PSEB Solutions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables
PSEB 10th Class Maths Solutions Chapter 3 Pair of Linear Equations in Two Variables
Ex 3.1
Question 1.
Aftab tells his daughter, “Seven years ago I was seven times as old as you were then. Also, three years from now, I shall
be three times as old as you will be” (Isn’t this interesting ?). Represent this situation algebraically and graphically.
Solution:
Let Aftabs present age = x years
and Aftab’s daughter’s present age = y years
Algebraical-Situation
According to 1st condition,
x – 7 = 7(y – 7)
or x – 7 = 7y – 49
or x – 7y + 42 = 0
According to 2nd condition,
x + 3 = 3(y + 3)
or x + 3 = 3y + 9
or x – 3y – 6 = 0
∴ Pair of Line’ar Equation in two variables are
x – 7y + 42 = 0
and x – 3y – 6 = 0
Graphical – Situation:
x – 7y + 42 = 0
x – 3y – 6 = 0
x = 7y – 42 ………….(1)
x = 3y + 6 …………..(2)
Putting y = 5 in (1), we get
Putting y = 0 in (2), we get
x = 7 × 5 – 42 = 35 – 42
x = -7
Putting y = 6 in (1), we get
x = 7 × 6 – 42 = 42 – 42 = 0
Putting y = 7 in (1), we get:
x = 7 × 7 – 42 = 49 – 42 = 7
Table
Plotting the points A (-7, 5), B (0, 6), C (7, 7) and drawing a line joining them, we get the graph of the equation x – 7y + 42 = 0
x = 3 × 0 + 6
x = 0 + 6 = 6
Putting y = 3 in (2), we get:
x = 3 × 3 + 6
= 9 + 6 = 15
Putting y = -2 in (2), we get:
x = 3 × -2 + 6
= -6 + 6 = 0
Table:
Plotting the points D (6, 0), E (15, 3), F (0, -2) and drawing a line joining them, we get the graph of the equation x – 3y – 6 = 0
From the graph it is clear that the two lines intersect at G (42, 12).
Hence, x = 42 and y = 12 is the solution of given pair of linear equations.
Question 2.
The coach of a cricket team buys 3 bats and 6 balls for 3900. Later, she buys another bat and 3 more balls of the same kind for 1300. Represent this situation algebraically and geometrically. [Pb. 2019, Set-A, B, C]
Solution:
Let cost of one bat = x
Cost of one hail = y
Algehraical – Situation
According to 1st condition,
3x + 6y = 3900
or x + 2y = 1300
According to 2nd condition,
1x + 3y = 1300
∴ Pair of linear equations in two variables are:
x + 2y = 13001
and x + 3y = 1300
Graphical – Situation:
x + 2y = 1300
x = 1300 – 2y …………..(1)
Putting y = 0 in (1), we get
x = 1300 – 2 × 0
x= 1300
Putting y = 500 in (1), we get :
= 1300 – 2 × 500
= 1300 – 1000 = 300
Putting y = 650 in (1), we get :
x = 1300 – 2 × 650
1300 – 1300 = 0
Table:
Plotting the points A (1300, 0), B (300, 500) and C (0, 650) drawing a line joining them we get the graph of the equation x + 2y = 1300.
x + 3y = 1300
x = 1300 – 3y …………….(2)
Putting y = 0 in (2), we get :
x = 1300 – 3 × 0 = 1300
Putting y = 500 in (2), we get :
x = 1300 – 3 × 500
= 1300 – 1500 = – 200
Putting y = 300 in (2), we get :
x = 1300 – 3 × 300
= 1300 – 900 = 400
Table:
Plotting the points A (1300, 0), E (-200, 500), F (400, 300) and drawing a line joining them we get the graph of the
equation.
x + 3y = 1300
From the graph it is clear that the two lines intersect at A (1300, 0).
Hence x = 1300 and y = 0 is the solution of given pair of linear equations.
Question 3.
The cost of 2 kg of apples and 1 kg of grapes on a day was found to be 160. After a month, the cost of 4 kg of apples and 2 kg of grapes is ₹ 300. Represent the situation algebraically and geometrically.
Solution:
Let cost of 1 kg apples = ₹ x
Cost of 1 kg grapes = ₹ y
Algebraical – Situation
According to 1st condition,
2x + 1y = 160
According to 2nd condition,
4x + 2y = 300
∴ Pair of linear equations in two variables
2x + y = 160
and 4x + 2y = 300
Graphical – Situation
2x + y = 160
2x = 160 – y
Plotting the points D (75, 0), E (50, 50), F (0, 150) and drawing a line joining them, we get the graph of equation
4x + 2y = 300
From the graph, it is clear that the two lines do not intersect anywhere i.e. they are parallel.
Ex 3.2
Question 1.
Form the pair of linear equations in the following problems, and find their solutions graphically.
(i) 10 students of Class X took part in a Mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz.
(ii) 5 pencils and 7 pens together cost ₹ 50, whereas 7 pencils and 5 pens together cost ₹ 46. Find the cost of one
pencil and that of one pen.
Solution:
(i) Let the number of boys in the Quiz = X
and the number of girls in the Quiz = y
Total number of students took part in Quiz = 10
x + y = 10
or x + y – 10 = 0
According to Question,
y = x + 4
or x = y – 4
Now, draw the graph of linear equations
x + y = 10
and x – y + 4 = 0
x + y = 10
or x = 10 – y
Putting y = 0 in (1), we get :
x = 10 – 0 = 10
Putting y = 7 in(1), we get:
x = 10 – 7 = 3
Putting y = 10 in (1) we get:
X = 10 – 10 = 0
Plotting the points A (10, 0), B (3, 7), C (0, 10) and drawing a line joining them we get the graph of the equation x + y = 10
Now x – y + 4 = 0
or x = y – 4
Putting y = 0 in (2), we get:
x = 0 – 4 = -4
Putting y = 7 in (2), we get:
x = 7 – 4 = 3
Putting y = 4 in (2), we get:
x = 4 – 4 = 0
Plotting the points D (-4, 0), B (3, 7), E (0, 4) and drawing a line joining them, we get the graph of the equation x – y + 4 = 0
From the graph it is clear that both the linear equations meets at a point B (3, 7).
∴ Point B (3, 7) is the graphic solution.
Hence, number of boys in the Quiz = 3
Number of girls in the Quiz = 7
Table:
Plotting the points E (6.5, 0), B (3, 5), F (9.5. -4) and drawing a line joining them, we get the graph of the equation.
7x + 5y = 46
From the graph, it is clear that both the linear equations meets at a point B (3, 5).
∴ point B (3, 5) is the graphic solution.
Hence, cost of one pencil = ₹ 3
Cost of one pen = ₹ 5
(iv) Given pair of linear equations is
5x – 3y= 11
and -10x + 6y = -22
Or 5x – 3y – 11 = 0
and -10x + 6y + 22 = 0
Here a1 = 5, b1 = -3, c1 = -11
a2 = -10, b2 = 6, c2 = 22
Hence given pair of linear equations is consistent.
Question 4.
Which of the following pairs of linear equations are consistent / inconsistent? If consistent, obtain the solution graphically.
(i) x + y = 5, 2x + 2y = 10
(ii) x – y = 8, 3x – 3y = 16
(iii) 2x + y – 6 = 0,4x – 2y – 4 = 0
(iv) 2x – 2y – 2 = 0,4x – 4y – 5= 0
Solution:
(1) Given pair of linear equations is
x + y = 5
and 2x + 2y = 10
Or x + y – 5 = 0
2x + 2y – 10 = 0
Here a1 = 1, b1 = 1, c1 = -5
a2 = 2, b2 = 2, c2 = -10
Now
x + y = 5
x = 5 – y ………….(1)
Putting y = 0 in (1), we get:
x = 5 – 0 = 5
Putting y = 3 in (1), we get
x = 5 – 3 = 2
Putting y = 5 in (1), we get
x = 5 – 5 = 0
Plotting the points A (5, 0), B (2, 3), C (0, 5) and drawing a line joining them, we ge the graph of the equation x + y = 5
2x + 2y = 10 Or 2 (x + y) = 10
Or x + y = 5
Or x = 5 – y
Putting y = 0 in (1), we get :
x = 5 – 0 = 5
Putting y = 2 in (2), we get :
x = 5 – 2 = 3
Putting y = 5 in (2), we get:
x = 5 – 5 = 0
Putting y = 2 in (2), we get :
x = 5 – 2 = 3
Putting y = 5 in (2), we get:
x = 5 – 5 = 0
Plotting the points A (5, 0), D (3, 2), C (0, 5) and drawing a line joining them, we get the graph of the equation 2x + 2y = 10
The graphs of two equations are coincident. Hence system of equations has infinitely many solutions i.e. consistent.
(ii) Given pair of linear equations is :
x – y = 8
and 3x – 3y = 16
Or
x – y – 8 = 0
and 3x – 3y – 16 = 0
Plotting the points D (1, 0), B (2, 2), E (0, -2) and drawing a line joining them, we get the graph of the equation
4x – 2y – 4 = 0
From the graph, it is clear that given system of equations meets at a point B (2, 2).
Hence, given pair of linear equations have unique solution.
Question 5.
Half the perimeter of a rectangular garden, whose length is 4 m more than its width is 36 m Find the dimensions of the garden.
Solution:
Let length of garden = x m
Width of garden =y m
Perimeter of garden = 2 [x + y] m
Half perimeter of garden = (x + y) m
According to 1st condition x = y +4
According to 2nd condition
x + y = 36
∴ Pair of linear equations is
x = y + 4
and x + y = 36
x = y + 4 ………………(1)
Putting y = 0 in (1), we get :
x = 0 + 4 = 4
Putting y = – 4 in (1), we get:
x = -4 + 4 = 0
Putting y = 16 in (1), we get :
x = 16 + 4 = 20
Plotting the points A (4, 0) B (0, -4), C (20, 16) and drawing a line joining them.
we get the graph of the equation.
x = y + 4
Now x + y = 36
x = 36 – y
Putting y = 12 in (2), we get:
x = 36 – 12 = 24
Putting y = 24 in (2), we get :
x = 36 – 24 = 12
Putting y = 16 in (2), we get:
x = 36 – 16 = 20
Plotting the points D (24, 12), E (12, 24) C(20, 16) and drawing a line joining them, we get the raph of the equation.
x + y = 36
From the graph. it is clear that pair of linear equations meet at a point C (20, 16).
∴ C (20, 16) i.e. x = 20 and y = 16 is the solution of linear equations.
Hence, length of garden = 20 m
Width of garden = 16 m
Another Method:
Let width of garden = x m
Lengh of garden = (x + 4) m
Perimeter of garden = 2 [Length + Width]
= 2 [x + x + 4] m
= 2[2x + 4]m
∴ Half perimeter of garden = (2x +4) m
According to Question,
2x + 4 = 36
or 2x = 36 – 4
or 2x = 32
or x = 32/16 = 2 m
Hence, width of garden = 16 m
and length of garden = (16 + 4)m = 20 m
Question 6.
Given the Linear equation 2x + 3y – 8 = 0, write another linear equadon in two variables such that the geometiical representation of the pair so formed is:
(i) intersecting lines
(ii) parallel lines
(iii) coincident lines
Solution:
Case I. For Intersecting Unes
Given linear equation is:
2x + 3y – 8 = 0
There are many another linear equations in two variahes which satisfies the condition of intersecing lines i.e.
Case II:
For Parallel Lines
Given linear equation is
2x + 3y – 8 = 0 ……………(1)
There are many other linear equation in two variables which satisfies the condition of parallel lines i.e.
Plotting the points G (2.5, 0), H (-2, 3), I (7, -3) and drawing a line joining them,
we get the graph of the equation
2 + 3y – 5 = 0.
Case III. For Coincident Lines
Given linear equation is
2x + 3y – 8 = 0
There are many other linear equations ¡n two variables which satisfies the condition of coincident lines i.e.
One of which is as follow :
6x+ 9y – 24 = 0
Now, draw the graph of linear equations (1) and (4).
Consider linear equation (4)
6x + 9y – 24 = o
or 3[2x + 3 – 8] = 0
or 2x + 3y – 8 = 0
∴ The points of both are same and line of both equations are same.
Question 7.
Drab tht graphs of the equations x – y + 1 = 0 and 3x + 2y – 12 = 0. Determine the coordinates of the vertices of the triangle formed b these lines and the x-axis and shade the triangular region. (Pb. 2018 Set I, II, III)
Solution:
Consider the pair of linear equation
x – y + 1 =0
and 3x + 2y – 12 = 0
x – y + 1 = 0
or x = y – 1
Putting y = 0 in (1), we get:
x = 0 – 1 = -1
Puning y = 3 in (1) we get:
x = 3 – 1 = 2
Putting y = 1 in (1) , we get:
x = 1 – 1 = 0
Plotting the points D(4, 0) B (2, 3), E (0, 6) and drawing a line oinin them, we get the graph of the equation 3x – 2y – 12 = 0
The vertices of the triangle formed by pair of linear equations and the x-axis are shaded in the graph. The triangle so formed is ∆ABD.Coordinates of the vertices of ∆ABD are
A(-1, 0), B(2, 3) and D(4, 0).
Now, length of Base AD = AO + OD = 1 + 4 = 5 units
Length of perpendicular BF = 3 units
∴ Area of ∆ABD = 1/2 × Base × altitude
= 1/2 × AD × BF
= (1/2 × 5 × 3) sq. units
= 15/2 = 7.5 sq. units
Ex 3.3
Question 1.
Solve the following pair of linear equations by the substitution method:
(i) x + y = 14
x – y = 4
(ii) s – t = 3
s/3 + t/2 = 6
(iii) 3x – y = 3
9x – 3y = 9
(iv) 0.2x + 0.3y = 1.3
0.4x + 0.5y = 2.3
(iii) Given pair of linear equation is:
3x – y = 3 .
and 9x – 3y = 9
From (1),
3x – 3 = y
Or y = 3x – 3 ………….(3)
Substitute this value of y in equation (2), we get :
9x – 3(3x – 3) = 9
Or 9x – 9x + 9 = 9
Or 9 = 9
This statement is true for all values of x. However, we do not get a specific value of x as a solution. Therefore we cannot obtain a specific value of y. This situation has arises because both the given equations are same. Therefore, equations (1) and (2) have infinitely many solutions.
= 18/9 = 2
Hence, x = 2 and y = 3
Question 2.
Solve 2x + 3y = 11 and 2x – 4y = -24 and hence find the value of ‘m’ for which y = mx + 3.
Solution:
Given pair of linear equations is:
2x + 3y = 11
and 2x – 4y = -24 …………(2)
From (2),
2x = 4y – 24
2x = 2 [2y – 12]
Or x = 2y – 12 …………..(3)
Substitute this value of x in (1), we get:
2 (2y – 12) + 3y = 11
Or 4y – 24 + 3y = 11
Or 7y = 11 + 24
Or 7y = 35
y = 35/7 = 5
Substitute this value of y in (3), we get:
x = 2(5) – 12 = 10 – 12 = -2
Now, consider y = mx + 3
Substitute the value of x = -2, y = 5, we get:
5 = m(-2) + 3
Or 5 – 3 = -2m
Or 2 = – 2m
Or -2m = 2
Or m = -1
Hence, x = -2, y = 5 and m = -1
Question 3.
Form the pair of linear equations for the following problems and find their solution by substitution method.
(i) The difference between two numbers is 26 and one number is three times the other. Find them.
(ii) The larger of two supplementary angles exceeds the smaller by 18 degrees. Find them.
(iii) The coach of a cricket team buys 7 bats and 6 balls for ₹ 3800. Later, she buys 3 bats and 5 balls for ₹ 1750. Find the cost of each bat and each ball.
(iv) The taxi charges in a city consist of a fixed charge together with the charge for 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 kilometre? How much does a person have to pay for travelling a distance of 25 km?
(v) A fraction becomes 9/11, if 2 is added to both the numerator and the denominator. If 3 ¡s added to both the numerator and the denominator it becomes 5/6. Find the fraction.
(vi) 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?
Solution:
(I) Let two number be x and y.
According to 1st condition,
x – y=26 ……….(1)
According to 2nd condition,
x = 3y …………(2)
Substitute this value of x in (1), we get :
3y – y = 26
Or 2y = 26
y = 26/2 = 13
Substitute this value of y in (2), we get:
x = 3 × 13 = 39
Hence, two numbers are 39, 13.
(ii) Let, required two supplementary angles are x, y and x > y
According to 1st condition,
x + y = 180 ………..(1)
According to 2nd condition,
x = y + 18 …………..(2)
Substitute this value of x in (1), we get:
y + 18 + y = 180
Or 2y = 180 – 18
or 2y =162
Or y = 162/2 = 81
Substitute this value of y in (2), we get:
x = 81 + 18 = 99
Hence, required angles are 99, 81.
(iv) Let the fixed charges for the taxi = ₹ x
and charges for travelling one km = ₹ y
According to 1st condition,
x + 10y = 105 …………..(1)
According to 2nd condition,
x + 15y = 155 ……………(2)
From (1),
x = 105 – 10y …………..(3)
Substitute the value of x in (2), we get
105 – 10y + 15y = 155
Or 5y = 155 – 105
Or 5y = 50
Or y = 50/5 = 10
Substitute the value of y in (3), we get:
x = 105 – 10 × 10
= 105 – 100 = 5
Hence, fixed charges for the taxi = ₹ 5
and charges for travelling one km = ₹ 10
Also, charges for travelling 25 km = ₹(10 × 25) + ₹ 5
= ₹[250 + 5] = ₹ 255
(v) Let numerator of given fraction = x
Denominator of given fraction = y
∴ Required fraction = x/y
Acccrding to 1st condition,
(vi) Let Jacob’s present age = x years
and Jacob son’s present age = y years
Five years hence
Jacob’s age = (x + 5) years
His son’s age = (y + 5)years
According to 1st condition.
x + 5 = 3(y + 5)
Or x + 5 = 3y + 15
Or x = 3y + 15 – 5
Or x = 3y + 10 ……………(1)
Five years ago
Jacobs age = (x – 5) years
His son’s age = (y – 5) years
According to 2nd condition.
x – 5 = 7(y – 5)
Or x – 5 = 7y – 35
Or x – 7y = -35 + 5
Or x – 7y = -30
Substitute the value of x from (1), we get:
3y + 10 – 7y = -30
– 4y = – 30 – 10
-4y = -40
y = 10
Substitute tins value of y in (1), we get:
x = 3 (10) + 10
= 30 + 10 = 40
Hence. Jacob and his son’s ages are 40 years and 10 years respecùveiy.
Ex 3.4
Question 1.
Solve the following pair of equations by the elimination method and the substitution method.
(i) x + y = 5 and 2x – 3y = 4
(ii) 3x + 4y = 10 and 2x – 2y = 2
(iii) 3x – 5y – 4= 0 and 9x = 2y + 7
(iv) x/2+2y/3 and x – y/3 = 3
Solution:
(i) Given pair of linear equations
x + y = 5 …………..(1)
and 2x – 3y = 4 ………….(2)
Elimination Method
Multiplying (1) by 2, we get:
2x + 2y = 10 …………..(3)
Now, (3) – (2) gives
Substitution method:
From (2),
2x = 2 + 2y
or x = y + 1 ………..(3)
Substitute this value of x in (1), we get :
3(y + 1) +4y = 10
or 3y + 3 + 4y = 10
or 7y = 10 – 3
or 7y = 7
or y = 1
Substitute this value of y in (3), we get :
x = 1 + 1 = 2
Hence, x = 2 and y = 1.
(iii) Given pair of linear equation is :
3x – 5y – 4 = 0 ……..(1)
and 9x = 2y + 7
or 9x – 2y – 7 = 0
Elimination Method:
Multiplying (1) by 3, we get:
9x – 15y – 12 = 0 ……………(3)
Now, (3) – (2) gives
Substitute this value of y in (1), we get :
3x + 4(-3) = -6
or 3x – 12 = -6
or 3x = -6 + 12
or 3x = 6
x = 6/3 = 2
Hence x = 2, y = – 3.
Substitution Method:
From (2), y = 3x – 9 …………..(4)
Substitute this value of y in (1), we get :
3x + 4(3x – 9) = -6
or 3x + 12x – 36 = -6
or 15x = -6 + 36
or 15x = 30
or x = 30/15 = 2
Substitute this value of x in (4), we get :
y = 3 (2) – 9
= 6 – 9 = -3
Hence x = 2, y = -3.
Question 2.
Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method:
(i) If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces to 1. It becomes if
we only add 1 to the denominator. What ¡s the fraction?
(ii) Five years ago, Nun was thrice as old as Sonu. Ten years later, Nun will be twice as old as Sonu. How old are Nun
and Sonti?
(iii) The sum of the digits of a two-digit nunaber ¡s 9. Also, nine times this number is twice the number obtained by reversing the order of the number. Find the number.
(iv) Meena went to a bank to withdraw ₹ 2000. She asked the cashier to give her ₹ 50 and ₹ 100 notes only. Meena got ₹ 25 notes in all. Find how many notes of ₹ 50 and ₹ 100 she received.
(v) 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.
or x = 3
Substitute this value of x in (2), we get :
2 × 3 – y – 1 = 0
or 6 – y – 1 = 0
or 5 – y = 0
or y = 5
Hence, required fraction is 3/5.
(ii) Let Nun’s present age = x years
Sonus present age = y years
Five years ago
Nun’s age = (x – 5) years
Sonus age = (y – 5) years
According to 1st condition,
x – 5 = 3(y – 5)
or x – 5 = 3y – 15
or x – 3y + 10 = 0 …………….(1)
Ten years later
Nun’s age = (x + 10) years
Sonu’s age = (y + 10) years
According to 2nd condition,
x + 10 = 2 (y + 10)
or x + 10 = 2y + 20
or x – 2y – 10 = 0
Now, (1) – (2) gives
or -y = -20
or y = 20
Substitute this value of y in (2), we get:
x – 2(20) – 10 = 0
or x – 40 – 10 = 0
or x = 50
Hence, Nun’s present age = 50 years
Sonu’s present age = 20 years.
(iii) Let unit’s digit = x
Ten’s digit = y
∴ Required Number = 10y + x
According to 1st condition,
x + y = 9 …………..(1)
On reversing
Unit’s digit = y
Ten’s digit = x
∴ Number = 10x + y
According to 2nd condition,
9[10y + x] = 2[10x + y]
or 90y + 9x = 20x + 2y
or 90y + 9x – 20x – 2y = 0
or -11x + 88y = 0
or x – 8y = 0 ………………(2)
Now, (2) – (1) gives
y = 1
Substitute this value of y in (2), we get :
x – 8 × 1 = 0
or x = 8
Hence, required number = 10y + x
= 10 × 1 + 8 = 18.
(iv) Let, Meena received number of Rs. 50 notes = x
also, Meena received number of Rs. 100 notes = y
According to 1st condition,
x + y = 25 ……………(1)
According to 2nd condition,
50x + 100y = 2000
or x + 2y = 40 ………………(2)
Now, (2) – (1) gives
Substitute this value of y in (1), we get:
x + 15 = 25
or x = 25 – 15 = 10
Hence, Meena received number of notes of
Rs. 50 and Rs. 100 are 10 and 15 respectively.
(v) Let fixed charges for first three days = ₹ x
An additional charge for each day thereafter = ₹ y
In case of Saritha,
x + 4y = 27 …………..(1)
In case of Susy,
x + 2y = 21
Now, (1) – (2) gives
or y = 6/2 = 3
Substitute this value of y in (2), we get:
x + 2(3) = 21
or x + 6 = 21
or x = 21 – 6 = 15
Hence, fixed charges for first three days and an additional charge for each day thereafter ₹ 15 and ₹ 3.
Ex 3.5
Question 1.
Which of the following pairs of linear equations has unique solution, no solution, or infinitély many solutions. In case there is a unique solution, find it by using cross multiplication method.
(i) x – 3y – 3 = 0
3x – 9y – 2 = 0
(ii) 2x + y = 5
3x + 2y = 8
(iii) 3x – 5y = 20
6x – 10y = 40
(iv) x – 3y – 7 = 0
3x – 3y – 15 = 0
Solution:
(i) Given pair of linear equation is:
x – 3y – 3 = 0
and 3x – 9y – 2 = 0
Here a1 = 1, b1 = -3, c1 = -3
a2 = 3, b2 = -9, c2 = -2
(ii) Given pair of linear equations
2x + y = 5
and 3x + 2y = 8
or 2x + y – 5 = 0
and 3x + 2y – 8=0
Here a1 = 2, b1 = 1, c1 = -5
a2 = 3,b2 = 2, c2 = 8
Now,
From I and III, we get:
y/1=1/1
⇒ y = 1
Hence, x = 2 and y = 1.
Question 2.
(i) For which values of a and b does the following pair of linear eqU1tions have an infinite number of solutions?
2x + 3y = 7
(a – b)x ÷ (a + b)y = 3a + b – 2
(ii) For which value of k wifi the following pair of linear equations have no solution?
3x + y = 1
(2k – 1) x + (k – 1) y = 2k + 1
Solution:
(i) Given pair of linear equation are
2x + 3y = 7
and (a – b)x + (a + b)y = 3a + b – 2
or 2x + 3y – 7 = 0
and (a – b)x + (a + b)y – (3a + b – 2) = 0
Here a1 = 2, b1 = 3, c1 = -7
a2 = a – b, b2 = a + b, c2 = -(3a + b – 2)
∵ System of equation have an infinite number of solutions.
Substitute the value of a from (1) in above, we get:
9b – 4 – 2b – 3 = 0
or 7b – 7 = 0
or 7b = 7
b = 1
Substitute this value of b in (1), we get
a = 9 × 1 – 4 = 9 – 4
a = 5
Hence a = 5 and b = 1
(ii) Given pair of linear equation are
3x + y = 1
and (2k – 1)x + (k – 1)y = 2k + 1
or 3x + y – 1 = 0
and(2k – 1)x + (k – 1)y – (2k + 1) = 0
Here a1 = 3, b1 = -1, c1 = -1
a2 = (2k – 1), b2 = k – 1, c2 = -(2k + 1)
∵ system of equations have no solution.
Question 3.
Solve the following pair of linear equations by the substitution and cross- multiplication methods:
8x + 5y = 9
3x + 2y = 4
Solution:
Given pair of linear equation is:
8x + 5y = 9 ………….(1)
3x + 2y = 4 …………..(2)
Substitution Method:
Cross-multiplication Method:
Given pair of linear equation is:
8x + 5y – 9 = o
and 3x + 2y – 4= 0
Here a1 = 8, b1 = 5, c1 = -9
a2 = 3, b2 = 2, c2 = -4
Now,
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 1000 as hostel charges whereas a student B, who takes food for 26 days, pays 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 differ it 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 units. Find the dimensions of the rectangle.
Solution:
(i) Let monthly fixed hostel charges = ₹ x
and cost of food per day = ₹ y
According to 1st condition
x + 20y = 1000 ………….(1)
According to 2nd condition
x + 26y = 1180 …………..(2)
Hence, monthly fixed hostel charges and cost of food per day are 400 and 30 respectively.
∴ Number of right questions = 15
Number of wrong questions = 5
Hence, total number of questions = [No. of right questions] + [No. of wrong questions]
=15 + 5 = 20.
(iv) Let speed of car at place A = x km/hour
and speed of car at place B = y km/hour
Distance between places A and B = 100 km
In case of 5 hours
Distance covered by car A = 5x km
[∵ Distance = Speed × Time]
Distance covered by car B = 5y km
According to I st condition,
5x – 5y = 100
or x – y = 20
or x – y – 20 = 0
In case of one hour
Distance covered by car A = x km
[∵ Distance = Speed × Time]
Distance covered by car B = y km
According to 2nd condition,
x + y = 100
or x + y – 100 = 00 ……………….(2)
Hence, length and breadth of rectangle are 17 units and 9 units respectively.
Ex 3.6
Question 1.
Solve the following pairs of equations by reducing them to a pair of linear equations:
Question 2.
Formulate the following problems as a pair of equations, and hence find their solutions.
(i) 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.
(ii) 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.
(iii) Roohi travels 300 km to her home party by train and partly by bus. She takes 4 hours if 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.
Solution:
(i) Let the speed of Ritu in still water = x km/hour
and the speed of current = y km/hour
∴ speed in upstream = (x – y) km/hour
and speed in downstream = (x + y) km/hour
Distance covered by Ritu in downstream in 2 hours = Speed × Time
= (x + y) × 2 km
According to 1st condition
2(x + y) = 20
x + y = 10 ……………..(1)
Distance covered by Rim in upstream in 2 hours
= Speed × Time
= 2(x – y)km
According to 2nd condition,
2(x – y) =4
x – y = 2 ……………….(2)
Now, (1) + (2) gives
Substitute this value of x in (1), we get:
6 + y = 10
y = 10 – 6 = 4
Hence, Ritu’s speed in still water = 6 km/hour
and speed of current = 4 km/hour.
or y = 80
Hence, speed of train and bus are 60 km/hour and 80 km/hour respectively.
Ex 3.7
Question 1.
The ages of two friends Ani and Biju differ by 3 years. Ani’s father Dharam is twice as old as Am and Biju Is twice as old as his sister Cathy. The ages of Cathy and Dharam differ by 30 years. Find the ages of Ani and Biju.
Solution:
Let Ani’s age = x years
and Biju’s age = y years
Dharam’s age = 2x years
Cathy’s age = years
According to 1st condition,
(Ani’s age) (Biju’s age) = 3
x – y = 3 ……………(1)
According to 2nd condition,
(Dharam’ age) – (Cathy’s age) = 30
2x – y/2 = 30
or 4x−y/2 = 30
or 4x – y = 60 ………….(2)
Now (2) – (1) gives,
Substitute this value of x in (1), we get:
19 – y = 3
or -y = 3 – 19
or -y = -16
or y = 16
Hence, Ani’s age = 19 years
Biju’s age =16 years.
Question 2.
One says, “Give me a hundred, friend! I shall then become twice as rich as you”. The other replies, “if you give me ten, I shall be six times as rich as you.” Tell me what is the amount of their (respective) capital ? [From the Bijaganita of Bhaskara II] [Hint: x + 100 = 2(y – 100), y + 10 = 6 (x – 10)].
Solution:
Let Capital of one friend = ₹ x
and capital of 2nd friend = ₹ y
According to 1st condition
x + 100 = 2(y – 100)
or x + 100 = 2y – 200
or x – 2y = -200 – 100
or x – 2y = -300 …………..(1)
According to 2nd condition
y + 10 = 6(x – 10)
or v-f 10 = 6x – 60
or 6x – y = 10 + 60
or 6x – y = 70 ……………(2)
Multiplying (1) by 6, we get
6x – 12y = – 1800 …………….(3)
Now, (3) – (2) gives
Substitute this value of y in (2), we get:
6x – 170 = 70
or 6x = 70 + 170
or 6x = 240
or x = 240/6 = 40
Hence, amount of their capital are 40 and 170 respectively.
Question 3.
A train covered a certain distance at a uniform speed. If the train would have been 10 km/b faster, It would have taken 2 hours less than the scheduled time. And, if the train were slower by 10 km/h; it would have taken 3 hours more than the scheduled time. Find the distance covered by the train.
Solution:
Let speed of train =x km/hour
and time taken by train =y hour
∴ Distance covered by train = (Speed) (Time) = (xy) km
According to 1st condition,
(x + 10)(y – 2) = xy
or xy – 2x + 10y – 20 = y
or -2x + 10y – 20 = 0
or x – 5y + 10 = 0
According to 2nd condition,
(x – 10) (y + 3) = xy
or xy + 3x – 10y – 30 = xy
or 3x – 10y – 30 = 0
Multiplying (1) by 3, we get:
3x – 15y + 30 = 0
Now, (3) – (2) gives
Substitute this value of y in (1), we get:
x – 5 × 12 + 10 = 0
or x – 60 + 10 = 0
or x – 50 = 0
or x = 50
Speed of train = 50 km/hour
Time taken by train = 12 hour
Hence, distance covered by train = (50 × 12) km = 600 km.
Question 4.
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.
Solution:
Let number of students in each row = x
and number of rows = y
Total number of students in the class = xy
According to 1st condition,
(x + 3) (y – 1) = xy
or xy – x + 3y – 3 = xy
or -x + 3y – 3 = 0
or x – 3y + 3 = 0
According to 2nd condition,
(x -3) (y + 2) = xy
or xy + 2x – 3y – 6= xy
or 2x – 3y – 6 = 0
Now, (2) – (1) gives
x = 9
Substitute this value of x in (1), we get:
9 – 3y + 3 = 0
or -3y + 12 = 0
or -3y = -12
or y = 12/3 = 4
∴ Number of students in each row = 9 and number of rows = 4
Hence, total number of students in the class = 9 × 4 = 36.
Question 5.
In a ∆ABC, ∠C = 3 ∠B = 2(∠A +∠B) find the three angles.
Solution:
In ∆ABC,
Given that, ∠C = 3∠B = 2(∠A + ∠B)
I II III
From II and III, we get:
3∠B =2(∠A+∠B)
or 3∠B = 2∠A + 2∠B
or 3∠B – 2∠B = 2∠A
or ∠B = 2∠A ……………..(1)
From I and II, we get:
∠C = 3∠B
or ∠C = 3(2∠A) [using (1)]
or ∠C = 6∠A …………….(2)
Sum of three angles of a triangle is 180°
∠A + ∠B + ∠C = 180°
or ∠A + 2∠A + 6∠A = 180°
or 9∠A = 180°
∠A = 180°/9 = 20°
Hence, ∠A = 20°: ∠B =2 x 20° = 40°; ∠C = 6 x 20° = 120°.
Question 6.
Draw the graphs of the equations 5x – y = 5 and 3x – y = 3. Determine the co-ordinates of the vertices of the triangle
formed by these lines and the y axis.
Solution:
Given pair of linear equation are 5x – y = 5 and 3x – y = 3
Consider,
5x – y = 5
or 5x = 5 + y
Putting y = 0 in (1), we get:
Plotting A (1, 0); D (0, -3); E (2, 3) on the graph. we get the equation of line 3x – y = 3. From the graph, it is clear that given lines intersect at A (1, 0). Triangle formed by these lines and y axis are shaded in the graph i.e. ∆ABD. Coordinates of the vertices of ∆ABD are A(1, 0); B(0, -5) and D(0, -3).
Question 7.
Solve the following pair of linear equations:
(i) px + qy = p – q
qx – py = p + q
(ii) ax + by = c
bx + ay = 1 + c
(iii) x/a−y/b = 0
ax + by = a2 + b2
(iv) (a – b)x + (a + b)y = a2 – 2ab – b2
(a + b) (x + y) = a2 + b2
(v) 152x – 378y = 74
-378x + 152y = -604
Solution:
(i) Given pair of linear equation are
px + qy = p – q …………(1)
and qx – py = p + q ………….(2)
Multiplying (1) by q and (2) by p, we get:
or y = -1
Substitute this value of y in (1), we get:
px + q(-1) = p – q
or px – q = p – q
or px = p – q + q
or px = p
or x = 1
Hence, x = 1 and y = -1.
(ii) Given pair of linear equation are
ax + by = c
bx + ay = 1 + c
ax + by – c = 0
bx + ay – (1 + c) = 0
(iv) Given pair of linear equation are
(a – b)x + (a + b)y = a2 – 2ab – b2
or ax – bx + ay + by = a2 – 2ab – b2 …………….(1)
and (a + b) (x + y) = a2 + b2
or ax + bx + ay + by = a2 + b2
Now, (1) – (2) gives
(v) Given pair of linear equation are
152x – 378y = – 74
and -378x + 152y = -604
or 76x – 189y + 37 = 0
and -189x + 76y + 302 = 0
Question 8.
ABCD is a cyclic quadrilateral (see Fig.). Find the angles of the cyclic quadrilateral.
Solution:
In cyclic quadrilateral ABCD,
∠A = (4y + 20); ∠B = 3y – 5; ∠C = 4x and ∠D = 7x + 5
Sum of opposite angles of a cyclic quadrilateral are of measure 180°.
∴ ∠A + ∠C = 180°
or 4y + 20 + (4x) = 180°
or 4x + 4y = 180° – 20
or 4x + 4y = 160
or x + y = 40
or y = 40 – x ……………..(1)
and ∠B + ∠D = 180°
or 3y – 5 + (7x + 5)= 180°
or 3y – 5 + 7x + 5 = 180°
or 7x + 3y = 180° …………….(2)
Substitute the value of y from (1) in (2). we get:
7x + 3(40 – x)= 180°
or 7x + 120 – 3x = 180°
or 4x = 180 – 120
or 4x = 60
x = 60/4 = 15
Substitute this value of x in (1 ), we get:
y = 40 – 15 = 25
∴ ∠A = 4y + 20 = 4 × 25 + 20 = 120°
∠B = 3y – 5 = 3 × 25 – 5 = 70°
∠C = 4x =4 × 15 = 60°
∠D = 7x + 5 = 7 × 15 + 5 = 110°
Hence, ∠A = 120°, ∠B = 70°; ∠C = 60° and ∠D = 110°.