CBSE Class 10 Mathematics Pair Of Linear Equations In 2 Variables Worksheet Set M

Pair of Linear Equations in Two Variables

(Key Points)

• An equation of the form ax + by + c = 0, where a, b, c are real nos. (a  0, b  0) i.e (a2+b2 ≠0) is called a linear equation in two variables x and y.

Ex : (i) x – 5y + 2 =0

(ii) 32 x – y =1

• The general form for a pair of linear equations in two variables x and y is

a1x + b1y + c1 = 0

a2x + b2y + c2 = 0

Where a1, b1, c1, a2, b2, c2 are all real nos and a1 0, b1 0, a2 0, b2 0.

Examples

• Graphical representation of a pair of linear equations in two variables:

a1x + b1y + c1 = 0

a2x + b2y + c2 = 0

(i) Will represent intersecting lines if

I.e. unique solution. And these types of equations are called consistent pair of linear equations.

Ex: x – 2y = 0

3x + 4y – 20 = 0

(ii) will represent overlapping or coincident lines if

i.e. Infinitely many solutions, consistent or dependent pair of linear equations

Ex: 2x + 3y – 9 = 0

4x + 6y – 18 = 0

(iii) will represent parallel lines if

i.e. no solution and called inconsistent pair of linear equations.

Ex: x + 2y – 4 = 0

2x + 4y – 12 = 0

• Algebraic methods of solving a pair of linear equations:

(i) Substitution method

(ii) Elimination Method

(iii) Cross multiplication method

Level - I

1. Find the value of ‘a’ so that the point(2,9) lies on the line represented by ax-3y=5

2. Find the value of k so that the lines 2x – 3y = 9 and kx-9y =18 will be parallel.

3. Find the value of k for which x + 2y =5, 3x+ky+15=0 is inconsistent

4. Check whether given pair of lines is consistent or not 5x – 1 = 2y, y = +

5. Determine the value of ‘a’ if the system of linear equations 3x+2y -4 =0 and ax – y – 3 = 0 will represent intersecting lines.

6. Write any one equation of the line which is parallel to 2x – 3y =5

7. Find the point of intersection of line -3x + 7y =3 with x-axis

8. For what value of k the following pair has infinite number of solutions.

(k-3)x + 3y = k

k(x+y)=12

9. Write the condition sothat a1x + b1y = c1 and a2x + b2y = c2 have unique solution.

Level - II

1. 5 pencils and 7pens together cost Rs. 50 whereas 7 pencils and 5 pens together cost Rs. 46. Find the cost of one pencil and that of one pen.

2. Solve the equations:

3x – y = 3

7x + 2y = 20

3. Find the fraction which becomes to 2/3 when the numerator is increased by 2 and equal to 4/7 when the denominator is increased by 4

4. Solve the equation:

px + qy = p – q

qx – py = p + q

5. Solve the equation using the method of substitution:

6. Solve the equations:

Where, x

7. Solve the equations by using the method of cross multiplication:

5x + 12y =7

Level - III

1. Draw the graph of the equations

4x – y = 4

4x + y = 12

Determine the vertices of the triangle formed by the lines representing these equations and the x-axis. Shade the triangular region so formed

2. Solve Graphically

x – y = -1 and

3x + 2y = 12

Choose the correct answer from the given options:

Question. The father’s age is six times his son’s age. Four years hence, the age of the father will be four times his son’s age. The present ages, (in years) of the son and the father are, respectively
(a) 4 and 24
(b) 5 and 30
(c) 6 and 36
(d) 3 and 24
Answer : C

Question. A purse contains 25 paise and 10 paise coins. The total amount in the purse is ₹ 8.25. If the number of 25 paise coins is one-third of the number of 10 paise coins in the purse, then the total number of coins in the purse is
(a) 60
(b) 40
(c) 80
(d) 72
Answer : A

Question. Aruna has only ₹ 1 and ₹ 2 coins with her. If the total number of coins that she has is 50 and the amount of money with her is ₹ 75, then the number of ₹ 1 and ₹ 2 coins are, respectively
(a) 35 and 15
(b) 35 and 20
(c) 15 and 35
(d) 25 and 25
Answer : A

Question. A fraction becomes 3/1 when 2 is subtracted from the numerator and it becomes 2/1 when 1 is subtracted from the denominator. The fraction is
(a) 2/5
(b) 5/18
(c) 4/13
(d) 7/15
Answer : D

Question. The difference between two numbers is 26 and the larger number exceeds thrice of the smaller number by 4. The numbers are
(a) 39, 13
(b) 12, 38
(c) 37, 11
(d) None of these
Answer : C

Question. A man has some hens and cows. If the number of heads be 48 and the number of feet equals 140, the number of hens will be
(a) 18
(b) 26
(c) 32
(d) 40
Answer : B

Question. If 3 chairs and 1 table costs ₹ 1500 and 6 chairs and 1 table costs ₹ 2400, the pair of linear equations to represent this situation is
(a) 6x + y = 1500, 3x + y = 2400
(b) x/3 + y = 1500, x/6 + y = 2400
(c) 3x + y = 1500, 6x + y = 2400
(d) None of these
Answer : C

Question. Meena went to a bank to withdraw ₹ 2,000. She asked the cashier to give her ₹ 50 and ₹ 100 notes only. Meena got 25 notes in all. How many notes of ₹ 50 and ₹ 100 she received?
(a) ₹ 50 : 10, ₹ 100 : 15
(b) ₹ 50 : 12, ₹ 100 : 10
(c) ₹ 50 : 15, ₹ 100 : 10
(d) None of these
Answer : A

Question. A two-digit number is obtained by either multiplying the sum of digits by 8 and then subtracting 5 or by multiplying the difference of digits by 16 and adding 3. The number is
(a) 23
(b) 34
(c) 83
(d) 119
Answer : C

Question. The sum of a two-digit number and the number obtained by interchanging the digits is 132. If the two digits differ by 2, the number is
(a) 45
(b) 75
(c) 85
(d) 115
Answer : B

Question. Two years ago, a father was five times as old as his son. Two years later, his age will be 8 more than three times the age of the son. The present age of father and son, respectively are
(a) 40 years, 12 years
(b) 30 years, 6 years
(c) 32 years, 8 years
(d) 42 years, 10 years
Answer : D

Question. A motor boat whose speed in still water is 18 km/h, takes 1 hour more to go 24 km upstream than to return downstream to the same spot. The speed of the stream is
(a) 60 km/hr
(b) 6 km/min
(c) 6 km/s
(d) 6 km/hr
Answer : B

Question. The two consecutive odd positive integers, sum of whose squares is 290 are
(a) 5, 13
(b) 11, 13
(c) 13, 17
(d) None of these
Answer : B

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