III, rearrange the terms in the given equation so that each equation appears in slope-intercept form, y = mx + b. Then graph the equation by using the y-intercept and the slope. Remember that • The "y" must be alone and positive. • The "b" is a constant that is not attached to either variable. The "b" represents the y-intercept of the line. • The "m" is the coefficient of x and represents the slope of the line. The slope is the ratio of the vertical change to the horizontal in the line from one point to another. 1. x+y=5 Mathematics Readiness Project 1997 Slope-intercept form: Page 21 Section 4: Using Slope-Intercept Form with Graphs II. 4x – 2y = 8 Slope-intercept form: III. 3x = 6y – 12 Slope-intercept form: Mathematics Readiness Project 1997 Student Page Page 22 Section 4: Using Slope-Intercept Form with Graphs Student Page In problems IV through VI, find a linear equation for the line shown. Begin by finding the y-intercept ("b"). Then use the two given points to identify the slope ("m"). Finally, write the equation in slope-intercept form. IV. b= m= Equation: V. b= m= Equation: VI. b= m= Equation: Mathematics Readiness Project 1997 Page 23 Section 5: Families of Linear Equations Teacher Page Common errors in graphing linear equations often result from 1. 2. 3. Confusing the slope (m) with the y-intercept (b). Inadvertently reversing the sign of the slope. Plotting the y-intercept on the wrong axis. If students are alerted to these common errors, they will be better prepared to avoid them. Studying families of related equations may raise students' awareness of these potential errors. Show the students how the graphs of y = 2x + 4 and y = –2x + 4 differ. This will help them recognize the significance of the "m" term. Show the students how the graphs of y = 2x + 4 and y = 2x – 4 differ. This will draw their attention to the significance of the "b" term. Mathematics Readiness Project 1997 Page 24 Section 5: Families of Linear Equations Teacher Page Show the students how the graphs of y = 2x + 4 and y = 4x + 2 differ. This will alert them to the problems associated with confusing the slope of a line with its y-intercept. The comparison charts in the Families of Linear Equations student pages will help reinforce these important concepts. Students should be encouraged to create their own comparison charts based on original equations. Mathematics Readiness Project 1997 Page 25 Section 5: Families of Linear Equations Teacher Page Answers to student page questions: I. II. III. Mathematics Readiness Project 1997 Page 26 Section 5: Families of Linear Equations Student Page For each of the pairs of equations below, complete the chart. Indicate the slope and the y-intercept of each equation and graph each line. Answer the question beneath the graphs using at least one complete sentence. I. Equation: y = 3x + 5 y = –3x + 5 Slope: y-intercept: Graph: What is the reason for the difference in the graphs of the two equations? Mathematics Readiness Project 1997 Page 27 Section 5: Families of Linear Equations Student Page II. Equation: y = 2x + 3 y = 2x – 3 Slope: y-intercept: Graph: What is the reason for the difference in the graphs of the two equations? Mathematics Readiness Project 1997 Page 28 Section 5: Families of Linear Equations Student Page III. Equation: y = –3x + 2 y = 2x – 3 Slope: y-intercept: Graph: What is the reason for the difference in the graphs of the two equations? Mathematics Readiness Project 1997 Page 29 Section 6: Graphing Linear vs. Non-Linear Equations Teacher Page Use the equation x2 – y = 4 to show the students that not all graphs are lines. Proceed as follows: First, generate a table of values for x2 – y = 4: x y –3 5 –2 0 –1 –3 0 –4 1 –3 2 0 3 5 Second, point out that although the x values are increasing by the same amount, namely 1, the y-values are changing by different amounts. For example, as x increases from 0 to 1 y increases by 1, while when x increases from 1 to 2 y increases by 3. Third, plot the points so that the students will see that the graph is not a line: Ask students if they know of other characteristics that would indicate that the graph would not turn out to be a line. Having the students work through the student pages of Graphing Linear vs. Non-Linear Equations will reinforce the fact that not all graphs are lines. Mathematics Readiness Project 1997 Page 30 Section 6: Graphing Linear vs. Non-Linear Equations Teacher Page Answers to student page questions: I. The equation –3x + y = 6 is linear. II. The equation 2xy = 8 is not linear. III. The equation 4y = 2x is linear. Mathematics Readiness Project 1997 Page 31 Section 6: Graphing Linear vs. Non-Linear Equations Student Page For each equation, a) Solve for y in the first box. b) Create a table of values based on the equation. c) Graph the equation. d) Indicate whether or not the equation is linear by writing "yes" or "no" in the first box. Equation Table of Values –3x + y = 6 x y 2xy = 8 x y 4y = 2x x y Graph Linear? Linear? Linear? Mathematics Readiness Project 1997 Page 32 Section 7: Creating, Graphing and Using Linear Equations Teacher Page It is important for students to recognize that equations are just ways of describing the special relationship that exists between two variables. It is therefore helpful to provide students with real-world applications of these ideas. Work through the following application with the students: Tickets for the school play go on sale Monday. Prior to this a total of 27 tickets have been purchased by members of the cast for their families. As the day progresses a total of 6 tickets are sold each hour. Thus, if "h" represents the number of hours that have gone by and if "t" represents the number of tickets that have been sold, we can create a table of values: h t 0 27 1 33 2 39 3 45 4 51 We also can write an equation that represents this situation: t = 6h + 27 Finally, we graph this equation. Draw the graph only for h between 0 and 6. Point out that the graph is a straight line. Show how to use the graph to estimate how many tickets will be sold after 10 hours. [The answer 87] The problems in the student pages of Creating, Graphing and Using Linear Equations provide the students with other applications of linear equations. Mathematics Readiness Project 1997 Page 33 Section 7: Creating, Graphing and Using Linear Equations Teacher Page Answers to student page questions: I. The equation is B = 15t. Monique will have 15 tests graded by the time She has $225 in her bank account. II. The equation is I = 5s + 200. Mr. Gonzalez's income will be $250 if he sells 10 shares of stock. Mathematics Readiness Project 1997 Page 34 Section 7: Creating, Graphing and Using Linear Equations Student Page For each situation below, a) Write an equation that describes the relationship between the two variables mentioned. State whether or not the equation is linear. b) Draw the graph of each equation in the space provided. c) Write two statements that are true based on your graph. d Use the graph to estimate the amount requested in the final statement. I. Monique earns $15 for every biochemistry test she grades as a teacher's assistant. She starts the semester with nothing in the bank and saves all of the money he earns grading tests. Use "B" to represent the amount of money in her bank account and "t" to represent the number of tests she grades. Equation: Statements based on the graph: Using your graph, estimate how many tests Monique will have graded by the time she has $225 in her bank account. Mathematics Readiness Project 1997 Page 35 Section 7: Creating, Graphing and Using Linear Equations Student Page II. Each week, Mr. Gonzalez earns $200 per week plus $5 for every share of stock he sells. His income is represented by "I" and the amount of stock he sells is represented by "s." Equation: Statements based on the graph: Using your graph, estimate Mr. Gonzalez's income if he sells 10 shares of stock. Mathematics Readiness Project 1997 Page 36 Section 8: Simple System of Equations Teacher Page Work through the following application with the students: Thomas has been offered a job where he can earn $5 for every hour he spends painting houses. If he starts with nothing in his bank account and doesn't spend any money all week, how much will be in his bank account? The answer obviously depends upon how many hours Thomas works. If "h" represents the number of hours he works and "a" represents the amount of money in his bank account, we know that a = 5h Suppose now that Thomas has a alternate job offer in which he would be given an initial payment of $12 and then would earn $3 an hour? Which job would give Thomas the most money after 10 hours? After 10 hours, the first job would give him 5× 10 = $50 and the second job would give him only 3 × 10 + 12 = $42. After how many hours of work will both jobs pay him the same amount? Complete the following table for the first equation, a = 5h: h 0 1 2 3 4 5 6 7 8 9 a 0 5 10 15 20 25 30 35 40 45 Complete the following table for the second equation, a = 3h + 12: h 0 1 2 3 4 5 6 7 8 9 a 12 15 18 21 24 27 30 33 36 39 Point out that the pair (6,30) appears in both tables and that 6 hours of work would pay him the same amount from both jobs. Draw the graphs of both equations on the same axes and point out that (6,30) is the point of intersection. Discuss which job pays more after 10 hours. Students should now complete the student pages of Simple Systems of Equations. Mathematics Readiness Project 1997 Page 37 Section 8: Simple System of Equations Teacher Page Answers to student page questions: I. (5,3) II. (1,5) III. (2,4) IV. (–1,6) Mathematics Readiness Project 1997 Page 38 Section 8: Simple System of Equations I. Student Page Consider the system of equations x – y = 2 x + y = 8 Complete the following table for the first equation, x – y = 2: Complete the following table for the second equation, x + y = 8: x x 1 2 3 4 5 6 7 y 1 2 3 4 5 6 7 y What is the common pair (x,y) on both tables? Verify that this pair (x,y) satisfies the first equation, x – y = 2. Verify that this pair (x,y) satisfies the second equation, x + y = 8. To the right, draw the graphs of both equations, x – y = 2 and x + y = 8. At what point (x,y) do the two lines intersect? Explain in a full sentence how this point (x,y) is related to the (x,y) found above. Mathematics Readiness Project 1997 Page 39 Section 8: Simple System of Equations Student Page In problems II, III and IV, make a table of values for each of the two given equations to answer the question. x+y=6 II. If , then y = y = 3x + 2 y=x+2 III. If , then x = y = 3x – 2 2x + y = 4 IV. If , then y = y = 3x + 9 Mathematics Readiness Project 1997 Page 40 Section 9: What Went Wrong? Teacher Page Students can avoid many future mistakes if they have the opportunity to critique the work of others, especially if that work contains frequently encountered errors. In this section the students are asked to describe the mistake and write the correct answer. Answers to student page questions: (c) is the correct answer. Mathematics Readiness Project 1997 Page 41 Section 9: What Went Wrong? I. Teacher Page Amy tried to answer the following question: Which of the following is a portion of the graph of y = –2x + 4? (a) (b) (d) (e) (c) She wrote (e) for her answer. Explain what Amy did wrong. Use complete sentences. What is the correct answer? II. Max wrote (a) for his answer to the same question. Explain what Max did wrong. Use complete sentences. Mathematics Readiness Project 1997 Page 42 Section 10: Exploring with a Graphing Calculator Teacher Page If you have graphing calculators available, then students can use them to to find the intersection of two graphs. Lead the class through the following graphing calculator activities: 1. Graph y = 2x – 3 on the graphing calculator. By tracing verify that the yintercept is at (0,–3) and that the xintercept is at (1.5,0). 2. If the graphing calculator has a "table" option, use the table to view a listing of (x,y) values for y = 2x – 3. Use these values to verify that the slope of the line is 2. 3. Graph y = –3x + 2 now on the graphing calculator, leaving the first graph, y = 2x – 3, "on." Mathematics Readiness Project 1997 x –1 0 1 2 3 4 y 5 –3 –1 1 3 5 Page 43 Section 10: Exploring with a Graphing Calculator Teacher Page 4. By tracing and zooming, verify that the point of intersection is (1,–1). 5. If the graphing calculator has a "table" option, use the table to verify that the coordinates (1,–1) appear on both tables. x –1 0 1 2 3 4 y = 2x – 3 5 –3 –1 1 3 5 x –1 0 1 2 3 4 y = –3x + 2 5 2 –1 –4 –7 –10 Students should now complete the student pages of Exploring with a Graphing Calculator. Mathematics Readiness Project 1997 Page 44 Section 10: Exploring with a Graphing Calculator Teacher Page Answers to student page questions: III. (1,2) IV. (–4,–4) Mathematics Readiness Project 1997 Page 45 Section 10: Exploring with a Graphing Calculator Student Page Use your graphing calculator to help do the following problems. I. Graph y = –x + 3. What is the x-intercept? Label it on your graph. What is the y-intercept? Label it on your graph. Give the coordinates of three other points that lie on this line: ( , ), ( , ), ( , ) What is the y-coordinate of the point with x-coordinate –4? II. Turn off the graph for #I. Graph –3x + y = –1. What is the x-intercept? Label it on your graph. What is the y-intercept? Label it on your graph. Give the coordinates of three other points that lie on this line: ( , ), ( , Mathematics Readiness Project 1997 ), ( , ) Page 46 Section 10: Exploring with a Graphing Calculator Student Page III. Turn on the graph for #I, namely y = –x + 3. On this set of axes, sketch the graph of y = –x + 3 and the graph of –3x + y = –1 What are the coordinates of the point where the two lines intersect? ( , ) Use "table" mode to view (x,y) tables for each graph. What pair of coordinates appears in the table for both graphs? ( , ) 1 y = 2 x – 2 IV. Use a graphing calculator to graph the equations and 5 6 – y = – 2 x then use the graphs and the calculator functions to identify the solution to the system. V. Imagine that your friend has called you on the phone to ask for help using a graphing calculator to find the solution to a system of two equations with two variables. On separate paper, write out your explanation. Remember, your friend cannot see your calculator over the phone, so your directions must be very specific and clear. Mathematics Readiness Project 1997 Page 47 Section 10: What Went Wrong? Mathematics Readiness Project 1997 Student Page Page 48

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