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Algebra I – Turner Chapter 7 Lesson Goal: Solve a system of two linear equations by graphing. Definition System of Equations: A system of equations consists of two or more equations in two or more variables. Definition Solution of a System of Equations: A solution of a system of equations in two variables is an ordered pair that makes every equation in the system true. 1. Consider the system: 5x y 8 and 3x 2 y 2 . a. Is (6, 3) a solution? b. Is (1, 3) a solution? c. Is ( 2, 2) a solution? 2. Consider this system of equations: x y 7 and x y 3 . Can you find some solutions by trial and error? y 8 7 6 5 4 3. Solve graphically: x y 7 and x y 3 . 3 2 1 -8 -7 -6 -5 -4 -3 -2 -10 1 2 3 4 5 6 7 8 x -1 -2 -3 -4 -5 -6 -7 -8 Algebra I – Turner Classwork – 7 4. Could a system of linear equations ever have more than one solution? Could it ever have no solutions? y 5. Solve graphically: 2 x 3 y 6 and 4 x 3 y 12 . 8 7 6 5 4 3 2 1 -8 -7 -6 -5 -4 -3 -2 -10 1 2 3 4 5 6 7 8 x -1 -2 -3 -4 -5 -6 -7 -8 y 1 8 6. Solve graphically: 3 y x 9 and y x 1 . 7 3 6 5 4 3 2 1 -8 -7 -6 -5 -4 -3 -2 -10 1 2 3 4 5 6 7 8 x -1 -2 -3 -4 -5 -6 -7 -8 2 Algebra I – Turner Classwork – 7 2 y 7. Solve graphically: y x 1 and x 4 . 8 3 7 6 5 4 3 2 1 -8 -7 -6 -5 -4 -3 -2 -10 1 2 3 4 5 6 7 8 x -1 -2 -3 -4 -5 -6 -7 -8 How many solutions would each system have? 8. y x 4 and y x 9 9. y 2x 4 and y 3x 6 10. y 6 x 2 and y 6x 2 Lesson Goal: Solve a system of two linear equations by substitution. Process Solving a System of Equations by Substitution 1. Solve one equation for one variable. 2. Substitute this expression into the other equation. 3. Solve the resulting equation. 4. Substitute this value into the expression obtained from step #1. 5. Write the solution set. 3 Algebra I – Turner Classwork – 7 11. Solve by substitution: 2 x y 9 and 7 x 3 y 1 . 12. Solve by substitution: 4 x y 15 and x y 10 . 13. Solve by substitution: 2 x 4 y 7 and 2 y x 1 . 4 Algebra I – Turner Classwork – 7 14. Solve by substitution: 5x 3 y 15 and x 3 y 7 . Lesson Goal: Solve a word problem using a system of linear equations. 15. A quilt maker sews both large and small quilts. A large quilt requires 8 yards of fabric while the small quilt requires 3 yards. How many of each size quilt did she make if she used a total of 90 yards of fabric to make 15 quilts? 16. A store is selling all shoes for $12 a pair and all slippers for $8 a pair. If Linda buys a total of 7 items and spends a total of $76, how many pairs of shoes and slippers did she buy? 5 Algebra I – Turner Classwork – 7 17. A toy maker produces wooden trains and wooden airplanes. Each train requires 3 ounces of paint and each airplane requires 5 ounces of paint. The toy maker has a gallon can of paint (64 ounces). If he wants to paint 14 toys, how many of each can he paint? 18. A girl has 15 coins, all nickels and dimes, with a total value of $1.20. Find the number of each kind of coin. Lesson Goal: Solve a system of two linear equations by addition. Review: Theorem Addition Property for Equality: If a b , then a c b c . Restatement: Theorem Addition Property for Equality: If a b and c d , then a c b d . Now consider it written in this form: ab () c d ac bd This is the form of the property that we are going to use in our solutions today. 6 Algebra I – Turner Classwork – 7 19. Solve by addition: x y 7 and x y 1. Then check. 20. Solve by addition: 2a b 15 and 2a 3b 5 . 21. Solve by addition: 5x 3 y 32 and 5x 7 y 8 . 22. A plumber charges a flat fee to work on a job plus an hourly rate. If he charged $150 for a 3 hour job and $300 for an 8 hour job, what was the flat fee and the hourly rate? 7 Algebra I – Turner Classwork – 7 Lesson Goal: Solve a system of two linear equations by addition and multiplication. 23. Solve by linear combinations: 3m 4n 2 and 5m n 12 . 24. Solve by linear combinations: 4r 14 5t and 3r 6t 9 . 25. Solve by linear combinations: 7 x 14 2 y and 28 4 y 14 x . 26. The winning average on a Junior High baseball team was 0.381. If the team had won 3 more games, then its winning average would have been 0.524 . Find the number of games the team won and the number of games it played. 8 Algebra I – Turner Classwork – 7 27. The difference between three times one number and a smaller number is 23. The sum of the smaller and twice the larger number is 27. Name the numbers. Lesson Goal: Solve interest problems using a system of linear equations. Using the Language of Investments Review: Formula Interest: Interest Principal Rate Time Investment problems use a variety of words to describe interest, principal, and rate. Here are some of the words you should associate with each: Principal always refers to the starting amount of money that is invested, deposited, or borrowed. Rate is sometimes referred to as interest, but it is easily identified as the rate because it is always expressed as a percentage. Interest is the amount of money that is earned by the investment. It can be referred to as the yield, dividend, growth, increase, return, earnings, or income. 28. Mr. Kirchner invested $8000, part earning 8% interest and the rest earning 10% interest per year. How much did he invest at each rate if his annual yield was $760? 9 Algebra I – Turner Classwork – 7 29. Mrs. Mooney wants to earn at least $288 in annual dividends. She plans to invest a certain sum of money at 12% and 3 times as much at 8%. What is the least amount she can invest at 12% to accomplish this goal? 30. Mrs. Miller deposited $1400, part at 9% and the rest at 12% annual interest. If both deposits earn the same annual income, how much has been deposited at each rate? 31. Miss Dilmer took out two loans to pay for a car. One loan required her to pay 10% annual interest while the other required her to pay 15% annual interest. Miss Dilmer borrowed $1400 more at the lower rate than she did at the higher rate. If the annual interest on the loan at 10% was $25 less than the interest on the loan at 15%, how much did she borrow at each rate? 10 Algebra I – Turner Classwork – 7 Lesson Goal: Solve word problems involving wind and water currents using a system of linear equations. Wind and Water Currents Application: Water Currents: When traveling upstream, the actual rate a boat is traveling equals the rate of the boat in still water minus the rate of the current. When traveling downstream, the actual rate a boat is traveling equals the rate of the boat in still water plus the rate of the current. Example: If Bob can row his boat at 8 km/hr in still water and he is rowing upstream against a current traveling at 3 km/h, then his boat is actually traveling up the river at 8 3 5 km/h. Example: If Bob can row his boat at 8 km/hr in still water and he is rowing downstream with a current traveling at 3 km/h, then his boat is actually traveling up the river at 8 3 11 km/h. Application: Wind: When traveling against the wind (called flying with a headwind), the actual rate an airplane is traveling (called the groundspeed) equals the rate of the airplane in still air (called the airspeed) minus the rate of the wind. When traveling with the wind (called flying with a tailwind), the actual rate an airplane is traveling equals the rate of the airplane in still air plus the rate of the wind. Example: If a plane travels at a still air rate (or airspeed) of 300 km/h against a 5 km/h head wind, then the actual rate the plane is traveling (or groundspeed) is 300 5 295 km/h. Example: If a plane travels at a still air rate (or airspeed) of 300 km/h with a 5 km/h tail wind, then the actual rate the plane is traveling (or groundspeed) is 300 5 305 km/h. 32. An airplane travels 1800 miles in 3 hours flying with the wind. On the return trip, flying against the wind, it takes 4 hours. Find the rate of the wind and the rate of the plane in still air? 11 Algebra I – Turner Classwork – 7 33. A barge travels 24 miles up the Ohio River in 3 hours. The return trip takes 2/3 that amount of time. Find the rate of the barge in still water and the rate of the current. 34. Len is planning a five-hour flight up to the mountains and back. He knows that he can fly in still air at 100 mi/h and finds that he has a 10 mph tail wind as he heads towards the mountains. How much time can he spend flying towards the mountains before he must turn back? 35. A rowing team rows downstream for half an hour, turns around, and rows back to their starting point in an hour and a quarter. If the current flows at 3 miles per hour, how fast does the team row, and how far did they travel downstream? 12 Algebra I – Turner Classwork – 7 Lesson Goal: Solve distance-rate-time word problems using a system of linear equations. 36. Two ships are sailing toward each other from an initial distance of 120 nautical miles apart. The rate of one ship is 4 knots greater than the rate of the other ship. If they meet in 3 hours, find the rate of each ship. 37. A motorist traveling 55 miles per hour is being pursued by a highway patrol car traveling 65 miles per hour. If the patrol car is 4 miles behind the motorist, how long will it take the patrol car to overtake the motorist? 38. Sally started home from school riding her bike at 20 mph. Part way home she hit a pothole and got a flat tire. So she walked her bike the rest of the way home at 2 mph. It took her an hour and 30 minutes to get home. If she lives 9 miles from school, for how many minutes did she ride her bike? 13 Algebra I – Turner Classwork – 7 14 Algebra I – Turner Classwork – 7 39. The steamship Empress Anne sailing due West at 32 knots passed the freighter Oregon which was sailing due East at 24 knots. In how many hours after the meeting will the ships be 448 nautical miles apart? Lesson Goal: Solve word problems involving mixtures using a single equation or a system of linear equations. 40. How many grams of water must be added to 40 grams of a 90% acid solution to produce a 50% acid solution? 41. How many kilograms of antifreeze must be added to 4 kilograms of a 10% antifreeze solution to produce a 20% solution? 15 Algebra I – Turner Classwork – 7 42. How many grams of water must be evaporated from 10 grams of a 40% antiseptic solution to produce a 50% solution? 43. Cream is 30% butterfat and milk is 4% butterfat. How much of each must a dairy mix together to get 100 quarts of a product that is 15% butterfat? 16 Algebra I – Turner Classwork – 7 Lesson Goal: Solve a word problem using a graph of a system of equations. 44. Suppose x represents the number of people who attend the Dapper Dan basketball game in Pittsburgh, Pa. Let C( x) 5x 40,000 represent the cost of hosting the game and R( x) 30x represent the total revenue taken in from the game. y a. Graph the functions on Fathom. Use the graph to answer the questions below. 100,000 90,000 80,000 70,000 60,000 50,000 b. For what attendance is revenue equal to operating 40,000 costs? This is called the break-even point. 30,000 20,000 c. For what attendance is revenue greater than 10,000 operating costs? x 0 400 800 1200 1600 2000 2400 2800 3200 d. What is the attendance when the revenue is less than operating costs, and how does that relate to profit? e. Write an equation or inequality that relates to each question above. 17 Algebra I – Turner Classwork – 7 45. In the United States whole milk consumption was 103 quarts annually per person in 1970 and fell by 3 quarts per person each year for the next 20 years. By contrast, low fat milk consumption was only 25 quarts annually per person in 1970, but rose by 2.25 quarts per person each year for the next 20 years. Let x represent the year, with x 0 corresponding to 1970. a. Write the linear function representing the consumption of whole milk. b. Write the linear function representing the consumption of low fat milk. c. Graph the two functions on Fathom. y 110 100 90 d. Write an equation that could be used to determine in what year the consumption of 80 whole milk was the same as the 70 consumption of low fat milk. 60 50 40 30 e. Use the graph to solve the equation. 20 10 x 0 2 4 6 8 1 12 14 16 18 20 0 f. Write an inequality that could be used to determine in what years the consumption of whole milk was less than the consumption of low fat milk. g. Use the graph to solve the inequality. 18 Algebra I – Turner Classwork – 7 Lesson Goal: Use a graph of two functions to solve a related equation or inequality. 46. A worm starts at the oak tree and moves away, heading for the elm tree at a constant rate of 13 meters per hour. At the same time a snail starts at the elm tree and moves toward the oak tree at a constant rate of 17 m/h. The two trees are 100 m apart. Let x be the number of hours the two creatures have been creeping. a. Draw a picture that illustrates what is happening in the problem. b. Write a function f (x) for the distance the worm is from the OAK tree in terms of x. c. Write a function g (x ) for the distance the snail is from the OAK tree in terms of x. d. Graph the two functions on Fathom. e. Write an equation that could be used to answer the question "When do the worm and the snail meet?" f. Use the graph to solve the equation. g. Write an inequality that could be solved to answer the question "When is the worm farther from the OAK tree than the snail is?" h. Use the graph to solve the inequality. 19 Algebra I – Turner Classwork – 7 47. A company has two tanks of water. One has a capacity of 2000 liters, but now has only 550 liters of water in it. The other tank has a capacity of 1600 liters and is now full. Suppose the company starts to pump water from the 1600 liter tank into the 2000 liter tank at a rate of 50 liters per hour. Let x be the amount of time elapsed. a. Write a function for the amount of water in the 2000-liter tank in terms of x. b. Write a function for the amount of water in the 1600-liter tank in terms of x. c. Graph the two functions on Fathom. d. Write an equation that could be used to determine when the two tanks have the same amount of water. e. Use the graph to solve the equation. f. Write an inequality that could be used to determine when the 2000-liter tank will contain more water than the 1600-liter tank. g. Use the graph to solve the inequality. h. Write an inequality that could be used to determine when the 2000-liter tank will be overflowing. i. Use the graph to solve the inequality. 20 Algebra I – Turner Classwork – 7 48. A cougar is lying down beside a tree and spots a fawn 132 meters away. The cougar starts running toward the fawn at a speed of 18 meters per second. At the same instant, the fawn starts running away at 11 meters per second. Let x be the number of seconds they have been running. a. Draw a picture that illustrates what is happening in the problem. b. Write a function f (x) for the distance the cougar is from the tree in terms of x. c. Write a function g (x ) for the distance the fawn is from the tree in terms of x. d. Graph the two functions on Fathom. e. Write an equation that could be used to determine when the cougar catches the fawn. f. Use the graph to solve the equation. g. Write an inequality that could be used to determine when the fawn is farther from the tree than the cougar. h. Use the graph to solve the inequality. 21