Algebra 1 Topic 10: Student Activity Sheet 1; use with Overview 1. Consider the equation y = 3x + 4. [OV, page 1] a. What can you know about the graph of the equation? Numerous answers are possible, such as: It is a line. It has slope 3. It has y-intercept 4. b. How does a point, (x,y), end up on the graph of this equation? The point must, upon substitution, make the equation true. 2. Consider the equation 2x + 2y = 24. [OV, page 2] a. What do you know about the graph of this equation? Numerous answers are possible, such as: It is a line. It has x-intercept 12. It has y-intercept 12. b. How does a point, (x,y) end up on the graph of this equation? The point must, upon substitution, make the equation true. Page 1 of 1
Student Activity Sheet 1; use with Overview 3. Show why the point (2,10) is a solution to the system of equations. [OV, page 3] y = 3x + 4 2x + 2y = 24 A solution to a system of equations is an ordered pair that will make both equations true. (2,10) is a solution to the given system as demonstrated below: 10 = 3(2) + 4 2(2) + 2(10) = 24 10 = 6 + 4 4 + 20 = 24 10 = 10 24 = 24 Page 2 of 2
Student Activity Sheet 2; use with Exploring Solving systems of equations in slope-intercept form 1. Desmond has determined the following information about his two options. Option 1, repairing his current gas-powered mower: The cost of repair will be $160, after which the cost to operate the repaired mower is estimated to be $45 per week. Option 2, buying a new energy-efficient electric-powered mower: The cost of the new mower will be $400, after which the cost to operate the new mower is estimated to be $15 per week. Create equations to model the options. Each equation should model the cost of that option, y, in dollars, as a function of time, x, in weeks. [EX1, page 2] 400 15 45 160 2. Desmond s best option will be that option with the smallest total cost over the rest of the mowing season. [EX1, page 3] a. Which option is best if there is little time left in the mowing season? In this situation, option 1 is best, because it has the smaller initial cost (of repair). b. Which option is probably best if the mowing season has just begun? In this situation option 2 is probably best, because it has the smaller operational cost per week. Page 1 of 3
Student Activity Sheet 2; use with Exploring Solving systems of equations in slope-intercept form c. It would be useful to know how many weeks it would take for the two options to result in the same total cost. How can Desmond figure that out? He can solve the system of equations: y = 45x + 160 3. What is the solution to the following system? [EX1, page 4] The solution is the point (8,520). y = 45x + 160 y = 15x + 400 y = 15x + 400 4. What does the solution you found in question 3 tell you about Desmond s decision with respect to his two options? [EX1, page 5] First, it tells us that after 8 weeks, both options will have a total cost of $520. In addition, it lets Desmond know that if there are more than 8 weeks left in the mowing season, it is more economical to buy a new mower. On the other hand, if there are fewer than 8 weeks left, repairing the old mower will be the less costly option. 5. The two ad options are $10 per week plus 12.5 per click $6.25 per week plus 20 per click Set up an equation to model the total cost for each option if Desmond runs the ad for one week. Then use your graphing calculator to help you determine how many clicks it would take for the two options to result in the same total cost. [EX1, page 7] A system of equations that models this situation is y = 10 + 0.125x y = 6.25 + 0.20x The total cost is the same for both options at 50 clicks. At that point the total cost for each option is $16.25. Therefore, the first option becomes less expensive than the second option after 50 clicks for one week. Page 2 of 3
Student Activity Sheet 2; use with Exploring Solving systems of equations in slope-intercept form 6. REINFORCE Andrew has challenged his cousin, Kyle, to a race. Since Andrew is older, Kyle insists on a head start. Andrew agrees and gives Kyle a 20-meter head start. Andrew runs at a constant rate of 8.5 meters per second, and Kyle runs at a constant rate of 6 meters per second. a. Set up a system of two linear equations whose solution will represent at what distance and at what time Andrew will overtake Kyle in the race. The system of equations is y = 20 + 6x y = 8.5x where x is the number of seconds since the race started, and y is the distance from the starting line. b. Solve the system using both a table and a graph. Interpret the solution. The solution is the point (8,68), which indicates that after 8 seconds both boys will be 68 meters from the starting line. Because Kyle had a 20-meter head start, he has actually run only 48 meters.. c. Show how both the table and the graph can be used to determine how long it takes before Andrew is 10 meters behind Kyle. Inspection of either a table or a graph will indicate that after 4 seconds Andrew will be 34 meters from the starting line and Kyle will be 44 meters from the starting line. The 10 meters can be seen visually by measuring the distance between the two graphs at x = 4. Page 3 of 3
Student Activity Sheet 3; use with Exploring Solving systems of equations in standard form 1. Desmond has lost his record sheet for the second week of his work. The only thing he remembers is that he received $240 for mowing 12 lawns that week. In the neighborhood where Desmond and his customers live, the houses are built on lots that come in two sizes: standard-sized interior lots and larger corner lots. Desmond charges $15 per standard-sized lot and $30 per large corner lot for his mowing services. If x represents the number of standard-sized lots mowed that week and y represents the number of large corner lots mowed that week, complete the puzzle to create a system of two linear equations to model this situation. [EX2, page 1] 240 15x + 30y 12 x + y Copyright 2014 Agile Mind, Inc. Page 1 of 3
Student Activity Sheet 3; use with Exploring Solving systems of equations in standard form 2. You found a system of equations to represent Desmond s missing records situation. [EX2, page 2] a. How is this system different from the systems of equations you solved in the last Exploring? The equations are in standard form, Ax + By = C, instead of slope-intercept form, y = mx + b. b. Will this difference prevent you from using a table and/or a graph to solve the system? No, our goal is still to find the coordinates of the point of intersection for the two linear graphs. c. How do you know whether to use slope-intercept form or standard form for the equations in system of linear equations? Use whichever is more convenient and best models the situation for either or both of the equations. Ultimately, it does not matter as long as the equations correctly model the problem posed. 3. Sketch the graph of each equation. [EX2, page 3] x + y = 12 15x + 30y = 240 14 12 10 x+y=12 8 6 4 15x+30y=240 2 5 10 15 [Replace with screenshot from EX2_4, once produced.] Copyright 2014 Agile Mind, Inc. Page 2 of 3
Student Activity Sheet 3; use with Exploring Solving systems of equations in standard form 4. As you saw in both the table and graph, the solution to the system is the point (8,4). What does this solution tell you about Desmond s missing records problem regarding how many lots of each size were mowed during the week? [EX2, page 4] The solution to the system, (8,4), indicates that $240 would be earned while mowing 12 different lots if 8 of the lots were standard-sized interior lots and 4 of the lots were the larger corner lots. Furthermore, this is the only solution (point) that satisfies both equations (belongs to both lines). 5. Suppose this time, Desmond charges $20 for the standard-sized interior lots and $25 for the larger corner lots. He received a total of $300 for mowing 14 lots during the past week. Using x to represent the number of interior lots mowed and y to represent the number of corner lots mowed, set up a system of two linear equations to model this situation. Then use your graphing calculator to help you determine how many lots of each type Desmond mowed during the week. [EX2, page 6] A system of equations that models this situation is x + y = 14 20x + 25y = 300 Desmond mowed 10 interior lots and 4 corner lots. 6. REINFORCE The sum of two numbers is 186. The difference between the same numbers is 32. Create a system of equations to model the situation, and then solve the system using a graphical strategy. In this situation, you are looking for two numbers. When you add the numbers, the answer is 186. When you subtract the numbers, the answer is 32. Let x and y represent the two numbers and write equations to represent the sum and the difference. x + y = 186 x y = 32 Using a graphing calculator, enter y = 186 x (or y = -x + 186) and y = x 32 (or y = -32 + x), and use the intersection function to find the solution (109,77). Copyright 2014 Agile Mind, Inc. Page 3 of 3
Student Activity Sheet 4; use with Exploring Systems of inequalities 1. Desmond needs to schedule the lawn mowing jobs each week so that he and Shelly can do all of the mowing and edging that is required. What are some issues that Desmond needs to consider? [EX3, page 1] Answers may vary. Possible answers include how much time they have for working, the cost of gasoline and other expenses, the location of the houses, how long each yard will take, how much to charge per yard, and how to get customers. 2. Based on their previous experience, Desmond and Shelly come up with the following time estimates: Standard-sized interior lot: 1 hour to mow and a half hour to edge Larger corner lot: 2 hours to mow and 45 minutes to edge Desmond can spend at most 30 hours a week mowing lawns. Shelly can only spend at most 12 hours per week edging. How can Desmond model this information as a system of two linear inequalities? [EX3, page 2] Standard-sized yards Large yards Constraints Number of yards x y Number of mowing hours per yard 1 2 30 Number of edging hours per yard 1 2 3 4 12 System of inequalities: x + 2y 30 1 x + 3 y 12 2 4 Page 1 of 7
Student Activity Sheet 4; use with Exploring Systems of inequalities 3. Determine whether the following combinations of sizes of lawns represent feasible numbers of lawns to mow and edge in any one week, given the time constraints represented by the system of inequalities. Explain your conclusions in terms of the amount of time Desmond and Shelly will work in each case. [EX3, page 3] a. 10 standard-size interior lawns and 8 large corner lawns Both inequalities are satisfied, so Desmond and Shelly can schedule 10 standard lawns and 8 corner lawns to be done in one week of mowing and edging, based on their time constraints. Notice that this schedule would require Desmond to mow for 26 hours during the week and Shelly to edge for 11 hours during the week. Each has a little extra time available but not used. b. 6 standard-size interior lawns and 15 large corner lawns Neither inequality is satisfied, so Desmond and Shelly cannot schedule 6 standard lawns and 15 corner lawns to be done in one week of mowing and edging, based on their time constraints. Notice that this schedule would require Desmond to mow for 36 hours during the week and Shelly to edge for 14.25 hours during the week. Neither has adequate time allotted for this schedule of lawns. c. 6 standard-size interior lawns and 12 large corner lawns Both inequalities are satisfied, so Desmond and Shelly can schedule 6 standard lawns and 12 corner lawns to be done in one week of mowing and edging, based on their time constraints. Notice that this schedule would require Desmond to mow for all 30 hours that he is available during the week and Shelly to edge for all 12 hours that she is available during the week. Neither has any extra time available but not used. d. 18 standard-size lawns and 6 large corner lawns The first inequality is satisfied, so Desmond has allocated enough mowing hours per week for this schedule of lawns; however, the second inequality is not satisfied, so Shelly has not allocated enough edging hours per week for this schedule of lawns. Therefore, they cannot schedule 18 standard lawns and 6 corner lawns to be done in one week of mowing and edging, based on their time constraints. Notice that this schedule would require Desmond to mow for all 30 hours that he is available, but it would also require Shelly to edge for 13.5 hours during the week 1.5 hours more than the 12 hours that she is available. Page 2 of 7
Student Activity Sheet 4; use with Exploring Systems of inequalities 4. Sketch the graph of x + 2y 30. [EX3, page 4] 18 16 14 12 10 8 6 (10, 8) 4 x + 2y = 30 2 10 20 30 5. Sketch the graph of 1 2 x + 3 4 y 12. [EX3, page 4] 18 16 14 12 1 2 x+ 3 4 y=12 10 8 6 (10, 8) 4 2 10 20 30 6. Sketch the graph of the system: [EX3, page 4] x + 2y 30 1 x + 3 y 12 2 4 18 16 14 12 10 8 6 1 2 x+ 3 4 y=12 (10, 8) 4 x + 2y = 30 2 10 20 30 Page 3 of 7
Student Activity Sheet 4; use with Exploring Systems of inequalities 7. Desmond makes two important observations. [EX3, page 5] a. Because neither variable can be negative, we really have a system of four inequalities, not just two inequalities. Write these two new inequalities, along with the original two inequalities, to show the complete system of four inequalities to which Desmond is referring. x + 2y 30 1 2 x + 3 4 y 12 x 0, y 0 b. Desmond continues, Also notice that the point (10,8) is below both of the lines, 1 3 x + 2y 30 and x + y 12. That agrees with the observation we made earlier 2 4 about the mowing schedule that point represents! Can you explain what Desmond means by this? When you checked the point (10,8) earlier, you discovered not only that 10 interior lawns and 8 corner lawns was a feasible combination for the mowing schedule, but also that neither Desmond nor Shelly would use all the time available for mowing and edging during the week. This corresponds to the point (10,8) falling below both of the lines. Page 4 of 7
Student Activity Sheet 4; use with Exploring Systems of inequalities 8. Complete the statements to explain the other points representing combinations of sizes of lawns. Use the answer choices provided. [EX3, page 6] Shelly above both below Desmond neither on Page 5 of 7
Student Activity Sheet 4; use with Exploring Systems of inequalities 9. Desmond and Shelly decide to allocate more time per week to their respective tasks. Desmond increases his mowing time to a maximum of 36 hours per week. Shelly increases her edging time to a maximum of 15 hours per week. Write the system of inequalities that represents the new constraints. Graph the system of inequalities, shading the solution set. Are the mowing schedules represented by the points (6,15) and (18,6) now feasible? [EX3, page 7] If x represents the number of standard-size interior lots and y represents the number of large corner lots, the system of inequalities is now: Desmond s time constraints x + 2y 36 Shelly s time constraints 1 x + 3 y 15 2 4 Common sense restrictions x 0, y 0 Page 6 of 7
Student Activity Sheet 4; use with Exploring Systems of inequalities 10. REINFORCE The souvenir shop at the ballpark sells signed baseballs for $2 each and signed miniature bats for $6 each. Jay can spend at most $12 to buy no more than 4 items. Create a system of inequalities to model this situation, and then graph the solution set. Which solution results in Jay spending all his money to buy 4 items? Let x represent the number of baseballs purchased and let y represent the number of bats purchased. A system of inequalities that represents this situation is x + y 4 2x + 6y 12 x 0, y 0 The solution set is shown below. The solution requested is the intersection of the two boundaries, (3,1), which represents buying 3 baseballs and 1 bat. 5 4 3 2 1 2 4 6 Page 7 of 7