Question 1:
Which of the following pairs of linear equations has unique solution, no solution or infinitely many solutions? In case there is a unique solution, find it by using cross multiplication method.
Therefore, the given sets of lines are parallel to each other. Therefore, they will not intersect each other and thus, there will not be any solution for these equations.
Therefore, they will intersect each other at a unique point and thus, there will be a unique solution for these equations.
By cross-multiplication method,
∴ x = 2, y = 1
Therefore, the given sets of lines will be overlapping each other i.e., the lines will be coincident to each other and thus, there are infinite solutions possible for these equations.
Therefore, they will intersect each other at a unique point and thus, there will be a unique solution for these equations.
By cross-multiplication,
∴
Question 2:
(i) For which values of a and b will the following pair of linear equations have an
infinite number of solutions?
(ii) For which value of k will the following pair of linear equations have no solution?
For infinitely many solutions,
Subtracting (1) from (2), we obtain
Substituting this in equation (2), we obtain
Hence, a = 5 and b = 1 are the values for which the given equations give infinitely many solution
For no solution,
Hence, for k = 2, the given equation has no solution.
Question 3:
Solve the following pair of linear equations by the substitution and cross-multiplication methods:
From equation (ii), we obtain
Substituting this value in equation (i), we obtain
Substituting this value in equation (ii), we obtain
Hence,
Question 4:
Form the pair of linear equations in the following problems and find their solutions (if they exist) by any algebraic method:
(i)A part of monthly hostel charges is fixed and the remaining depends on the number of days one has taken food in the mess. When a student A takes food for 20 days she has to pay Rs 1000 as hostel charges whereas a student B, who takes food for 26 days, pays Rs 1180 as hostel charges. Find the fixed charges and the cost of food per day.
(ii)A fraction becomes
when 1 is subtracted from the numerator and it becomes
when 8 is added to its denominator. Find the fraction.
(iii)Yash scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect answer, then Yash would have scored 50 marks. How many questions were there in the test?
(iv) Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds, they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?
(v)The area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square units. Find the dimensions of the rectangle.
(i)Let x be the fixed charge of the food and y be the charge for food per day.
According to the given information,
Subtracting equation (1) from equation (2), we obtain
Substituting this value in equation (1), we obtain
Hence, fixed charge = Rs 400
And charge per day = Rs 30
(ii)Let the fraction be.
According to the given information,
Subtracting equation (1) from equation (2), we obtain
Putting this value in equation (1), we obtain
Hence, the fraction is.
(iii)Let the number of right answers and wrong answers be x and y
respectively.
According to the given information,
Subtracting equation (2) from equation (1), we obtain
x = 15 (3)
Substituting this in equation (2), we obtain
Therefore, number of right answers = 15
And number of wrong answers = 5
Total number of questions = 20
(iv)Let the speed of 1st car and 2nd car be u km/h and v km/h.
Respective speed of both cars while they are travelling in same direction =)
km/h
(
Respective speed of both cars while they are travelling in opposite directions i.e., travelling towards each other = (
) km/h
According to the given information,
Adding both the equations, we obtain
Substituting this value in equation (2), we obtain
v = 40 km/h
Hence, speed of one car = 60 km/h and speed of other car = 40 km/h
(v) Let length and breadth of rectangle be x unit and y unit respectively.
Area = xy
According to the question,
By cross-multiplication method, we obtain
Hence, the length and breadth of the rectangle are 17 units and 9 units respectively.
According to the given information,
Subtracting equation (1) from equation (2), we obtain
Substituting this value in equation (1), we obtain
Hence, fixed charge = Rs 400
And charge per day = Rs 30
(ii)Let the fraction be.
According to the given information,
Subtracting equation (1) from equation (2), we obtain
Putting this value in equation (1), we obtain
Hence, the fraction is.
(iii)Let the number of right answers and wrong answers be x and y
respectively.
According to the given information,
Subtracting equation (2) from equation (1), we obtain
x = 15 (3)
Substituting this in equation (2), we obtain
Therefore, number of right answers = 15
And number of wrong answers = 5
Total number of questions = 20
(iv)Let the speed of 1st car and 2nd car be u km/h and v km/h.
Respective speed of both cars while they are travelling in same direction =)
Respective speed of both cars while they are travelling in opposite directions i.e., travelling towards each other = (
According to the given information,
Adding both the equations, we obtain
Substituting this value in equation (2), we obtain
v = 40 km/h
Hence, speed of one car = 60 km/h and speed of other car = 40 km/h
(v) Let length and breadth of rectangle be x unit and y unit respectively.
Area = xy
According to the question,
By cross-multiplication method, we obtain
Hence, the length and breadth of the rectangle are 17 units and 9 units respectively.