This document discusses pairs of linear equations in two variables. It defines them algebraically and geometrically as two straight lines that can intersect, be parallel, or coincide. Pairs can be solved graphically by drawing the lines or algebraically using substitution, elimination, or cross multiplication methods. The document provides examples of pairs of linear equations and their solutions. It also includes 40 multiple choice questions related to pairs of linear equations and their properties.

1. 4. Pair of Linear
Equations in two
Variables

2. 4. Pair of Linear Equations in two Variables
Page 2
Pair of Linear Equations in two variables
1. A pair of Linear equations in two variables x and y can be represented
algebraically as where
and are eal numbers such that .
2. Graphically or geometrically a pair of linear equations
in two variables represent a pair of
straight lines which are
i. Intersecting
ii. Parallel
iii. coincident
3. A pair of linear equations in two variables can be solved by the
i. Graphical method
ii. Algebraic method
4. To solve a pair of linear equations in two variables by graphical method, we first
draw the lines represented by them.
i. If the pair of lines intersect at a point, then we say that the pair is
consistent and the coordinates of the point provide us the unique
solution.
ii. If the pair of lines are parallel, then the pair has no solution and is called
inconsistent pair of equations.
iii. If the pair of lines are coincident, then it has infinitely many solutions,
each point on the line being of solution. In this case, we say that the pair
of linear equations is consistent with infinitely many solutions.
5. To solve a pair of linear equations in two variables algebraically, we have
following methods.
i. Substitution method
ii. Elimination method
iii. Cross multiplication method
6. If is a pair of linear equations in two
variables x and y such that
i. then the pair of linear equations is consistent with a unique
solution
ii. , then the pair of linear equations is inconsistent.
iii. , then the pair of linear equations is consistent with infinitely
many solutions

3. 4. Pair of Linear Equations in two Variables
Page 3
Multiple choice questions
1. 5 pencils and 7 pens together cost Rs. 50 where as 7 pencils and 5pens together
cost Rs.46 then the cost of 1 pencil is ________rupees.
(a) 1 (b)2 (c)3 (d)4
2. The area of a rectangle gets reduced by 80 sq. units if its length is reduced by 5
units and breadth is increased by 2 units. If we increase the length by 10 units
and decrease the breadth by 5 units, the area will increases by 50Sq units, then
the length of the rectangle= ________units
(a) 20 (b)21 (c)19 (d)None
3. The cost of 2 kg apples and 1 kg of grapes on a day was found to be Rs.160. After
a month, the cost of 4 Kg of apples and 2kg of grapes is the Rs.300. Which of the
following equations represent this situations
(a) (b)
(c) (d)None
4. The coach of a cricket team buys 3 bats and 6 balls for Rs.3900. Later he buys
another bat and 3 more balls of the same kind for Rs.1300which of the following
equations represent this situation.
(a) (b) 300
(c) (d)None
5. 10 students of class 10th took part in Mathematics quiz. If the number of girls is 4
more than the number of boys. Which of the following pairs represent the
situation.
(a) (b)
(c) (d)None
6. Rubina went to a bank to withdraw Rs.2000. She asked the cashier to give the
cash in Rs.50 and Rs.100 notes only. Rubina got 25 notes in all. Then number of
Rs.50 those Rubina got
(a) 12 (b)11 (c)10 (d)9
7. The ratio of incomes of two persons is 9:7 and the ratio of their expenditures is
4:3. If each of them manages to save rs.2000 per month, their monthly incomes
are
(a) 18000, 14000 (b)36000,28000 (c)9000,7000 (d)27000, 21000

4. 4. Pair of Linear Equations in two Variables
Page 4
8. Two angles are complementary. The larger angle is less than twice the
measure of the smaller angle. Then the greater angle is
(a) (b) (c) (d)
9. If and where and then the values of and
are
(a) (b) (c) (d) None
10. If and then
(a) (b) (c) (d) None
11. If and then
(a) (b) (c) (d) 4
12. If and then
(a) (b) (c) (d) none
13. If and where and then the value of
(a) (b) (c) (d) 2
14. If the system of linear equations and has a unique
solution then
(a) (b) (c) (d)None
15. If the system of linear equations and has no
solution then
(a) (b) (c) (d)
16. If the system of linear equations and has infinitely
many solutions then
(a) (b) (c) (d)
17. If the system of linear equations and represent
coincidental lines then
(a) (b) (c) (d)6
18. If the system of linear equations and has
infinitely many solutions then ___and
(a) (b) (c) (d)None

5. 4. Pair of Linear Equations in two Variables
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19. If the system of linear equations and have a unique
solution which of the following is true
(a) (b) (c) (d)None
20. If the system of linear equations and has no
solution then the value of
(a) (b) (c) (d)-6
21. The sum of the numerator and denominator of a fraction is 12. If the
denominator is increased by 3, the fraction becomes . Then the fraction is
(a) (b) (c) (d)
22. On selling a T.V. at 5% gain and a fridge at 10% gain, a shopkeeper gains
Rs.2000. But if he sells the T.V at 10% gain and a fridge at 5% loss, he gains
Rs.1500 on the transaction. Then the original price of T.V. is Rs. _______
(a) (b)25000 (c) (d)None
23. The sum of a two digit number and the number formed by interchanging the
digits is 132. If 12 is added to the number, the new number becomes 5 times the
sum of the digits. Then the number is
(a) 8 (b) (c) (d)both (a)&(b)
24. If twice the son’s age in years is added to the father’s age the sum is . But if
twice the father’s age is added to the son’s age the sum is . Then the age of
son is
(a) 10 years (b)15 years (c) years (d) None
25. Ten years later, A will be twice as old as B and five years ago, A was three times
as old as B. then the present ages of A and B in years
(a) 50, 20 (b)40,30 (c) (d) None
26. Five years ago, Kamala was thrice old as Sonia. Ten years later, Kamala will
twice as old as Sonia. Then the age of Sonia
(a) 35 (b) 30 (c) (d) 20
27. The value of for which the system of equations and
has no solution
(a) (b) (c) (d)none

6. 4. Pair of Linear Equations in two Variables
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28. The area of the triangle formed by the line with coordinate
axes is _______sq units
(a) (b)2ab (c) (d)
29. The area of the triangle formed by the lines and is ____ sq units
(a) (b) (c) (d)72
30. The area of the triangle formed by the lines and is _____ sq
units
(a) (b) (c) (d)None
31. The denominator of a fraction is 4 more than twice the numerator. When both
the numerator and denominator are decreased by 6 , then the denominator
becomes 12 times the numerator then the fraction is
(a) (b) (c) (d)None
32. A fraction becomes if 1 is added to both numerator and denominator. If,
however, 5 is subtracted from both numerator and denominator the fraction
becomes . then that fraction is
(a) (b) (c) (d)
33. 3 bags and 4 pens together cost Rs. 257 where as 4 bags and 3 pens together cost
Rs. 324. Then the cost of 1 book and 2 pens is
(a) (b) (c) (d)154
34. 4 chairs and 3 tables cost Rs. 2100 and 5 chairs and 2 tables cost Rs. 1750. Then
the cost of a chair
(a) (b) (c) (d)300
35. If and the value of
(a) (b) (c) (d)
36. The larger of two supplementary angles exceeds the smaller by . Then the
larger angle is
(a) (b) (c) (d)
37. The system of linear equations and
(a) intersect at a point (b)parallel
(c) coincident (d)None

7. 4. Pair of Linear Equations in two Variables
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38. The system of linear equations and
(a) intersect at a point (b)parallel
(c) coincident (d)None
39. If a pair of linear equations in two variables is consistent, then the lines
represented by two equations are
(a) intersecting (b)parallel
(c) coincident (d)intersecting or coincident
40. If the system of equations and has
infinitely many solutions, then and
(a) (b) (c) (d)
Answers
1 2 3 4 5 6 7 8 9 10
c a c a a c a d b a
11 12 13 14 15 16 17 18 19 20
d b a c b a b c a a
21 22 23 24 25 26 27 28 29 30
a c a b a d a c b d
31 32 33 34 35 36 37 38 39 40
a b c a b a a b d b