X-MATHEMATICS
CHAPTER – 3
PAIR OF LINEAR EQUATIONS IN TWO
VARIABLES
CLASS DISCUSSION AND HOME WORKSHEET
Consistent System
A system of simultaneous linear equations is said to be consistent if it has at least one solution.
Inconsistent System
A system of simultaneous linear equations is said to be inconsistent if it has no solution.
Nature of solutions of simultaneous linear equations
Based on the coefficients:
Let 𝑎1 𝑥 + 𝑏1 𝑦 + 𝑐1 = 0, 𝑎2 𝑥 + 𝑏2 𝑦 + 𝑐2 = 0 be a system of linear equations.
𝑎
𝑏
Case (i): 𝑎1 ≠ 𝑏1. In this case, the given system is consistent. This implies that the system has a
2
2
unique solution. Graphically, the two lines intersect at exactly at one point which is the solution.
Case (ii):
𝑎1
𝑎2
𝑏
𝑐
= 𝑏1 ≠ 𝑐1. In this case, the given system is inconsistent. This implies that the
2
2
system has no solution. Graphically, the two lines are parallel
Case (iii):
𝑎1
𝑎2
𝑏
𝑐
= 𝑏1 = 𝑐1 , in this case, the given system is independent and consistent. This
2
2
implies that the system has infinitely many solutions. Graphically, the two lines are coinciding
(overlapping)
Algebraic methods for solving simultaneous linear equations:
Substitution Method
Elimination Method
Cross-Multiplication Method
3.1 Class Discussion Questions
1.
Solve the following pair of equations by substitution method:
(i) 𝑥 − 2𝑦 = 1, 3𝑥 − 5𝑦 = 5
(ii) 2𝑥 + 3𝑦 = 1, 3𝑥 − 2𝑦 = −2
(iii) 0.2𝑥 + 0.3𝑦 = 1.3, 0.4𝑥 + 0.5𝑦 = 2.3
2.
Use elimination method to find all possible solutions of the following pair of linear equations:
(i) 𝑥 + 𝑦 = 5, 2𝑥 − 3𝑦 = 4
(ii) 2𝑥 + 3𝑦 = 1, 3𝑥 − 2𝑦 = −2
(iii) 2𝑥 − 3𝑦 = 5, 5𝑥 + 3𝑦 = 2
(vi) 2𝑥 − 3𝑦 = 5, 5𝑥 − 3𝑦 = 2
3.
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.
(i) 𝑥 − 2𝑦 = 1, 3𝑥 − 5𝑦 = 5
(ii) 2𝑥 + 3𝑦 = 1, 3𝑥 − 2𝑦 = −2
4.
Solve 2𝑥 + 3𝑦 = 11 and 2𝑥 − 4𝑦 = −24 and hence find the value of ′𝑚′ for which
𝑦 = 𝑚𝑥 + 3.
5.
Solve for 𝑥 𝑎𝑛𝑑 𝑦:
3𝑥
2
−
5𝑦
3
= −2,
𝑥
3
𝑦
13
2
6
+ =
3.1 Home Work
6.
Solve the following pair of equations by substitution method:
(i) 𝑥 + 𝑦 = 14, 𝑥 − 𝑦 = 4
𝑠
𝑡
(ii) 𝑠 − 𝑡 = 3, 3 + 2 = 6
(iii) 3𝑥 − 𝑦 = 3, 9𝑥 − 3𝑦 = 9
(iv) 0.2𝑥 + 0.3𝑦 − 0.11 = 0, 0.7𝑥 − 0.5𝑦 + 0.8 = 0
7.
Use elimination method to find all possible solutions of the following pair of linear equations:
(i) 3𝑥 + 4𝑦 = 10 𝑎𝑛𝑑 2𝑥 − 2𝑦 = 2
(ii) 3𝑥 − 5𝑦 − 4 = 0 𝑎𝑛𝑑 9𝑥 = 2𝑦 + 7
8.
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.
(i) 3𝑥 − 5𝑦 = 20 𝑎𝑛𝑑 6𝑥 − 10𝑦 = 40
(ii) 𝑥 − 3𝑦 − 7 = 0 𝑎𝑛𝑑 3𝑥 − 3𝑦 − 15 = 0
3.2 Class Discussion Questions
9.
For which values of 𝑝 does the pair of equations given below has unique solution?
4𝑥 + 𝑝𝑦 + 8 = 0 𝑎𝑛𝑑 2𝑥 + 2𝑦 + 2 = 0
10.
For which value of 𝑘 will the following pair of linear equations have no solution?
3𝑥 + 𝑦 = 1 𝑎𝑛𝑑 (2𝑘 − 1)𝑥 + (𝑘 − 1)𝑦 = 2𝑘 + 1.
11.
For what values of 𝑘 will the following pair of linear equations have infinitely many solution?
𝑘𝑥 + 3𝑦 − (𝑘 − 3) = 0 𝑎𝑛𝑑 12𝑥 + 𝑘𝑦 − 𝑘 = 0
12.
For which values of 𝑎 𝑎𝑛𝑑 𝑏 does the following pair of linear equations have an infinite number
of solutions?
2𝑥 + 3𝑦 = 7 𝑎𝑛𝑑 (𝑎 − 𝑏)𝑥 + (𝑎 + 𝑏)𝑦 = 3𝑎 + 𝑏 − 2
13.
Find the value of 𝑚 for which the pair of linear equation, 2𝑥 + 3𝑦 − 7 = 0 and (𝑚 − 1)𝑥 +
(𝑚 + 1)𝑦 = (3𝑚 − 1) has infinitely many solutions.
3.2 Home Work
14.
For each of the following systems of equations determine the value of 𝑘 for which the given
system of equations has a unique solution
2𝑥 − 3𝑦 = 1
𝑥 − 𝑘𝑦 = 2
(i)
(ii)
𝑘𝑥 + 5𝑦 = 7
3𝑥 + 2𝑦 = −5
15.
For each of the following system of equations determine the values of 𝑘 for which the given
system has no solution.
3𝑥 − 4𝑦 + 7 = 0
2𝑥 − 𝑘𝑦 + 3 = 0
(i)
(ii)
𝑘𝑥 + 3𝑦 − 5 = 0
3𝑥 + 2𝑦 − 1 = 0
16.
For each of the following systems of equations determine the value of 𝑘 for which the given
system of equations has infinitely many solutions.
(𝑘 − 3)𝑥 + 3𝑦 = 𝑘
5𝑥 + 2𝑦 = 𝑘
(i)
(ii)
10𝑥 + 4𝑦 = 3
𝑘𝑥 + 𝑘𝑦 = 12
3.3 Class Discussion Questions
(Equations Reducible to a pair of linear equations in two variables)
17.
Solve the following pair of equations:
2
3
5
4
(i) 𝑥 + 𝑦 = 13, 𝑥 − 𝑦 = −2
(iii)
2
3
4
9
√
√
√
√
+
𝑥
= 2,
𝑦
−
𝑥
= −1
𝑦
1
1
(ii) 2𝑥 − 2𝑦 + 2 = 0,
4
(iv) 𝑥 + 3𝑦 = 14,
3
𝑥
1
1
− 2𝑦 − 1 = 0
𝑥
− 4𝑦 = 23
(v)
7𝑥−2𝑦
= 5,
𝑥𝑦
1
8𝑥+7𝑦
𝑥𝑦
1
5
= 15
3
2
1
10
2
5
(vi) 𝑥+1 − 𝑦−1 = 2 ; 𝑥+1 + 𝑦−1 = 2, where 𝑥 ≠ 1, 𝑦 ≠ 1
1
1
(vii) 3x+y + 3𝑥−𝑦 = 4 , 2(3𝑥+𝑦) − 2(3𝑥−𝑦) =
−1
8
3.3 Home Work
18.
Solve the following pair of equations:
6
3
5
1
(i) 𝑥−1 − 𝑦−2 = 1, 𝑥−1 − 𝑦−2 = 2, where x ≠ 1, y ≠ 2
1
1
1
1
(ii) 2x + 3𝑦 = 2, 3𝑥 + 2𝑦 =
13
6
(iii) 6x + 3y = 6xy, 2x + 4y = 5xy
10
2
15
5
(iv) 𝑥+𝑦 + 𝑥−𝑦 = 4; 𝑥+𝑦 − 𝑥−𝑦 = −2
3.4 Class Discussion Questions (Special Type Questions)
20.
152𝑥 − 378𝑦 = −74
−378𝑥 + 152𝑦 = −604
𝑝𝑥 + 𝑞𝑦 = 𝑝 − 𝑞
Solve for 𝑥 and 𝑦: 𝑞𝑥 − 𝑝𝑦 = 𝑝 + 𝑞
21.
Solve for 𝑥 𝑎𝑛𝑑 𝑦: (𝑎 − 𝑏)𝑥 + (𝑎 + 𝑏)𝑦 = 𝑎2 − 2𝑎𝑏 − 𝑏 2 , (𝑎 + 𝑏)(𝑥 + 𝑦) = 𝑎2 + 𝑏 2 .
19.
Solve for 𝑥 and 𝑦;
3.4 Home Work
22.
Solve 𝑥 𝑎𝑛𝑑 𝑦:
99𝑥 + 101𝑦 = 499
.
101𝑥 + 99𝑦 = 501
𝑥
23.
Solve for 𝑥 𝑎𝑛𝑑 𝑦:
24.
Solve for 𝑥 𝑎𝑛𝑑 𝑦:
𝑎
𝑦
−𝑏 =0
𝑎𝑥 + 𝑏𝑦 = 𝑎2 + 𝑏 2
𝑎𝑥 + 𝑏𝑦 = 𝑐
𝑏𝑥 + 𝑎𝑦 = 1 + 𝑐
3.5 Class Discussion Questions (Word Problems)
25.
The larger of two supplementary angles exceeds the smaller by 18 degrees. Find them.
26.
A fraction becomes 11. If 2 is added to both the numerator and the denominator. If 3 is added to
9
5
both the numerator and the denominator becomes 6. Find the fraction.
27.
Five years hence, the age of Jacob will be three times that of his son. Five years ago, Jacob’s age
was seven times that of his son. What are their present ages?
28.
The taxi charges in a city consist of a fixed charge together with the charge of the distance
covered. For a distance of 10 km, the charge paid is ₹105 and for a journey of 15 km, the charge
paid is ₹155. What are the fixed charges and the charge per km? How much does a person have
to pay for travelling a distance of 25 km?
29.
A lending library has a fixed charge for the first three days and an additional charge for each day
thereafter. Saritha paid ₹27 for a book kept for seven days, while Susy paid ₹21 for the book,
she kept for five days. Find the fixed charge and the charge for each extra day.
30.
The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number
obtained by reversing the order of the digits. Find the number.
31.
The ratio of incomes of two persons is 9:7 and the ratio of their expenditure is 4:3. If each of
them manages to save ₹2000 per month, find their monthly incomes.
32.
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?
33.
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.
34.
A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km
upstream and 55 km down-stream. Determine the speed of the stream and that of the boat in still
water.
35.
Roohi travels 300 km to her home partly by train and partly by bus. She takes 4 hours of she
travels 60 km by train and the remaining by bus. If she travels 100 km by train and the remaining
by bus, she takes 10 minutes longer. Find the speed of the train and the bus separately.
36.
A train moving with uniform speed for a certain distance takes 6 hours less if its speed be
increased by 6 km/hour. It would have taken 6 hours more, had its speed been decreased by
4km/hour. Find the distance travelled of the journey and the speed of the train.
37.
The students of a class are made to stand in rows. If 3 students are extra in a row, there would
be 1 row less. If 3 students are less in a row, there would be 2 rows more. Find the number of
students in the class.
38.
2 women and 5 men can together finish an embroidery work in 4 days, while 3 women and 6
men can finish it in 3 days. Find the time taken by 1 woman alone to finish the work, and also
that taken by 1 man alone.
39.
ABCD is a cyclic quadrilateral. Find the angles of the cyclic quadrilateral.
3.5 Home Work
40.
If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces to 1. It
1
becomes 2 if we only add 1 to the denominator. What is the fraction.
41.
A fraction becomes 3 when 1 is subtracted from the numerator and it becomes 4 when 8 is added
to its denominator. Find the fraction.
42.
Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu.
How old are Nuri and Sonu?
43.
The sum of a two-digit number and the number obtained by reversing the digits is 66. If the
digits of the number differ by 2, find the number. How many such numbers are there?
44.
Meena went to a bank to withdraw RS 2000. She asked the cashier to give her RS 50 and RS
100 notes only. Meena got 25 notes in all. Find how many notes of ₹50 and ₹100 she received.
45.
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 best?
1
1
46.
A motor boat can travel 30 km upstream and 28 km downstream in 7 hours. It can travel 21 km
upstream and return in 5 hours. Find the speed of the boat in still water and the speed of the
stream.
47.
Ritu can row downstream 20 km in 2 hours and upstream 4 km in 2 hours. Find her speed of
rowing in still water and the speed of the current.
48.
A lending library has a fixed charge for the first two days and an additional charge for each day
thereafter. Abdul paid RS 30 for a book kept for 6 days while Kaira paid RS 45 for a book kept
for 9 days. Find the fixed charge and the charge for each extra day.
49.
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 RS 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.
50.
The ratio of incomes of two persons is 11:7 and ratio of their expenditure is 9:5. If each of them
manages to saves RS 400 per month, find their monthly incomes.
51.
From a bus stand in Bangalore, if we buy 2 tickets o Malleswaram and 3 tickets to Yeshwanthpur,
the total cost is RS 46; but if we buy 3 tickets to Malleswaram and 5 tickets to Yeshwanthpur
the total cost is RS 74. Find the fares from the bus stand to Malleswaran and to Yeshwanthpur.
52.
Ramesh travels 760 km to his home partly by train and partly by car. He takes 8 hours if he
travels 160 km, by train and the rest by car. He takes 12 minutes more if the travels 240 km by
train and the rest by car. Find the speed of the train and car respectively.
53.
2 Men and 7 boys can do a piece of work in 4 days. It is done by 4 men and 4 boys in 3 days.
How long would it take for one man or one boy to do it.
Nature of solution of simultaneous linear equations
Based on Graph
Case (i)
The lines intersect at a point. The point of intersection is the unique solution of the two equation.
In this case, the pair of equations is consistent.
Case (ii)
The lines coincide. The pair of equations has infinitely many solutions – Each point on the line
is a solution. In this case, the pair of equations is dependent (which is consistent).
Case (iii)
The lines are parallel. The pair of equations has no solution. In this case, the pair of equations is
inconsistent.
3.6 Class Discussion Questions
54.
𝑎
𝑏
𝑐
On comparing 𝑎1 , 𝑏1 𝑎𝑛𝑑 𝑐1 find out whether the lines representing the following pairs of linear
2
2
2
equations intersect at a point, are parallel or coincident:
(i) 5𝑥 − 4𝑦 + 8 = 0
and
7𝑥 + 6𝑦 − 9 = 0
(ii) 9𝑥 + 3𝑦 + 12 = 0
and
18𝑥 + 6𝑦 + 24 = 0
(iii) 2𝑥 + 3𝑦 = 10
and
4𝑥 + 6𝑦 = 15
55.
𝑎
𝑏
𝑐
On comparing 𝑎1 , 𝑏1 𝑎𝑛𝑑 𝑐1 find out whether the lines representing the following pairs of linear
2
2
2
equations are consistent or inconsistent.
(i) 𝑥 + 2𝑦 = 5
and
2𝑥 − 3𝑦 = 7
3
5
(ii) 2 𝑥 + 3 𝑦 = 7
and
9𝑥 − 10𝑦 = 14
56.
Given the linear equation 2𝑥 + 3𝑦 − 8 = 0, write another linear equation in two variables such
that the geometrical representation of the pair so formed is:
(i)
Intersecting line
(ii) Parallel lines
(iii) Coincident lines
57.
Check graphically whether the pair of equations: 𝑥 + 3𝑦 = 6 and 2𝑥 − 3𝑦 = 12 is consistent.
If so, solve them graphically.
58.
Solve the following system graphically: 𝑥 + 𝑦 = 3, 2𝑥 + 5𝑦 = 12.
59.
Draw the graphs of the equations 𝑥 − 𝑦 + 1 = 0 and 3𝑥 + 2𝑦 − 12 = 0. Determine the
coordinates of the vertices of the triangle formed by these lines and the line of 𝑦 = 0
(i.e., 𝑥 −axis), and shade the triangular region. Also, find the area of triangle.
3.6 Home Work Questions
60.
𝑎
𝑏
𝑐
On comparing 𝑎1 , 𝑏1 𝑎𝑛𝑑 𝑐1 , find out whether the lines representing the following pairs of linear
2
2
2
equations intersect at a point, are parallel or coincident.
(i) 6𝑥 − 3𝑦 + 10 = 0
and
2𝑥 − 𝑦 + 9 = 0
(ii) −𝑥 + 2𝑦 = −5
and
2𝑥 − 4𝑦 = 10
61.
𝑎
𝑏
𝑐
On comparing 𝑎1 , 𝑏1 𝑎𝑛𝑑 𝑐1 find out whether the lines representing the following pairs of linear
2
2
2
equations are consistent or inconsistent.
(i) 2𝑥 − 3𝑦 = 8
and
4𝑥 − 6𝑦 = 9
(ii) 5𝑥 − 3𝑦 = 11 − 10𝑥 + 6𝑦 = −22
4
(iii) 3 𝑥 + 2𝑦 = 8 2𝑥 + 3𝑦 = 12
62.
Solve the following graphically:
(i) 𝑥 + 𝑦 = 3
2𝑥 + 5𝑦 = 12
(ii) 2𝑥 + 𝑦 − 3 = 0 2𝑥 − 3𝑦 − 7 = 0
63.
Show graphically that 3𝑥 − 5𝑦 − 20 = 0 and 6𝑥 − 10𝑦 = −40 is inconsistent.
64.
Solve the following graphically and find the co-ordinates of the points where the lines meet the
axis denoted against each
(i) 2𝑥 − 5𝑦 + 4 = 0, 2𝑥 + 𝑦 − 8 = 0 (𝑥 − 𝑎𝑥𝑖𝑠)
(ii) 𝑥 + 2𝑦 − 7 = 0, 2𝑥 − 𝑦 − 4 = 0 (𝑦 − 𝑎𝑥𝑖𝑠)
65.
The coach of a cricket team buys 3 bats and 6 balls for Rs 3900. Later she buys another bar and
3 more balls of the same kind for RS 1300. Represent this situation algebraically and
geometrically.