Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Quartic function wikipedia , lookup
Quadratic equation wikipedia , lookup
Cubic function wikipedia , lookup
Elementary algebra wikipedia , lookup
Signal-flow graph wikipedia , lookup
System of polynomial equations wikipedia , lookup
History of algebra wikipedia , lookup
Test on Topic 15 System of Equations Test on Topic 15 System of Equations Determine which of the following points, A(2,1) , B(1, – 3) and C(– 1 , 3) if any, satisfy both pairs of equations. 1. 4x – 7 4 y = 3x – 2y = 2. Determine which of the following points, A(1,1) , B(– 1, – 3) and C(0, 1) if any, satisfy both pairs of equations. y = 2x – 1 2x – 4y = – 2 3(a). On the grid opposite graph 2x + 4y = 0 by plotting 3 points. x 0 2 4 3(b). y y On the grid opposite graph y = 2x – 5 by plotting 3 points. x 0 x y 0 1 2 3.(c) 4. Solution to the system of equations is? Identify each system of linear equations below as consistent, inconsistent or dependant. Also indicate the number of solutions that would occur. Page | 1 Test on Topic 15 System of Equations 5. Express each of the system of equations below in the slope intercept form ( y = ….. form). And without drawing graphs identify each system of linear equations below as consistent, inconsistent or dependant. Also indicate the number of solutions that would occur. (a) 2y – 3x = 4x = 6. 7. 8. 9. (b) 4y – 8x = 2y – 6 = 4 8y – 7 12 4x Express each of the system of equations below in the slope intercept form ( y = ….. form). And without drawing graphs identify each system of linear equations below as consistent, inconsistent or dependant. Also indicate the number of solutions that would occur. (a) y – 2x = 4 2y = 4 – 2x (b) x – 2y = 2x = Solve the system of equations y = 2x + 3y = Solve the system of equations Solve the system of equations 10. Solve the system of equations y y x – 2y 3x + y 12. Solve the system of equations 13. Solve the system of equations 14. Solve the system of equations 3x – 3 13 (c) x – 4 4y – 8 = = 8 8 = = by using the substitution method. 2x + 3 4x – 8 = = –8 y – 2x by using the addition method. by using the substitution method. 6 4 by using the addition method. 5x – 2y = 5 = –7 y – 3x by using the substitution method. 2x – 2y = 4x + 3y = 4 1 by using the addition method. 5x – 2y = 3x + 2y = –8 24 4y 2x by using the substitution method. 5x – 2y = 2x + 4y = 2x – 2y = 3 = 11. Solve the system of equations 10 4y + 20 by using the addition method. 15. Solve the system of equations 2x – 2y = 4x + 3y = 4 1 by using the addition method. 16. Solve the system of equations 5x – 2y = 3x + 2y = 10 14 by using the addition method. 17. Solve the system of equations 2x – 3y = 3x + 4y = –3 21 by using the addition method. 18. Solve the system of equations 2x – 2y = 4x + 3y = 4 1 by using the addition method. Page | 2 Test on Topic 15 System of Equations 19. In 1997, France was the most visited country in the world and U.S.A. was the second most visited. A total of 116 million people visited these countries. If 18 million more visited France than the U.S.A. how many people visited each country. 20. The plumber charges a fixed rate for turning up at your house plus a charge per hour for the work done From previous jobs that the plumber has done it was found that a 4 hour job will cost a total of $283 while a 6 hour job will cost a total of $387. What is the fixed rate and the charge per hour? 21. The Acme printing company charges a fixed setup fee x, plus a fixed amount per page y. John has used this company before and after looking up his receipts he finds that his last two printing jobs cost the following. A 2000 page job cost him $150, while a 3000 page job cost him $200. (a) Set up a system of equations to reflect the above information. (b) Solve the system of equations and so find how much the fixed setup fee x is and how much is the charge per page y? (c) He needs to have 4500 copies of a flyer made, what should the Acme Printing company charge him to do this job. 22. A nutritionist finds that a large order of fries has 20 grams more than twice the fat content of a Haggis Burger. The difference in fat content between the fries and a Haggis Burger is 100 grams. What is the fat content of a large order of fries and a Haggis Burger? 23. By weight one alloy of brass is 70% Copper and 30% Zinc. Another Alloy is 40% Copper and 60% Zinc. How many grams of each alloy would need to be melted and combined to obtain 600 grams of a brass alloy that is 60% Copper and 40% Zinc? 24. A truck rental agency charges a daily fee plus a mileage fee. Julie was charged $85 for two days and 100 miles and Christina was charged $165 for 3 days and 400 miles. What 8is the agency’s daily fee and what is the mileage fee? 25. A pet store owner wants to make 600 pounds of mixed bird seed, The sunflower seeds cost $2.30 per pound and the peanut seeds cost $1.40 per pound. The total cost of all the seed used in the mixture was $1200. How much of each seed was used? 26. The fat content of a haggis is more than that for fries. The total fat content for a haggis and fries is 190 grams. The amount of fat in a haggis is 40 grams more than twice the amount of fat that is in fries. Find the amount of fat that is in a haggis and in a portion of fries? Page | 3 Test on Topic 15 System of Equations Test on Topic 15 System of Equations Solutions 1. Determine which of the following points, A(2,1) , B(1, – 3) and C(– 1 , 3) if any, satisfy both pairs of equations. y = 3x – 2y = For A(2,1) For C(– 1 , 3) 2. = = = 4x – 7 4(2) – 7 1 (yes) For A(2,1) y = –3 = –3 = 4x – 7 4(1) – 7 –3 (yes) For B(1, – 3) y 3 3 4x – 7 4(– 1) – 7 – 11 (no) y 1 1 For B(1, – 3) 4x – 7 4 = = = 3x – 2y = 3(2) – 2(1) = 4 = 4 4 4 (yes) 3x – 2y = 3(1) – 2(– 3) = 9 = 4 4 4 (no) For C(– 1 , 3) 3x – 2y = 3(– 1) – 2(3) = –9 = 4 4 4 (no) Determine which of the following points, A(1,1) , B(– 1, – 3) and C(0, 1) if any, satisfy both pairs of equations. y = 2x – 1 2x – 4y = – 2 For A(1,1) y = = = 2x – 1 2(1) – 1 1 true 2x – 4y = 2(1) – 4(1) = –2 = –2 –2 – 2 true y –3 = –3 = = 2x – 1 2(–1) – 1 – 3 true 2x – 4y = 2(–1) – 4(–3) = 10 = –2 –2 – 2 false x = 1 and y = 1 1 1 For B(– 1, – 3) For C(0,1) x = –1 and y = –3 x = 0 and y = 1 y 1 1 = = = 2x – 1 2(0) – 1 –1 false So the point A(1,1) satisfies the pair of equations. Page | 4 Test on Topic 15 System of Equations 3(a). On the grid opposite graph 2x + 4y = 0 by plotting 3 points. x 0 2 4 3(b). 3.(c) 4. On the grid opposite graph y = 2x – 5 by plotting 3 points. x y 0 –5 1 –3 2 –1 0 x 2x + 4y = 0 Solution is (2, – 1) Identify each system of linear equations below as consistent, inconsistent or dependant. Also indicate the number of solutions that would occur. Consistent # of solutions = 1 5. y = 2x – 5 y y 0 –1 –2 Inconsistent # of solutions = o dependant # of solutions = infinite Express each of the system of equations below in the slope intercept form ( y = ….. form). And without drawing graphs identify each system of linear equations below as consistent, inconsistent or dependant. Also indicate the number of solutions that would occur. (a) 2y – 3x = 4x = 2y – 3x = 2y = y = Answer : 4 8y – 7 4 3x + 4 3 x2 2 4x = 4x + 7 = 1 7 x = 2 2 8y – 7 8y y These two lines have different slopes, so the equations are consistent and there will be one solution Page | 5 Test on Topic 15 System of Equations (b) 4y – 8x = 2y – 6 = 12 4x 4y – 8x = 4y = y = Answer : 6. 2y – 6 = 2y = y = 12 8x + 12 2x 3 These two lines are identical, so the equations are dependant and there will be infinite solutions. Express each of the system of equations below in the slope intercept form ( y = ….. form). And without drawing graphs identify each system of linear equations below as consistent, inconsistent or dependant. Also indicate the number of solutions that would occur. (a) y – 2x = 4 2y = 4 – 2x (b) x – 2y = 2x = 10 4y + 20 x – 2y – 2y y 10 – x + 10 ½x – 5 y – 2x = 4 y = 2x + 4 2y = y = 4 – 2x 2–x consistent 1 solution 7. 4x 4x + 6 2x + 3 Solve the system of equations Solution: Substitute y = 3x – 3 into the equation Substitute x = 2 into equation = = = (c) x – 4 4y – 8 2x = 4y + 20 2x – 20 = 4y ½x–5 = y dependant infinite solution = = 4y 2x x–4 = ¼x–1= 4y y 4y – 8 4y y 2x 2x + 8 ½x + 2 = = = inconsistent 0 solutions y = 3x – 3 by using the substitution method. 2x + 3y = 13 2x + 3y 2x + 3(3x – 3) 2x + 9x – 9 11x – 9 11x x = = = = = = 13 13 13 13 22 2 y = 3x – 3 = 3(2) – 3 = 3 So the solution is x = 2 and y = 3 or (2,3) Page | 6 Test on Topic 15 System of Equations 8. 5x – 2y = 2x + 4y = Solve the system of equations Solution: 5x – 2y = 2x + 4y = 8 8 equation1….. multiply by 2 equation 2 Substitute x = 2 into equation 2x + 4y 2(2) + 4y 4 + 4y 4y y = = = = = 8 8 by using the addition method. 10x – 4y = 2x + 4y = 12x = x = 16 8 24 2 add equationd 8 8 8 4 1 So the solution is x = 2 and y = 1 or (2,1) 9. Solve the system of equations y Put y = 2x + 3 into the equation Put x = 5.5 into the equation y 2x + 3 3 11 5.5 y y y y = 2x + 3 4x – 8 by using the substitution method. = = = = = 4x – 8 4x – 8 2x – 8 2x x = = = = 4x – 8 4(5.5) – 8 22 – 8 14 2x – 2y = – 8 3 = y – 2x 10. Solve the system of equations Rearrange the equation 3 = y = Solution is (5.5,14) by using the substitution method. y – 2x so that it is in the form y = ….. 3 = y – 2x 3 + 2x = y Put y = 3 + 2x into the equation 2x – 2y 2x – 2(3 + 2x) 2x – 6 – 4x – 2x – 6 – 2x x Put x = 1 into equation y = 3 + 2x = 3 + 2(1) = 5 = = = = = = –8 –8 –8 –8 –2 1 Solution is (1,5) Page | 7 Test on Topic 15 System of Equations 11. Solve the system of equations x – 2y 3x + y = = 6 4 multiply by 2 Put x = 2 into 3x + y 3(2) + y = 6+y y = Solution is (2, – 2 ) x – 2y 3x + y = = 4 x – 2y 6x + 2y 7x x 6 8 14 2 = = = = 6 by using the addition method. = 4 4 = 4 –2 12. Solve the system of equations Re arrange 5 = y – 3x Substitute y = 3x + 5 into to get 5x – 2y = 5 –7 by using the substitution method. = y – 3x 5 y – 3x y y – 3x 5 3x + 5 = = = 5x – 2y 5x – 2(3x + 5) 5x – 6x – 10 – x – 10 –x x = = = = = = –7 –7 –7 –7 3 –3 Use x = – 3 in equation y = 3x + 5 = 3(– 3) + 5 = – 9 + 5 = – 4 The solution to the above system of equations is (– 3, – 4) 2x – 2y = 4x + 3y = 13. Solve the system of equations Equation 1: Equation 2 :` 2x – 2y = 4x + 3y = Use x = 1 in the equation 4 1 4 1 by using the addition method. 6x – 6y 8x + 6y 14x x multiply by 3 multiply by 2 4x + 3y 4(1) + 3y 4 + 3y 3y y = = = = = = = = = 12 2 14 1 (add) 1 1 1 –3 –1 The solution to the above system of equations is (1, – 1) Page | 8 Test on Topic 15 System of Equations 5x – 2y = 3x + 2y = 14. Solve the system of equations 5x – 2y 3x + 2y 8x x = = = = –8 24 16 2 –8 24 by using the addition method. (add) Use x = 2 in the equation 3x + 2y 3(2) + 2y 6 + 2y 2y y = = = = = 24 24 24 16 8 The solution to the above system of equations is (2, 8) 2x – 2y = 4x + 3y = 15. Solve the system of equations Equation 1: Equation 2 :` 2x – 2y = 4x + 3y = Use x = 1 in the equation 4 1 3x + 5 into by using the addition method. 6x – 6y 8x + 6y 14x x multiply by 3 multiply by 2 4x + 3y 4(1) + 3y 4 + 3y 3y y The solution to the above system of equations is Re arrange 5 = y – 3x to get 5 = y – 3x = y = Substitute y = 4 1 = = = = = = = = = 12 2 14 1 (add) 1 1 1 –3 –1 (1, – 1) y – 3x 5 3x + 5 5x – 2y 5x – 2(3x + 5) 5x – 6x – 10 – x – 10 –x x = = = = = = –7 –7 –7 –7 3 –3 Use x = – 3 in equation y = 3x + 5 = 3(– 3) + 5 = – 9 + 5 = – 4 The solution to the above system of equations is (– 3, – 4) Page | 9 Test on Topic 15 System of Equations 16. Solve the system of equations 5x – 2y = 3x + 2y = 10 14 8x x = = = = = = = 14 14 14 5 2.5 2x – 3y = 3x + 4y = –3 21 Add the equations Put x = 3 into equation 3x + 2y 3(3) + 2y 9 + 2y 2y y by using the addition method. 24 3 So the Solution is (3,2.5) 17. Solve the system of equations 2x – 3y = 3x + 4y = – 3 (multiply by 4) 21 (multiply by 3) Add the equations: Put x = 3 into the equation 3x + 4y 3(3) + 4y 9 + 4y 4y y 8x – 12y 9x + 12y 17x x = = = = = by using the addition method. = = = = – 12 63 51 3 21 21 21 12 3 Answer: Solution to system of equations is (3,3) 2x – 2y = 4x + 3y = 18. Solve the system of equations Equation 1: Equation 2 :` 2x – 2y = 4x + 3y = Use x = 1 in the equation 4 1 4 1 by using the addition method. 6x – 6y 8x + 6y 14x x multiply by 3 multiply by 2 4x + 3y 4(1) + 3y 4 + 3y 3y y = = = = = = = = = 12 2 14 1 (add) 1 1 1 –3 –1 The solution to the above system of equations is (1, – 1) Page | 10 Test on Topic 15 System of Equations 19. In 1997, France was the most visited country in the world and U.S.A. was the second most visited. A total of 116 million people visited these countries. If 18 million more visited France than the U.S.A. how many people visited each country. Let x y = = x+y x–y 2x x the number of visitors to France (in millions) the number of visitors to U.S.A. (in millions) = = = = 116 18 134 67 (add) Use x = 67 in equation x+y 67 + y y = = = 116 116 49 So France has 67 million visitors and U.S.A. has 49 million visitors 20. The plumber charges a fixed rate for turning up at your house plus a charge per hour for the work done From previous jobs that the plumber has done it was found that a 4 hour job will cost a total of $283 while a 6 hour job will cost a total of $387. What is the fixed rate and the charge per hour? Let x y = = fixed amount charge per hour x + 4y x + 6y = = 283 387 Rearrange x + 4y = 283 Use x = 283 – 4y into equation x = 283 – 4y to get x + 6y 283 – 4y + 6y 283 + 2y 2y y = = = = = 387 387 387 104 52 Use y = 52 in equation x = 283 – 4y = 283 – 4(52) = 283 – 208 = 75 So the plumber had a fixed amount of $75 plus he charged $52 per hour. Page | 11 Test on Topic 15 System of Equations 21. The Acme printing company charges a fixed setup fee x, plus a fixed amount per page y. John has used this company before and after looking up his receipts he finds that his last two printing jobs cost the following. A 2000 page job cost him $150, while a 3000 page job cost him $200. We set up a system of equations to reflect the above information. (b) 2000 page job cost him $150 gives us the equation 3000 page job cost him $200 gives us the equation x + 2000y x + 3000y = = 150 200 (c) Solve the system of equations and so find how much the fixed setup fee x is and how much is the charge per page y? x + 2000y = 150 can be written as x = – 2000y + 150 Substitute x = – 2000y + 150 into the equation Substitute y = 0.5 into the equation x x x x + 3000y – 2000y + 150 + 3000y 150 + 1000y 100y y = = = = = 200 200 200 50 0.5 = – 2000y + 150 = – 2000(0.5) + 150 = 50 So set up f = x = $50 and cost pr page = y = $0.50 (d) He needs to have 4500 copies of a flyer made, what should the Acme Printing company charge him to do this job. Cost = = = x + 4500y 5 0 + 4500(0.5) $275 Page | 12 Test on Topic 15 System of Equations 22. A nutritionist finds that a large order of fries has 20 grams more than twice the fat content of a Haggis Burger. The difference in fat content between the fries and a Haggis Burger is 100 grams. What is the fat content of a large order of fries and a Haggis Burger? Let x y = = number of grams of fat in an order of fries number of grams of fat in an Haggis Burger The information that A large order of fries has 20 grams more than twice the fat content of a Haggis Burger, can be translated into x = 2y + 20 The information The difference in fat content between the fries and a Haggis Burger is 100 grams. Can be translated into x – y = 100 ( x is larger than y so this is the correct order) Put these equations together and solve. x = 2y + 20 x – y = 100 Put y = 2x + 20 into the equation x–y 2y + 20 – y y + 20 y = = = = 100 100 100 80 Put y = 80 into x = 2y + 20 = 2(80) + 20 = 180 Answer: The solution to this problem is and x = number of grams of fat in an order of fries = 180 grams y = number of grams of fat in an Haggis Burger = 80 grams Page | 13 Test on Topic 15 System of Equations 23. By weight one alloy of brass is 70% Copper and 30% Zinc. Another Alloy is 40% Copper and 60% Zinc. How many grams of each alloy would need to be melted and combined to obtain 600 grams of a brass alloy that is 60% Copper and 40% Zinc? Let x = amount of alloy 1 used Let y = amount of alloy 2 used The information that alloys combined to obtain 600 grams of a brass alloy gives us the equation x + y = 600 The information that alloys combined to give us 60% copper gives us the equation 0.7x + 0.4y = 0.6(600) which in turn becomes 0.7x + 0.4y = 360 Put these equations together and solve. x+y = 0.7x + 0.4y = Rearrange x+y y Put y = – x + 600in to the equation 0.7x + 0.4y 0.7x + 0.4(– x + 600) 0.7x – 0.4x + 240 0.3x + 240 0.3x x 600 360 = = 600 – x + 600 = = = = = = 360 360 360 360 120 400 Put x = 400 in to y = – x + 600 = – 400 + 600 = 200 Answer: The solution to this problem is x = amount of alloy 1 used = 400 grams and y = amount of alloy 2 used = 200 grams 24. A truck rental agency charges a daily fee plus a mileage fee. Julie was charged $85 for two days and 100 miles and Christina was charged $165 for 3 days and 400 miles. What 8is the agency’s daily fee and what is the mileage fee? Julie was charged $85 for two days and 100 miles Christina was charged $165 for 3 days and 400 miles – 6d – 300m 6d + 800m 500m m Put m = 0.15 into the equation 2d + 100m = 85 2d + 15 = 85 2d = 70 d = 35 So solution is d = $35 per day and m = $0.15 per mile 2d + 100m 3d + 400m = = 85 165 multiply by – 3 multiply by 2 Add the equations 2d + 100m 3d + 400m = = = = = = 85 165 – 255 330 75 0.15 Page | 14 Test on Topic 15 System of Equations 25. A pet store owner wants to make 600 pounds of mixed bird seed, The sunflower seeds cost $2.30 per pound and the peanut seeds cost $1.40 per pound. The total cost of all the seed used in the mixture was $1200. How much of each seed was used? x = amount of sunflower seed used and y = amount of peanut seed used Equation two is formed by using the fact that he “wants to make 600 pounds of mixed bird seed” This corresponds to the equation x + y = 600 Equation one is formed by the fact that “total cost of all the seed used in the mixture was $1200” This corresponds to the equation 2.3x + 1.4y = 1200 x+y = 2.3x + 1.4y = 2.3x + 1.4y 2.3x + 1.4(– x + 600) 2.3x – 1.4x + 840 0.9x + 840 0.9 x x = = = = = = Also written as y = – x + 600 600 1200 1200 1200 1200 1200 360 400 Substitute x = 400 into the equation y = – x + 600 = – 400 + 600 = 200 So amount of sunflower seed used = 400 pounds and amount of peanut seed used is 200 pounds 26. The fat content of a haggis is more than that for fries. The total fat content for a haggis and fries is 190 grams. The amount of fat in a haggis is 40 grams more than twice the amount of fat that is in fries. Find the amount of fat that is in a haggis and in a portion of fries? Let x = fat content of a haggis y = fat content of fries The total fat content for a haggis and fries is 190 grams gives us the equation x+y = 190 The amount of fat in a haggis is 40 grams more than twice the amount of fat that is in fries gives us the equation. x = 2y + 40 Solve by the substitution method: put x = 2y + 40 into equation x+y 2y + 40 + y 3y + 40 3y y = = = = = 190 190 190 150 50 Put y = 50 into equation x+y = 190 x+ 50 = 190 x = 140 The Solution is the fat content of a haggis = 140 grams and the fat content of the fries = 50 grams Page | 15