Outside the high school program / System of equations 56 exercises (100% corrected)

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1. Solving systems E.5110 1 Consider the matrix A defined by: A = 3 − 4 1 − 1 Determine the inverse of the matrix A . 2 Consider the following system of equations : 3 x − 4 y = 1 x − y = 1 a Translate this system of equations into a matrix rela-tionship. b Deduce the set of solutions of this system. E.5111 1 Consider the following two matrices : A = 1 − 2 − 2 2 1 2 2 0 1 ; B = 1 2 − 2 2 5 − 6 − 2 − 4 5 Show that these two matrices are inverses of each other. 2 Solve the following systems of equations : x − 2 y − 2 z = 0 2 x + y + 2 z = − 9 2 x + z = − 8 ; x + 2 y − 2 z = − 2 2 x + 5 y − 6 z = − 7 − 2 x − 4 y + 5 z = 8 E.6869 We consider the matrix A defined by: A = 1 1 − 2 3 1 a Effect the calculation: 4 · A − A 2 b En deduce the expression of the inverse matrix of the matrix A . c Donner the expression of the matrix: B = 4 5 · I 2 − 1 5 · A . 2 Consider the two matrices X and Y defined by: X = x y ; Y = 3 1 Using the question 1 , resolve the system : x + y = 3 − 2 · x + 3 · y = 1 2. Problem situations E.5786 In a café, here are two orders and their bill amounts : Invoice: 4.5 e . Invoice: 10.8 e In this café, what are the prices of a croissant and a coffee? E.5787 At a coffee shop, here are two orders and their bill amounts : Invoice: 11 e . Invoice: 14.3 e In this café, what are the prices of a croissant and a can? E.5788 At a coffee shop, here are two orders and their bill amounts : Invoice A : 15.20 e . Invoice B : 10 e What are the prices, in this café, for a coffee and a can? E.5797 In a café, here are two orders and their bill amounts : Invoice 1 : 12.2 e . Invoice 2 : 13 e In this café, what are the prices of a coffee and a can? 3. Introduction E.4178 Consider the equation ( E ) with two unknowns : ( E ) : 2 x − y = 3 1 Which of the pairs below verify the equation : a (2 ; 1) b ( − 4 ; 2) c (3 ; 3) 2 Give two other couples verifying this equality. E.5492 Consider the system ( E ) of two equa-tions with two unknowns : ( E ) : 3 x − y = 5 x + y = 3 Which of the following pairs are solutions of the system ( E ) : a (1 ; − 2) b (2 ; 1) c ( − 3 ; 6) https://chingmath.fr chapExoCorrec/5110 sacados/5110 chapExoCorrec/5111 sacados/5111 chapExoCorrec/6869 sacados/6869 chapExoCorrec/5786 sacados/5786 chapExoCorrec/5787 sacados/5787 chapExoCorrec/5788 sacados/5788 chapExoCorrec/5797 sacados/5797 chapExoCorrec/4178 sacados/4178 chapExoCorrec/5492 sacados/5492
E.3758 Justify each of your answers. 1 − 2 is he solution of the inequation : 3 x +12 < 4 − 2 x ? 2 − 2 is he solution of the equation : ( x − 2)(2 x +1)=0 ? 3 − 2 is it solution of the equation : x 3 + 8 = 0 ? 4 Is the couple ( − 2 ; 1) solution of the system : 2 x + 3 y = − 1 x + 5 y = 3 4. Solving with linear combinations E.992 Jean went to the ˇ tiendita ı on his street and bought 2 cans of soda and 1 pack of popcorn for 19 pesos. The next day, he bought 2 cans and 2 packs of popcorn for 27 pesos. 1 How much does a packet of popcorn cost? 2 Deduce the price of a can of soda. 3 Check that the values found verify the statement. E.5543 A schoolboy makes two purchases : 3 pencils and 2 black pens for 10.80 pesos 1 pencil and 1 black pen for 4.80 pesos. 1 a What would have been the price of 3 pencils and 3 black pens? b Deduce the price of a black pen. 2 Determine the price of a pencil. 3 Check that the prices found verify the conditions in the statement. E.5545 Consider the system ( S ) defined by: ( S ) : 3 x + 3 y = 9 4 x + 3 y = 10 Solve the system ( S ) . 5. Resolution by substitution E.993 The distance of the ˇ Paris-Mexico ı route is double that of the ˇ Paris-Ougadougou ı route. An air hostess traveled 24 000 km in one week by doing : a round trip ˇ Paris-Mexico ı ; a round trip ˇ Paris-Ougadougou ı. Determine the distance separating Paris from these two capi-tals. E.5546 A binder costs 1.80 e de more than a notebook. Knowing that 3 binders and 2 notebooks cost 11.40 e , give the price of a binder and a notebook. 6. Solving systems E.1003 1 Solve the system : ( S ) : 3 x + 2 y = 66 x + 3 y = 57 2 Verify that, for the solution ( x ; y ) found, we have x y = 4 5 . E.1008 1 Solve the system : ( S ) : 10 x − 3 y = 35 5 x − 4 y = − 20 2 Show that the values found for x and y verify the follow-ing condition : 8 x − 5 y − 5 = 3 x + 20 y + 20 E.997 1 Solve by the method of linear combinations the following system : ( S ) : 3 x + 2 y = 5 4 x + 3 y = 3 2 Solve the following system by the substitution method : ( T ) : 3 x + y = 16 8 x − 5 y = 12 7. Resolution and modeling https://chingmath.fr chapExoCorrec/3758 sacados/3758 chapExoCorrec/992 sacados/992 chapExoCorrec/5543 sacados/5543 chapExoCorrec/5545 sacados/5545 chapExoCorrec/993 sacados/993 chapExoCorrec/5546 sacados/5546 chapExoCorrec/1003 sacados/1003 chapExoCorrec/1008 sacados/1008 chapExoCorrec/997 sacados/997
E.1010 1 Solve the system : x − y = 8 7 x + 5 y = 104 2 A library buys 7 DVDs and 5 books. The total price is 104 euros. A book costs 8 euros less than a DVD. a How much does a DVD cost? b How much does a book cost? E.1005 1 Solve the system ( S ) : x + 3 y = 2 250 2 x + y = 2750 2 For the purchase of a T-shirt and 3 caps, André paid 2 250 F . For the purchase of 2 T-shirts and 1 cap, Maeva paid 2 750 F . Determine the price of a T-shirt and a cap. Note : prices are given in Polynesian francs ( FP ). For infor-mation 1 euro is worth approximately 119.33 FP E.996 1 Solve the system : x + y = 60 10 x + 3 y = 355 2 For a flower bed, a landscaper buys a batch of 60 plants consisting of roses at 10 e pièce and irises 3 e pièce. The invoice amount for this purchase is 355 e . How many plants of each kind does he buy? E.1004 1 Solve the following system : ( S ) : x + y = 104 x − y = 8 2 Matéo and Simon, who are 8 years apart, add up their ages and find 104 years. Knowing that Matéo is the younger, calculate the age of each of these two people. E.995 1 Solve the following system : 4 x + 3 y = 206 2 x + 2 y = 114 2 At a show, the family A , consisting of 4 adults and 3 children, paid 206 euros. For the same show, the B family, made up of 2 adults and 2 children, paid 114 euros. How much will the C family pay, given that it consists of 3 adults and 2 children? E.1002 1 Solve the system : 6 x + 5 y = 57 3 x + 7 y = 55.5 2 To organize photos, a store offers two types of storage : albums and boxes. Léa buys 6 boxes and 5 albums and pays 57 euros ; Hugo buys 3 boxes and 7 albums and pays 55.50 euros. What is the price of a box? What is the price of an album? E.1006 1 Solve the system : 6 x + 5 y = 25 2 x + 3 y = 11 2 Peter and Julius buy goldfish and yellowfish from the same specialist store. To buy 6 goldfish and 5 yellowfish, Peter spends 25 euros. To buy 2 goldfish and 3 yellow fish, Jules spends 11 euros. a What is the price of a goldfish? b What is the price of a yellow fish? The procedure followed will be explained on the copy. E.1009 1 Solve the following system : 8 x + 3 y = 39.5 7 x + 9 y = 50.5 2 A one-hour sea tour is offered to two groups of tourists. The first group, comprising 8 adults and 3 children, pays 39.50 euros. The second, consisting of 7 adults and 9 children, pays 50.50 euros. How much does a ticket cost for an adult? For a child? E.1011 1 Resolve the system : 3 x + 2 y = 50.30 x + 3 y = 32.75 2 At the nursery ˇFruitfleurı, a customer buys 3 orange trees and 2 lemon trees for 50.30 euros. Another cus-tomer pays 32.75 euros for 1 orange tree and 3 lemon trees. Let x be the price of an orange tree and y the price of a lemon tree. a Write a system of two equations that represents the problem. b Calculate the price of an orange tree and the price of a lemon tree. E.2510 1 Solve the following system : 4 x + 3 y = 206 2 x + 2 y = 114 2 At a show, the family A , consisting of 4 adults and 3 children, paid 206 euros. For the same show, the B family, made up of 2 adults and 2 children, paid 114 euros. How much will the C family pay, given that it consists of 3 adults and 2 children? 8. Systems of equations with relative and rational numbers E.5549 Consider the following system : ( S ) : 2 x + 3 y = 10 5 x + 10 y = 20 Solve the system ( S ) . https://chingmath.fr chapExoCorrec/1010 sacados/1010 Bordeaux - Septembre 2004 chapExoCorrec/1005 sacados/1005 chapExoCorrec/996 sacados/996 chapExoCorrec/1004 sacados/1004 chapExoCorrec/995 sacados/995 chapExoCorrec/1002 sacados/1002 Nice, Aix-Marseille, Corse, Montpellier, Toulouse - Juin 2006 chapExoCorrec/1006 sacados/1006 chapExoCorrec/1009 sacados/1009 Paris - Juin 2006 chapExoCorrec/1011 sacados/1011 Guadeloupe - Septembre 2005 chapExoCorrec/2510 sacados/2510 Juin 2003 - Dijon chapExoCorrec/5549 sacados/5549
E.5550 Consider the system ( S ) of equa-tions : x − 3 y = 8 4 x + y = − 7 Solve the system ( S ) . E.5551 Consider the system ( S ) of equa-tions : 2 x − 3 y = 1 2 x + y = 0 Solve the system ( S ) . E.5552 Consider the system ( S ) of equa-tions : 2 x − 3 y = 3 4 x + 3 y = − 2 Solve the system ( S ) . 9. Modeling and solving E.2467 A zoo offers two entrance fees : one for adults and another for children. A group of four children and one adult pays 22 euros. This data can be expressed by the equation with two un-knowns : 4 x + y = 22 noted ( E 1 ) . 1 What does the unknown x represent and what does the unknown y represent in this equation? Another group consisting of six children and three adults pays 42 euros. 2 Translate this information into a second equation noted ( E 2 ) , depending on the two unknowns x and y . 3 Solve the system consisting of the two equations ( E 1 ) and ( E 2 ) above. 4 What is the price of an entry for a child and what is the price of an entry for an adult? E.2471 A first-grader runs errands for her classmates : the first time, she buys 5 pencils and 2 erasers for 10.90 euros ; the second time, she buys 8 pencils and 3 erasers for 17.20 euros. Using a system of equations, help the CP student find the price of each item. E.842 At a nursery, Moetia bought three orange and two lemon trees for 14 000 F and Orai paid 13 500 F for two orange and three lemon trees. Using a system of two equations with two unknowns, deter-mine the price of one orange and one lemon tree. E.2472 A hardware store customer looks at the following two receipts : 6 kilograms of varnish and 4 liters of wax cost 95 e . 3 kilograms of varnish and 3 liters of wax cost 55.50 e . Determine the price of a kilogram of varnish and the price of a liter of wax. E.994 In a restaurant, a couple order 1 pizza and 2 fruit juices and pay 11 euros. At the next table, friends order 5 pizzas and 9 fruit juices and pay 53 euros. All the pizzas have the same price and all the fruit juices have the same price. We call x the price in euros of a pizza and y the price in euros of a fruit juice. 1 Write a system of equations translating the data. 2 Calculate the price of a pizza and the price of a fruit juice. E.4179 On the outward journey, a train consists of two identical locomotives and ten tank cars of the same model, and this train is then 152 m long. After emptying the contents of all the tank cars, we unhook one locomotive and add two empty tank cars. After these changes, the train thus formed measures 160 long. We’re looking for the length x of a locomotive and the length y of a tank car. 1 Write a system of two equations with two unknowns rep-resenting the situation. 2 Solve the system : x + 5 y = 76 x + 12 y = 160 3 Deduct the length in meters of a locomotive and that of a tank car. 10. Graphic resolution E.998 1 Consider the following functions : the linear function f : x ↦−→ 2.25 x affine function g : x ↦−→ 2 x + 30 On a sheet of graph paper, draw, in a frame of refer-ence ( O ; I ; J ) the straight lines ( D f ) and ( D g ) which https://chingmath.fr chapExoCorrec/5550 sacados/5550 chapExoCorrec/5551 sacados/5551 chapExoCorrec/5552 sacados/5552 chapExoCorrec/2467 sacados/2467 chapExoCorrec/2471 sacados/2471 chapExoCorrec/842 sacados/842 chapExoCorrec/2472 sacados/2472 chapExoCorrec/994 sacados/994 Groupe Nord - Juin 2003 - 3 points chapExoCorrec/4179 sacados/4179 Asie Juin 2009 chapExoCorrec/998 sacados/998
-4-3-2-1234I-3-2-123JO(dAB represent the functions f and g respectively. Place the origin of the reference frame at the bottom left of the sheet of graph paper. We’ll take 1cm for 10 units on the abscissa and 1 cm for 15 units on the ordinate. 2 Graphically solve the following system : y = 2.25 · x prix de Transport Express y = 2 · x + 30 prix d’Inter Transport 3 Using a graphical reading made in question 3. , specify in which case the manufacturer should choose the company Inter Transport. E.999 ok 11. Functions and systems of equations E.975 In the reference frame O ; I ; J be-low, is given the straight line ( d ) representative of an affine function f . The algebraic expression of the affine function f is of the form : f ( x ) = a × x + b The aim of the exercise is to determine the values of the two numbers a and b . 1 Give the coordinates of points A and B . First method : 2 Using the points A and B , determine the directing coef-ficient of the affine function f . 3 Using the coordinates of the point A or the point B , de-termine the value of the number b . Write the complete expression of the function f . Second method : 4 Justify that the two numbers a and b verify the two equa-tions below : − 2 a + b = − 1 ; 2 a + b = 2 5 Solve the system of equations : − 2 a + b = − 1 2 a + b = 2 Write the complete expression of the function f . E.2651 The plane is provided with an orthonormal reference frame ( O ; I ; J ) . The unit of length is the centimeter. 1 Let f be an affine function verifying: f (4) = − 2 ; f (0) = 6 a Determine the expression of the function f . b Plot the graphical representation of the function f . 2 Let g be the affine function defined by: g ( x )= 1 2 x +1 . a Construct the straight line ( d ) graphically representing the function g . b Show that C ( − 4 ; − 1) belongs to ( d ) and place the point C . 3 a Solve the following system of equations by calcula-tion : y = − 2 x + 6 y = 1 2 x + 1 b Explain how the result can be found graphically. 12. A little further on E.1001 Find two numbers, knowing their sum 2003 and their difference 51 https://chingmath.fr chapExoCorrec/999 sacados/999 chapExoCorrec/975 sacados/975 -4-3-2-1234I-3-2-123JO(dAB chapExoCorrec/2651 sacados/2651 Extrait du probleme - polynesie - Septembre 2005 chapExoCorrec/1001 sacados/1001
verremétal Bijou no1Bijou no2Bijou no3 E.4335 Jewelry is made from trian-gles, all of which have the same shape. Some triangles are made of glass, others of metal. Three examples are shown below. The glass triangles are shown in white ; the metal triangles are shown in grey. All metal triangles have the same price ; all glass triangles have the same price. The jewel n o 1 costs 11 e ; the jewel n o 2 costs 9.10 e . How much does jewelry n o 3 cost? If the work is not completed, still leave a trace of the search. It will be taken into account in the grading. 13. System of linear equations E.551 1 Solve the system : x − y = 8 7 x + 5 y = 104 2 A library buys 7 DVDs and 5 books. The total price is 104 euros. A book costs 8 euros less than a DVD. a How much does a DVD cost? b What is the price of a book? E.4720 1 Solve the following system : 8 x + 3 y = 39.5 7 x + 9 y = 50.5 2 A one-hour sea tour is offered to two groups of tourists. The first group, consisting of 8 adults and 3 children, pays 39.50 euros. The second, consisting of 7 adults and 9 children, pays 50.50 euros. How much does a ticket cost for an adult? For a child? E.4721 1 Solve the following system : 4 x + 3 y = 206 2 x + 2 y = 114 2 At a show, the family A , consisting of 4 adults and 3 children, paid 206 euros. For the same show, the B family, made up of 2 adults and 2 children, paid 114 euros. How much will the C family pay, given that it consists of 3 adults and 2 children? E.539 A student buys two pens and a note-book from a stationery store for a total of 3.5 e . He returns to this store a second time to buy for 1 pen and 3 notebooks, of the same model and price, for an overall cost of 6.75 e . Determine the price of a pen and a notebook in this stationery store. E.549 On the Lyon-Marseille train line: A TGV departs from Lyon bound for Marseille at 9 h 30 and travels at a constant speed of 300 km = h . A long-distance train departs from Marseille for Lyon at 9 h and travels at a constant speed of 150 km = h . At what time will the two trains pass each other? (The dis-tance between Lyon and Marseille is 255 km ) Hint: Let x denote the elapsed time in hours starting at 9 h 30 . Let L ( x ) be the distance traveled by the train departing from Lyon and arriving in Marseille at time x . Let M ( x ) be the distance remaining at time x for the train departing from Marseille and heading to Lyon to travel. E.4725 1 Consider the following system of linear equations : ( S 1 ) : 45 · x + 36 · y + 24 = 0 75 · x + 60 · y + 40 = 0 Determine the number of solutions of this system. Justify your approach. 2 Consider the following system of linear equations : ( S 2 ) : 9 x − 6 y − 18 = 0 21 x − 14 y + 42 = 0 Determine the number of solutions of this system. Justify your approach. 3 Consider the following system of linear equations : ( S 3 ) : 2 · x + 3 · y − 1 = 0 − 3 · x − 5 · y + 3 = 0 a Determine the number of solutions in this system. Jus-tify your approach. b Determine the solution pair of this system. 14. Projected orthogonal https://chingmath.fr chapExoCorrec/4335 sacados/4335 verremétal Bijou no1Bijou no2Bijou no3 chapExoCorrec/551 sacados/551 chapExoCorrec/4720 sacados/4720 chapExoCorrec/4721 sacados/4721 chapExoCorrec/539 sacados/539 chapExoCorrec/549 sacados/549 chapExoCorrec/4725 sacados/4725
(d(dA verremétal Bijou no1Bijou no2Bijou no3 E.8551 In the grid below, consider the two straight lines ( d ) and ( d ) and a point A : 1 Place the point M orthogonally projected from the point A onto the line ( d ) . 2 Place the point N orthogonally projected from the point A onto the line ( d ) . 15. Unclassified financial years E.5693 Arthur empties his piggy bank and finds that he has 21 bills. He has 5 e bills and 10 e bills for a total sum of 125 e . How many bills of each kind does he have? If the work is not finished, leave a trace of the search anyway. It will be taken into account in the assessment. E.5789 Jewelry is made from triangles, all of which have the same shape. Some triangles are made of glass, others of metal. Three examples are shown below. The glass triangles are shown in white ; the metal triangles are shown in grey. All metal triangles have the same price ; all glass triangles have the same price. The jewel n o 1 costs 11 e ; the jewel n o 2 costs 9.10 e . How much does jewelry n o 3 cost? If the work is not completed, still leave a trace of the search. It will be taken into account in the grading. E.2678 1 − 2 is the solution to the inequation : 3 x +12 < 4 − 2 x ? Justify. 2 − 2 is a solution of the equation : ( x − 2)(2 x +1)=0 ? Justify. 3 − 2 is a solution of the equation : x 3 +8=0 ? Justify. 4 Is the couple ( − 2 ; 1) solution of the system : 2 x + 3 y = − 1 x + 5 y = 3 E.1843 1 Let f be the affine function whose representative line passes through the points A ( − 3 ; 1) and B (4 ; − 4) . The expression of the function f is given by: f ( x ) = a × x + b where a; b ∈ R a Justify that the integers a and b verify the following system of equations : 1 = − 3 a + b − 4 = 4 a + b b Solve this system to obtain the algebraic expression of the function f . 2 Consider the affine function g whose representation passes through the points C ( − 1 ; 1) and D (2 ; − 3) : a Justify that the directing coefficient of the function g has value − 4 3 . b Determine the algebraic expression of the function g . https://chingmath.fr chapExoCorrec/8551 sacados/8551 (d(dA chapExoCorrec/5693 sacados/5693 chapExoCorrec/5789 sacados/5789 verremétal Bijou no1Bijou no2Bijou no3 chapExoCorrec/2678 sacados/2678 chapExoCorrec/1843 sacados/1843
ABCMN E.2169 1 Solve the system : x − 3 y = 0 x − y = 4.5 2 In the triangle ABC below, we give : AB =6 cm ; BC =9 cm M is the point of [ AB ] such that : AM =2 cm . The line parallel to ( BC ) passing through M cuts [ AC ] at N . a Calculate MN . b Give the value of AN AC 3 Assume [ NC ] measures 4.5 cm and set AN = y and AC = x . a Establish the equalities: x − y =4.5 ; x − 3 y =0 b Calculate AN and AC , possibly using questions 1 and 3 a . Note: calculations are possible even if questions 1 and 3 a have not been processed. https://chingmath.fr chapExoCorrec/2169 sacados/2169 Limoges - 2000 - 5 points ABCMN