Grade 10 Linear Equations in Two Variables

Transcription

1 ID : ae-10-linear-equations-in-two-variables [1] Grade 10 Linear Equations in Two Variables For more such worksheets visit Answer t he quest ions (1) 12 chairs and 10 tables cost Dhs4700 and 9 chairs and 10 tables cost Dhs4400. Find the cost of one chair and one table separately. (2) The f ather's age is 5 times his daughter's age. Af ter 26 years, the age of the f ather will be 7/4 times his daughter's age. Find the present ages of the daughter and the f ather. (3) Which of the f ollowing equations has a unique solution? 3x + 2y + 5 0, 15x + 10y x + 2y + 5 0, 15x + 10y x + 2y + 5 0, 18x + 12y x + 2y , 15x + 4y (4) Givon and Jabir have some toys. Givon says to Jabir, "f f you give me 18 of your toys, I will have twice the number of toys lef t with you". Jabir replies, "if you give me 18 of your toys I will have the same number of toys lef t with you". Find the the number of toys with Givon and Kareem separately. (5) 10 chairs and 7 tables cost Dhs3550 and 5 chairs and 7 tables cost Dhs2300. Find the cost of 2 chairs and 6 tables. (6) Bibi has only Dhs20 and Dhs10 notes with her. If the total number of notes that she has is 17 and the amount of money with her is Dhs230, then the number of Dhs20 and Dhs10 notes are, respectively Choose correct answer(s) f rom given choice (7) Which of the f ollowing equations has an inf inite number of solutions? a. 3x + 4y + 1 0, 12x + 16y b. 3x + 4y + 1 0, 9x + 12y c. 3x + 4y + 1 0, 9x + 12y d. 3x + 4y + 5 0, 9x + 5y (8) If f ollowing pair of equations have inf initely many solutions, f ind the value of a and b. -10x - 6y -2 (2a + b)x + (2a + 3b)y 1 a. a 3, b -2 b. a 4, b -1 c. a 2, b -1 d. a 3, b -1 (9) Three years ago Farid was f our times older than his daughter. Af ter three years, Farid will be 4 years more than two times the age of his daughter. Find the present age of Farid and his daughter. a. 24 and 9 years b. 22 and 7 years c. 23 and 8 years d. 21 and 6 years

2 ID : ae-10-linear-equations-in-two-variables [2] (10) Aiesha is three times as old as her son. Af ter 19 years, Aiesha will be two times as hold as her son. Find the current age of Aiesha and her son. a. 57 and 19 years b. 63 and 21 years c. 60 and 20 years d. 54 and 18 years (11) There are two numbers. If f our times the larger of two numbers is divided by the smaller one, we get 6 as quotient and 10 as remainder. If six times the smaller of two numbers is divided by the larger one, we get 3 as quotient and 21 as remainder. Find the numbers. a. 29 and 18 b. 31 and 19 c. 29 and 20 d. 33 and 18 (12) The pair of equations -3x + 2y and 9x - 6y have a. no solution b. exactly two solutions c. inf initely many solutions d. a unique solution (13) For which value of a, f ollowing pair of linear equations have no solution. a x + y a 2 and x + a y 1 a. 0 b. -1 c. 1 d. 2 (14) The pair of equations x 35 and y -25 graphically represents lines which are a. coincident b. intersecting at (35, -25) c. intersecting at (-25, 35) d. parallel (15) If a pair of linear equations is consistent, then the lines will be a. always coincident b. always intersecting c. intersecting or coincident d. parallel 2016 Edugain ( All Rights Reserved Many more such worksheets can be generated at

3 Answers ID : ae-10-linear-equations-in-two-variables [3] (1) Dhs100 and Dhs350 Let the cost of a chair and table be Dhs x and Dhs y respectively. It is given that 12 x + 10 y x + 10 y 4400 On multiplying f irst equation by 1 and second equation by 1 12 x + 10 y x + 10 y 4400 Subtract second equation f rom f irst equation 1 > 3 x 300 > x (300)/(3) Dhs100 Substitute this value in f irst equation > y 4700 > 10 y > y (3500)/(10) Dhs350 (2) 6 years and 30 year Let's assume, the present of the daughter be 'x'. It is given that the f ather's age is 5 times his daughter's age, theref ore the present age of the f ather is 5x. Af ter 26 years, the age of the daughter x + 26, the age of the f ather 5x + 26 It is also given that af ter 26 years, the age of the f ather will be 7/4 times his daughter's age. Theref ore, 5x /4(x + 26) 5x - 7/4(x) 7/4(26) - 26 x 6 Theref ore, the present age of the daughter 6 years, the present age of the f ather 5x years.

4 (3) 3x + 2y , 15x + 4y ID : ae-10-linear-equations-in-two-variables [4] Equations a 1 x + b 1 y + c 1 0 and a 1 x + b 1 y + c 1 0 have unique solution if, a 1 a 2 b 1 b 2 If we try this condition on given f our equations, it is only satisf ied by pair 3x + 2y , 15x + 4y , since Theref ore answer is 3x + 2y , 15x + 4y (4) 126 and 90 items Let the number of toys with Givon and Jabir be x and y. It is given that x (y - 18) x - 18 y + 18 On subtracting second equation f rom f irst equation (y - 18) - (y + 18) 36 y - 54 y 5 18 y 90 Substitute value of y in f irst equation, x (90-18) x 126

5 (5) Dhs1400 ID : ae-10-linear-equations-in-two-variables [5] Let the cost of a chair and table be Dhs x and Dhs y respectively. It is given that 10 x + 7 y x + 7 y 2300 On multiplying f irst equation by 1 and second equation by 1 10 x + 7 y x + 7 y 2300 Subtract second equation f rom f irst equation 1 > 5 x 1250 > x (1250)/(5) Dhs250 Substitute this value in f irst equation > y 3550 > 7 y > y (1050)/(7) Dhs150 Step 5 Price of 2 chairs and 6 tables > 2x + 6y > Dhs1400

6 (6) 6 and 11 ID : ae-10-linear-equations-in-two-variables [6] Lets assume number Dhs20 notes x number Dhs10 notes y Is is given that total number of notes is 17, x + y (1) It is also given that total amount of money is Dhs230, (20)x + (10)y (2) Now multiply Eq. (1) by 20 and subtract is f rom Eq (2), (20-20)x + (10-20)y ( ) (-10)y -110 y (-110)/(-10) y 11 Step 5 Now substiture value of y in Eq. (1), x x x 6 Step 6 Theref ore number of Dhs20 notes 6, and number of Dhs10 notes 11 (7) b. 3x + 4y + 1 0, 9x + 12y Equations a 1 x + b 1 y + c 1 0 and a 1 x + b 1 y + c 1 0 have inf initely many solutions if, a 1 a 2 b 1 b 2 c 1 c 2 If we try this condition on given f our set of equations, it is only satisf ied by pair 3x + 4y + 1 0, 9x + 12y + 3 0, since Theref ore answer is 3x + 4y + 1 0, 9x + 12y + 3 0

7 (8) d. a 3, b -1 ID : ae-10-linear-equations-in-two-variables [7] From the equations, -10x - 6y -2 and (2a + b)x + (2a + 3b)y 1, we notice that, a 1-10, b 1-6, c 1-2 and a 2 2a + b, b 2 2a + 3b, c 2 1 It is given that, the equations -10x - 6y -2 and (2a + b)x + (2a + 3b)y 1 have inf initely many solutions. Theref ore, a 1 a 2 b 1 b 2 c 1 c a + b -6 2a + 3b -2 1 On comparing f irst two terms, a + b 2a + 3b By cross multiplying both sides, we get, -10(2a + 3b) -6(2a + b) -20a -30b -12a -6b -8a 24b a 24 b -8 a -3 b Now, lets compare f irst and last term, a + b 1 By cross multiplying both sides, we get, -10(1) -2(2a + b) -10-4a -2b On substituting a -3 b, -10-4(-3 b) -2b b -1 Step 5 a -3 b Step 6 Theref ore, the value of a 3, and the value of b -1.

8 ID : ae-10-linear-equations-in-two-variables [8] (9) c. 23 and 8 years Let x and y be the current age of Farid and his daughter respectively. It is given that, Three years ago Farid was f our times older than his daughter (x - 3) 4 (y - 3) x - 4y -9 (1) It is also given that, af ter three years Farid will be 4 years more than two times the age of his daughter (x + 3) 2 (y + 3) + 4 x - 2y 7 (2) On subtracting eq. 1 f rom 2 2y 16 y 8 years On substituting value of y in eq. 1 x - 4(8) -9 x 23 years Step 5 Thref ore, present age of Farid x 23 years present age of his daughter y 8 years

9 (11) b. 31 and 19 ID : ae-10-linear-equations-in-two-variables [9] Let the larger number be x and smaller number be y We know that Dividend (Divisor Quotient) + Remainder T heref ore we can write relationship provided as, 4x 6y + 10 > 4x - 6y y 3x + 21 > 3x - 6y On solving these two equations we get x 31 and y 19 (12) c. inf initely many solutions Lets calculate ratio of coef f icients of these equations a 1-3 a 2 9 b 1 b 2 c 1 c On comparing these ratios we observe a 1 a 2 b 1 b 2 c 1 c 2 T heref ore these equations have inf initely many solutions (13) b. -1

10 (14) b. intersecting at (35, -25) ID : ae-10-linear-equations-in-two-variables [10] Vertical line in f ollowing graph represents equation x 35, while vertical line in graph represents equation y -25 It can be seen in graph that two lines intersects at point (35, -25) (15) c. intersecting or coincident