Page 1 Exercise 3F 1. Write the number of solutions of the following pair of linear equations: x + 2y -8=0, 2x + 4y = 16 Sol: The given equations are x + 2y 2x + 4y Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where a1 = 1, b1 = 2, c1 = -8, a2 = 2, b2 = 4 and c2 = -18 Page 2 Exercise 3F 1. Write the number of solutions of the following pair of linear equations: x + 2y -8=0, 2x + 4y = 16 Sol: The given equations are x + 2y 2x + 4y Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where a1 = 1, b1 = 2, c1 = -8, a2 = 2, b2 = 4 and c2 = -18 Now = = = = = = = = Thus, the pair of linear equations are coincident and therefore has infinitely many solutions. 2. Find the value of k for which the system of linear equations has an infinite number of solutions. 2x + 3y 7 = 0, (k 1)x + (k + 2)y=3k Sol: The given equations are 2x + 3y (k 1)x + (k + 2)y Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where a1 = 2, b1 = 3, c1 = -7, a2 = k 1, b2 = k + 2 and c2 = -3k For the given pair of linear equations to have infinitely many solutions, we must have = = = = = , = and = 2(k + 2) = 3(k 1), 9k = 7k + 14 and 6k = 7k 7 k = 7, k = 7 and k = 7 Hence, k = 7. 3. Find the value of k for which the system of linear equations has an infinite number of solutions. 10x + 5y (k 5) = 0, 20x + 10y k = 0. Sol: The given pair of linear equations are 10x + 5y (k 20x + 10y Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where a1 = 10, b1 = 5, c1 = -(k 5), a2 = 20, b2 = 10 and c2 = -k For the given pair of linear equations to have infinitely many solutions, we must have Page 3 Exercise 3F 1. Write the number of solutions of the following pair of linear equations: x + 2y -8=0, 2x + 4y = 16 Sol: The given equations are x + 2y 2x + 4y Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where a1 = 1, b1 = 2, c1 = -8, a2 = 2, b2 = 4 and c2 = -18 Now = = = = = = = = Thus, the pair of linear equations are coincident and therefore has infinitely many solutions. 2. Find the value of k for which the system of linear equations has an infinite number of solutions. 2x + 3y 7 = 0, (k 1)x + (k + 2)y=3k Sol: The given equations are 2x + 3y (k 1)x + (k + 2)y Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where a1 = 2, b1 = 3, c1 = -7, a2 = k 1, b2 = k + 2 and c2 = -3k For the given pair of linear equations to have infinitely many solutions, we must have = = = = = , = and = 2(k + 2) = 3(k 1), 9k = 7k + 14 and 6k = 7k 7 k = 7, k = 7 and k = 7 Hence, k = 7. 3. Find the value of k for which the system of linear equations has an infinite number of solutions. 10x + 5y (k 5) = 0, 20x + 10y k = 0. Sol: The given pair of linear equations are 10x + 5y (k 20x + 10y Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where a1 = 10, b1 = 5, c1 = -(k 5), a2 = 20, b2 = 10 and c2 = -k For the given pair of linear equations to have infinitely many solutions, we must have = = = = = 2k 10 = k k = 10 Hence, k = 10. 4. Find the value of k for which the system of linear equations has an infinite number of solutions. 2x + 3y=9, 6x + (k 2)y =(3k 2 Sol: The given pair of linear equations are 2x + 3y 9 = 0 6x + (k 2)y (3k Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where a1 = 2, b1 = 3, c1 = -9, a2 = 6, b2 = k 2 and c2 = -(3k 2) For the given pair of linear equations to have infinitely many solutions, we must have = = = , k = 11, k = 11, 3(3k 2) Hence, k = 11. 5. Write the number of solutions of the following pair of linear equations: x + 3y 4 = 0, 2x + 6y 7 = 0. Sol: The given pair of linear equations are x + 3y 2x + 6y Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where a1 = 1, b1 = 3, c1 = -4, a2 = 2, b2 = 6 and c2 = -7 Now = = = Page 4 Exercise 3F 1. Write the number of solutions of the following pair of linear equations: x + 2y -8=0, 2x + 4y = 16 Sol: The given equations are x + 2y 2x + 4y Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where a1 = 1, b1 = 2, c1 = -8, a2 = 2, b2 = 4 and c2 = -18 Now = = = = = = = = Thus, the pair of linear equations are coincident and therefore has infinitely many solutions. 2. Find the value of k for which the system of linear equations has an infinite number of solutions. 2x + 3y 7 = 0, (k 1)x + (k + 2)y=3k Sol: The given equations are 2x + 3y (k 1)x + (k + 2)y Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where a1 = 2, b1 = 3, c1 = -7, a2 = k 1, b2 = k + 2 and c2 = -3k For the given pair of linear equations to have infinitely many solutions, we must have = = = = = , = and = 2(k + 2) = 3(k 1), 9k = 7k + 14 and 6k = 7k 7 k = 7, k = 7 and k = 7 Hence, k = 7. 3. Find the value of k for which the system of linear equations has an infinite number of solutions. 10x + 5y (k 5) = 0, 20x + 10y k = 0. Sol: The given pair of linear equations are 10x + 5y (k 20x + 10y Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where a1 = 10, b1 = 5, c1 = -(k 5), a2 = 20, b2 = 10 and c2 = -k For the given pair of linear equations to have infinitely many solutions, we must have = = = = = 2k 10 = k k = 10 Hence, k = 10. 4. Find the value of k for which the system of linear equations has an infinite number of solutions. 2x + 3y=9, 6x + (k 2)y =(3k 2 Sol: The given pair of linear equations are 2x + 3y 9 = 0 6x + (k 2)y (3k Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where a1 = 2, b1 = 3, c1 = -9, a2 = 6, b2 = k 2 and c2 = -(3k 2) For the given pair of linear equations to have infinitely many solutions, we must have = = = , k = 11, k = 11, 3(3k 2) Hence, k = 11. 5. Write the number of solutions of the following pair of linear equations: x + 3y 4 = 0, 2x + 6y 7 = 0. Sol: The given pair of linear equations are x + 3y 2x + 6y Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where a1 = 1, b1 = 3, c1 = -4, a2 = 2, b2 = 6 and c2 = -7 Now = = = = = = Thus, the pair of the given linear equations has no solution. 6. Find the values of k for which the system of equations 3x + ky = 0, 2x y = 0 has a unique solution. Sol: The given pair of linear equations are 2x Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where a1 = 3, b1 = k, c1 = 0, a2 = 2, b2 = -1 and c2 = 0 For the system to have a unique solution, we must have = Hence, . 7. The difference of two numbers is 5 and the difference between their squares is 65. Find the numbers. Sol: Let the numbers be x and y, where x y. Then as per the question x x 2 y 2 Dividing (ii) by (i), we get = = 13 Now, adding (i) and (ii), we have 2x = 18 x = 9 Substituting x = 9 in (iii), we have 9 + y = 13 y = 4 Hence, the numbers are 9 and 4. Page 5 Exercise 3F 1. Write the number of solutions of the following pair of linear equations: x + 2y -8=0, 2x + 4y = 16 Sol: The given equations are x + 2y 2x + 4y Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where a1 = 1, b1 = 2, c1 = -8, a2 = 2, b2 = 4 and c2 = -18 Now = = = = = = = = Thus, the pair of linear equations are coincident and therefore has infinitely many solutions. 2. Find the value of k for which the system of linear equations has an infinite number of solutions. 2x + 3y 7 = 0, (k 1)x + (k + 2)y=3k Sol: The given equations are 2x + 3y (k 1)x + (k + 2)y Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where a1 = 2, b1 = 3, c1 = -7, a2 = k 1, b2 = k + 2 and c2 = -3k For the given pair of linear equations to have infinitely many solutions, we must have = = = = = , = and = 2(k + 2) = 3(k 1), 9k = 7k + 14 and 6k = 7k 7 k = 7, k = 7 and k = 7 Hence, k = 7. 3. Find the value of k for which the system of linear equations has an infinite number of solutions. 10x + 5y (k 5) = 0, 20x + 10y k = 0. Sol: The given pair of linear equations are 10x + 5y (k 20x + 10y Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where a1 = 10, b1 = 5, c1 = -(k 5), a2 = 20, b2 = 10 and c2 = -k For the given pair of linear equations to have infinitely many solutions, we must have = = = = = 2k 10 = k k = 10 Hence, k = 10. 4. Find the value of k for which the system of linear equations has an infinite number of solutions. 2x + 3y=9, 6x + (k 2)y =(3k 2 Sol: The given pair of linear equations are 2x + 3y 9 = 0 6x + (k 2)y (3k Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where a1 = 2, b1 = 3, c1 = -9, a2 = 6, b2 = k 2 and c2 = -(3k 2) For the given pair of linear equations to have infinitely many solutions, we must have = = = , k = 11, k = 11, 3(3k 2) Hence, k = 11. 5. Write the number of solutions of the following pair of linear equations: x + 3y 4 = 0, 2x + 6y 7 = 0. Sol: The given pair of linear equations are x + 3y 2x + 6y Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where a1 = 1, b1 = 3, c1 = -4, a2 = 2, b2 = 6 and c2 = -7 Now = = = = = = Thus, the pair of the given linear equations has no solution. 6. Find the values of k for which the system of equations 3x + ky = 0, 2x y = 0 has a unique solution. Sol: The given pair of linear equations are 2x Which is of the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where a1 = 3, b1 = k, c1 = 0, a2 = 2, b2 = -1 and c2 = 0 For the system to have a unique solution, we must have = Hence, . 7. The difference of two numbers is 5 and the difference between their squares is 65. Find the numbers. Sol: Let the numbers be x and y, where x y. Then as per the question x x 2 y 2 Dividing (ii) by (i), we get = = 13 Now, adding (i) and (ii), we have 2x = 18 x = 9 Substituting x = 9 in (iii), we have 9 + y = 13 y = 4 Hence, the numbers are 9 and 4. 8. The cost of 5 pens and 8 pencils together cost Rs. 120 while 8 pens and 5 pencils together cost Rs. 153. Find the cost of a 1 pen and that of a 1pencil. Sol: Then as per the question 5x + 8y = 120 Adding (i) and (ii), we get 13x + 13y = 273 Subtracting (i) from (ii), we get 3x 3y = 33 x Now, adding (iii) and (iv), we get 2x = 32 x = 16 Substituting x = 16 in (iii), we have 16 + y = 21 y = 5 9. The sum of two numbers is 80. The larger number exceeds four times the smaller one by 5. Find the numbers. Sol: Let the larger number be x and the smaller number be y. Then as per the question x = 4y + 5 x Subtracting (ii) from (i), we get 5y = 75 y = 15 Now, putting y = 15 in (i), we have x + 15 = 80 x = 65 Hence, the numbers are 65 and 15. 10. A number consists of two digits whose sum is 10. If 18 is subtracted form the number, its digits are reversed. Find the number. Sol: Let the ones digit and tens digit be x and y respectively. Then as per the questionRead More

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