国外原版美国加州中学教材5

FPO OVERMATTER Solving Systems of Linear Equations Solving Systems of Linear Equations Make this Foldable to record information about solving systems of equations and inequalities. Begin with five sheets of grid paper. 1 Fold each sheet in half along the width. 3 Stack the sheets and staple to form a booklet. 2 Unfold and cut four rows from left side of each sheet, from the top to the crease. 4 Label each page with a lesson number and title. 250 Chapter 5 Solving Systems of Linear Equations -œ Û ˆ˜ }Ê -Þ ÃÌ i“ Ãʜ v �ˆ ˜i >À Ê µÕ >Ì ˆœ ˜Ã Ê x‡£Ê�À>«…ˆ˜}-ÞÃÌi“Ãʜv µÕ>̈œ˜Ã 5 Real-World Link TREES The tallest redwood trees in the world are in Humboldt County, California. The tallest redwood tree is 370 feet tall. In ideal conditions, a redwood tree could grow to a height of 420 feet. The height of these trees can be modeled by systems of linear equations. • Standard 9.0 Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets. (Key, CAHSEE) Key Vocabulary elimination (p. 266) substitution (p. 260) system of equations (p. 253) 250-251 COCH05-877852 250250-251 COCH05-877852 250 10/3/06 3:38:52 PM10/3/06 3:38:52 PM GET READY for Chapter 5 Diagnose Readiness You have two options for checking Prerequisite Skills. Option 2 Option 1 Take the Quick Check below. Refer to the Quick Review for help. Chapter 5 Get Ready For Chapter 5 251 Y X" ­ä]Ê�ή Õ«ÊÎÊ՘ˆÌà Àˆ}…ÌÊ{Ê՘ˆÌà EXAMPLE 1 Graph y = 3 _ 4 x - 3. Step 1 The y-intercept is -3. So, graph (0, -3). Step 2 The slope is 3 _ 4 . From (0, -3), move up 3 units and right 4 units. Draw a dot. Step 3 Draw the line. Solve each equation or formula for the variable specified. (Lesson 2-8) 8. 4x + a = 6x, for x 9. 8a + y = 16, for a 10. 7bc - d _ 10 = 12, for b 11. 7m + n _ q = 2m, for q EXAMPLE 2 Solve 2y _ 3s = 2y _ 13x for x. 2y _ 3s = 2y _ 13x Original equation 2y · 3s = 2y · 13x Find the cross products. 6ys = 26xy Simplify. 6ys _ 26y = x Divide each side by 26y. 3s _ 13 = x Simplify. Simplify each expression. If not possible, write simplified. (Lesson 1-6) 12. (3x + y) - (2x + y) 13. (7x - 2y) - (7x + 4y) 14. MOWING Jake and his brother charge x dollars to cut and y dollars to weed an average lawn. Simplify the expression that gives the total amount that their business earns in a weekend if Jake cuts and weeds 7 lawns and his brother cuts and weeds 10 lawns. EXAMPLE 3 Simplify 3(x - y) - (x - y). If not possible, write simplified. 3(x - y) - (x - y) Original expression = 3x - 3y - x + y Distributive Property = 2x - 2y Combine like terms. = 2(x - y) Factor out a 2. Graph each equation. (Lesson 3-3) 1. y = 1 2. y = -2x 3. y = 4 - x 4. y = 2x + 3 5. y = 5 - 2x 6. y = 1 _ 2 x + 2 7. HOUSES The number on Craig’s house is 7. The numbers of the houses on his block increase by 2. Graph the equation that models the house numbers on Craig’s block. Take the Online Readiness Quiz at ca.algebra1.com. 250-251 COCH05-877852 251250-251 COCH05-877852 251 10/3/06 3:39:02 PM10/3/06 3:39:02 PM EXPLORE 5-1 252 Chapter 5 Solving Systems of Linear Equations You can use a spreadsheet to investigate when two quantities will be equal. Enter each formula into the spreadsheet and look for the row in which both formulas have the same result. EXAMPLE Bill Winters is considering two job offers in telemarketing departments. The salary at the first job is $400 per week plus 10% commission on Mr. Winters’ sales. At the second job, the salary is $375 per week plus 15% commission. For what amount of sales would the weekly salary be the same at either job? Enter different amounts for Mr. Winters’ weekly sales in column A. Then enter the formula for the salary at the first job in each cell in column B. In each cell of column C, enter the formula for the salary at the second job. The spreadsheet shows that for sales of $500 the total weekly salary for each job is $450. EXERCISES For Exercises 1–4, use the spreadsheet of weekly salaries above. 1. If x is the amount of Mr. Winters’ weekly sales and y is his total weekly salary, write a linear equation for the salary at the first job. 2. Write a linear equation for the salary at the second job. 3. Which ordered pair is a solution for both of the equations you wrote for Exercises 1 and 2? a. (100, 410) b. (300, 420) c. (500, 450) d. (900, 510) 4. Use the graphing capability of the spreadsheet program to graph the salary data using a line graph. At what point do the two lines intersect? What is the significance of that point in the real-world situation? 5. How could you find the sales for which Mr. Winters’ salary will be equal without using a spreadsheet? A B C 1 3 4 5 6 7 8 9 10 11 2 12 13 Job Salaries.xls Sales Salary 1 Salary 2 0 100 200 300 400 500 600 700 800 900 1000 400 410 420 430 440 450 460 470 480 490 500 375 390 405 420 435 450 465 480 495 510 525 Sheet 1 Sheet 2 Sheet 3 Spreadsheet Lab Systems of Equations Preparation for Standard 9.0 Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets. (Key, CAHSEE) Interactive Lab ca.algebra1.com 252-258 CH05L1-877852 252252-258 CH05L1-877852 252 9/22/06 10:46:04 AM9/22/06 10:46:04 AM 5-1 If x is the number of years since 2000 and y is units sold in millions, the following equations represent the sales of CD singles and music videos. CD singles: y = 34.2 - 14.9x music videos: y = 3.3 + 4.7x The point at which the graphs of the two equations intersect represents the time when the CD units sold equaled the music videos sold. The ordered pair of this point is a solution of both equations. Source: The Recording Industry Association of America Number of Solutions Two equations, such as y = 34.2 - 14.9x and y = 3.3 + 4.7x, together are called a system of equations. A solution of a system is an ordered pair that satisfies both equations. A system of two linear equations can have no, one, or an infinite number of solutions. • If the graphs intersect or coincide, the system of equations is consistent. That is, it has at least one ordered pair that satisfies both equations. • If a consistent system has exactly one solution, it is independent. If it has infinite solutions, it is dependent. • If the graphs are parallel, the system of equations is said to be inconsistent. There are no ordered pairs that satisfy both equations. Lesson 5-1 Graphing Systems of Equations 253 Graphing Systems of Equations 1˜ˆÌÃÊ܏`ʜvÊ �ÊȘ}ià iµÕ>ÃÊ՘ˆÌÃÊ܏`ʜv “ÕÈVÊۈ`iœÃ �Ê-ˆ˜}iÃÊ>˜`Ê�ÕÈVÊ6ˆ`iœÃÊ->ià £ Ó Î { x È Ç n ÊÊx ä £ä £x Óä 1 ˜ˆ Ìà Ê- œ ` ­ˆ˜ ʓ ˆ ˆœ ˜Ã ®Ê Óx Îä Îx 9i>ÀÃÊ-ˆ˜ViÊÓäää YÊ�ÊÎ{°ÓÊ�Ê£{°™X X Y YÊ�ÊΰÎÊ�Ê{°ÇX Graphing Systems of Equations Graph of a System y xO y xO y xO Number of Solutions exactly one solution infinitely many no solutions Terminology consistent and independent consistent and dependent inconsistent Main Ideas • Determine whether a system of linear equations has no, one, or infinitely many solutions. • Solve systems of equations by graphing. Standard 9.0 Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets. (Key, CAHSEE) New Vocabulary system of equations consistent independent dependent inconsistent Reading Math Simultaneous Equations Systems of equations are also known as simultaneous equations, because a solution consists of values for the variables that satisfy all of the equations at the same time, or simultaneously. Animation ca.algebra1.com 252-258 CH05L1-877852 253252-258 CH05L1-877852 253 9/22/06 10:46:07 AM9/22/06 10:46:07 AM 254 Chapter 5 Solving Systems of Linear Equations Look Back To review graphing linear equations, see Lesson 3-3. EXAMPLE Number of Solutions Use the graph at the right to determine whether each system has no solution, one solution, or infinitely many solutions. a. y = -x + 5 y = x - 3 Since the graphs are intersecting lines, there is one solution. b. y = -x + 5 2x + 2y = -8 Since the graphs are parallel, there are no solutions. 1A. 2x + 2y = -8 1B. y = 2x + 14 y = -x - 4 y = -x + 5 Solve By Graphing One method of solving systems of equations is to carefully graph the equations on the same coordinate plane. EXAMPLE Solve a System of Equations Graph each system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it. a. y = -x + 8 y = 4x - 7 The graphs appear to intersect at (3, 5). Check by replacing x with 3 and y with 5. CHECK y = -x + 8 y = 4x - 7 5 � -3 + 8 5 � 4(3) - 7 5 = 5 � 5 = 5 � The solution is (3, 5). b. x + 2y = 5 2x + 4y = 2 The graphs are parallel lines. Since they do not intersect, there are no solutions to this system of equations. Notice that the lines have the same slope but different y-intercepts. Recall that a system of equations that has no solution is said to be inconsistent. 2A. x - y = 2 2B. y = -2x - 3 3y + 2x = 9 2x + y = -3 y x O y � 2x � 14 y � �x � 4 y � x � 3 y � �x � 5 2x � 2y � �8 y x O y � �x � 8 y � 4x � 7 (3, 5) y x O x � 2y � 5 2x � 4y � 2 Personal Tutor at ca.algebra1.com Write and Solve a System of Equations SPORTS The number of girls participating in high school soccer and track and field has steadily increased during the past few years. Use the information in the table to predict the year in which the number of girls participating in these two sports will be the same. High School Sport Number of Girls Participating in 2004 (thousands) Average Rate of Increase (thousands per year) soccer 309 8 track and field 418 3 Source: National Federation of State High School Associations Words Variables Equations Number of girls participating equals rate of increase times number of years after 2004 plus number participating in 2004. Let y � number of girls competing. Let x � number of years after 2004. soccer: y track and field: y � � 8 3 � � x x � � 309 418 Graph the equations y = 8x + 309 and y = 3x + 418. The graphs appear to intersect at (22, 485). Check by replacing x with 22 and y with 485 in each equation. CHECK y = 8x + 309 y = 3x + 418 485 = 8(22) + 309 485 = 3(22) + 418 485 = 485 � 485 ≈ 484 � The solution means that approximately 22 years after 2004, or in 2026, the number of girls participating in high school soccer and track and field will be the same, about 485,000. 3. GARDENS A rectangular garden has a border around it consisting of 60 bricks. The width of the border has 2 _ 3 the number of bricks as the length. How many bricks are along one length of the garden? Use the graph to determine whether each system has no solution, one solution, or infinitely many solutions. Example 1 (p. 254) 1. y = x - 4 y = 1 _ 3 x - 2 3. x - y = 4 y = x - 4 2. y = 1 _ 3 x + 2 y = 1 _ 3 x - 2 4. x - y = 4 y = - 1 _ 3 x + 4 y xO y � � x � 413 y � x � 213 y � x � 213 x � y � 4 y �x � 4 Y X/ ­ÓÓ]Ê{nx® Y� ��nX� ��Îä™ Y� ��ÎX� ��{£n £Ón{ £È Óä Ó{ Ón ÎÓä {ää {nä xÈä Ó{ä £Èä nä Real-World Link In 2004, 2.9 million girls participated in high school sports. This was an all-time high for female participation. Source: National Federation of State High School Associations Lesson 5-1 Graphing Systems of Equations 255Extra Examples at ca.algebra1.com Graph each system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it. 5. y = 3x - 4 6. x + y = 2 y = -3x - 4 y = 4x + 7 7. x + y = 4 8. 2x + 4y = 2 x + y = 1 3x + 6y = 3 9. GEOMETRY The length of the rectangle is 1 meter less than twice its width. What are the dimensions of the rectangle? Use the graph to determine whether each system has no solution, one solution, or infinitely many solutions. 10. x = -3 11. y = -x - 2 y = 2x + 1 y = 2x - 4 12. y = 2x + 1 13. y = 2x + 1 2y - 4x = 2 y = 2x - 4 14. y + x = -2 15. 2y - 4x = 2 y = -x - 2 y = 2x - 4 Graph each system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it. 16. y = -6 17. x = 2 18. y = 1 _ 2 x 4x + y = 2 3x - y = 8 2x + y = 10 19. y = -x 20. y = 2x + 6 21. x - 2y = 2 y = 2x - 6 y = -x - 3 3x + y = 6 22. x + y = 2 23. 3x + 2y = 12 24. 2x + 3y = 4 2y - x = 10 3x + 2y = 6 -4x - 6y = -8 25. 2x + y = -4 26. 4x + 3y = 24 27. 3x + y = 3 5x + 3y = -6 5x - 8y = -17 2y = -6x + 6 SAVINGS For Exercises 28 and 29, use the following information. Monica and Max Gordon each want to buy a scooter. Monica has already saved $25 and plans to save $5 per week until she can buy the scooter. Max has $16 and plans to save $8 per week. 28. In how many weeks will Monica and Max have saved the same amount of money? 29. How much will each person have saved at that time? HOMEWORK For Exercises 10–15 16–27 28–31 See Examples 1 2 3 � wPerimeter � 40 m y xO 2y � 4x � 2 y � x � �2 x � �3 y � 2x � 1 y � 2x � 4 y � �x � 2 y � �3x � 6 Example 2 (p. 254) Example 3 (p. 255) 256 Chapter 5 Solving Systems of Linear Equations 252-258 CH05L1-877852 256252-258 CH05L1-877852 256 9/22/06 10:46:13 AM9/22/06 10:46:13 AM BALLOONING For Exercises 30 and 31, use the information in the graphic at the right. 30. In how many minutes will the balloons be at the same height? 31. How high will the balloons be at that time? Is your answer reasonable? Explain. Graph each system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it. 32. y = 0.6x - 5 33. 6 - 3 _ 8 y = x 2y = 1.2x 2 _ 3 x + 1 _ 4 y = 4 ANALYZE GRAPHS For Exercises 34–36, use the graph at the right. 34. Which company had the greater profit during the ten years? 35. Which company had a greater rate of growth? 36. If the profit patterns continue, will the profits of the two companies ever be equal? Explain. POPULATION For Exercises 37–39, use the following information. The U.S. Census Bureau divides the country into four sections. They are the Northeast, the Midwest, the South, and the West. The populations and rates of growth for the Midwest and the West are shown in the table. Section 2000 Population (millions) Average Rate of Increase (millions per year) Midwest 64.4 0.3 West 63.2 1.0 Source: U.S. Census Bureau 37. Write an equation to represent the population of the Midwest for the years since 2000. 38. Write an equation to represent the population of the West for the years since 2000. 39. Graph the population equations. Assume that the rate of growth of each of these areas remained the same. Estimate the solution and interpret what it means. 40. CHALLENGE The solution of the system of equations Ax + y = 5 and Ax + By = 20 is (2, -3). What are the values of A and B? Justify your reasoning. 41. OPEN ENDED Write three equations such that they form a system of equations with y = 5x - 3. The systems should have no, one, and infinitely many solutions, respectively. Lesson 5-1 Graphing Systems of Equations 257 Balloon 1 is 10 meters above the ground, rising 15 meters per minute. Balloon 2 is 150 meters above the ground, descending 20 meters per minute. Yearly Profits 2 4 6 8 2 4 Pr o fi t ( m ill io n s o f d o lla rs ) 6 8 Year 0 Widget Company Gadget Company H.O.T. Problems You can graph a system of equations to predict when men’s and women’s Olympic times will be the same. Visit ca.algebra1.com to continue work on your project. EXTRA See pages 727, 748. Self-Check Quiz at ca.algebra1.com PRACTICEPRACTICE 252-258 CH05L1-877852 257252-258 CH05L1-877852 257 9/22/06 10:46:14 AM9/22/06 10:46:14 AM 42. REASONING Determine whether a system of two linear equations with (0, 0) and (2, 2) as solutions sometimes, always, or never has other solutions. Explain. 43. Writing in Math Use the information on page 253 to explain how graphs can be used to compare the sales of two products. Include an estimate of the year in which the CD units sold equaled the music videos sold. Then determine the reasonableness of your solution in the context of the problem. 44. A buffet restaurant has one price for adults and another price for children. The Taylor family has two adults and three children, and their bill was $40.50. The Wong family has three adults and one child. Their bill was $38. Which system of equations could be used to determine the buffet price for an adult and the price for a child? A x + y = 40.50 C 2x + 3y = 40.50 x + y = 38 3x + y = 38 B 2x + 3y = 40.50 D 2x + 2y = 40.50 x + 3y = 38 3x + y = 38 45. REVIEW Francisco has 3 dollars more than 1 _ 4 the number of dollars that Kayla has. Which expression represents how much money Francisco has? F 3 ( 1 _ 4 k ) G 3 - 1 _ 4 k H 1 _ 4 k + 3 J 1 _ 4 + 3k Write the slope-intercept form of an equation for the line that passes through the given point and is parallel to the graph of each equation. (Lesson 4-7) 46. y xO (0, 2) 47. y xO (1, 1) 48. BIOLOGY The table shows the date of the month that 10 students were born, and their heights. Draw a scatter plot and determine what relationship exists, if any, in the data. Explain. (Lesson 4-6) Date of Birth 12 28 24 15 3 11 20 5 3 9 Height (in.) 60 58 62 60 59 64 66 65 67 62 PREREQUISITE SKILL Solve each equation for the variable specified. (Lesson 2-8) 49. 12x - y = 10x, for y 50. 6a + b = 2a, for a 51. 7m - n _ q = 10, for q 258 Chapter 5 Solving Systems of Linear Equations 252-258 CH05L1-877852 258252-258 CH05L1-877852 258 9/22/06 10:46:15 AM9/22/06 10:46:15 AM Extend 5-1 Graphing Calculator Lab 259 Graphing Calculator Lab Systems of Equations EXTEND 5-1 You can use a graphing calculator to solve a system of equations. Solve the system of equations. State the decimal solution to the nearest hundredth. 2.93x + y = 6.08 8.32x - y = 4.11 Step 1 Solve each equation for y. Enter them into the calculator. 2.93x + y = 6.08 First equation 2.93x + y - 2.93x = 6.08 - 2.93x Subtract 2.93x from each side. y = 6.08 - 2.93x Simplify. 8.32x - y = 4.11 Second equation 8.32x - y - 8.32x = 4.11 - 8.32x Subtract 8.32x from each side. -y = 4.11 - 8.32x Simplify. (-1)(-y) = (-1)(4.11 - 8.32x) Multiply each side by -1. y = -4.11 + 8.32x Simplify. Step 2 Enter these equations in the Y = list and graph. KEYSTROKES: Review on pages 162–163. Step 3 Use the CALC menu to find the point of intersection. KEYSTROKES: 2nd [CALC] 5 ENTER ENTER ENTER The solution is approximately (0.91, 3.43). EXAMPLE EXERCISES Use a graphing calculator to solve each system of equations. Write decimal solutions to the nearest hundredth. 1. y = 3x - 4 2. y = 2x + 5 3. x + y = 5.35 y = -0.5x + 6 y = -0.2x - 4 3x - y = 3.75 4. 0.35x - y = 1.1