Linear Equations in Two Variables Class 10

Ex 3.1, 1 teachoo.com
Form the pair of linear equations in the following problems & 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.
Let Number of Girls who took part in the quiz be x
& Number of Boys who took part in the quiz be y

Given

Total 10 students took part in the quiz

-. Number of girls + Number of boys = 10

xty=10 (1)

Ex 3.1, 1 teachoo.com
Form the pair of linear equations in the following problems &
find their solutions graphically
(ii} 5 pencils and 7 pens 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
Let the Cost of one Pencil be Rs x

& Cost of one Pen be Rs y
Given that

5 pencils and 7 pens together cost Rs 50

5 x (Cost of pencil} + 7 x (Cost of pens) = 50

5x + 7y =50 (1)

Ex 3.1, 2 teachoo.com
On comparing the ratios “ , & 7 , find out whether the lines
representing the following pair of linear equations intersect at a
point, parallel or coincident
(i) Sx-4y+8=0;7x+6y-—9=0
5x—4y+8=0 (1)
7x + 6y-9=0 (2)
5x -4y +8=0 7x+6y-9=0
Comparing with a,x + b,y +c, =0 | Comparing with a,x + b,y+c,=0
~a,=5,b,=-4,¢,=8 “a =7,b,=6,c,=-9

teachoo.com
Ex 3.1, 2
On comparing the ratios “ > & 7 , find out whether the lines
representing the following pair of linear equations intersect at a
point, parallel or coincident
(ii) 9x + 33y +12 = 0; 18x + 6y+ 24-0
9x + 3y +12=0 (1)
18x + 6y + 24=0 ...(2)
9x +3y+12=0 18x + 6y + 24=0
Comparing with a,x + b,y + c,=0 | Comparing with a,x + boy + c,=0
a,=9,b6,=3,¢,=12 “a, =18,b,=6,¢,=24

teachoo.com
Ex 3.1, 2
On comparing the ratios “ , 7 & 7 , find out whether the lines
representing the following pair of linear equations intersect at a
point, parallel or coincident
(iii} 6x -3y +10=0;2x-y+9=0
6x-3y+10=0 (1)
2x-y+9=0 wn (2)
6x-3y+10=0 2x-y+9=0
Comparing with a,x + b,y +c, =0 | Comparing with a,x + b,y + c, =0
“a, =6,b,=-3,¢,=10 “a,=2,b,=-1,c,=9

teachoo.com
Ex 3.1, 3
On comparing the ratios “ , 7 & 7 , find out whether the
following pair of linear equations are consistent, or inconsistent.
(i) 3x+2y=5;2x-3y=7
3x+2y-5=0 (1)
2x—-3y-7=0 (2)
3x+2y-5=0 2x-3y-7=0
Comparing with a,x + b,y + c, = 0 | Comparing with a,x + b,y +c, =0
“a, =3,b,=2,¢,=-5 “a =2,b,=-3,¢),=-7

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Ex 3.1, 3
On comparing the ratios “ , & 7 , find out whether the
following pair of linear equations are consistent, or inconsistent.
(ii) 2x - 3y = 8; 4x - 6y =9
2x-3y-8=0 (1)
4x -6y-9=0 (2)
2x -—3y-8=0 4x -6y-9=0
Comparing with a,x + b,y + c, = 0 | Comparing with a,x + b,y+c,=0
“a,=2,b,=-3,¢,=-8 “a, =4,b,=-6,¢,=-9

Ex 3.1, 3 teachoo.com

On comparing the ratios “ , 7 & 7 , find out whether the

following pair of linear equations are consistent, or inconsistent.

way 3S

(ili) 5x + Gv = 7; 9x - 10y = 14

3 5

9x-—10y-14=0 (2)

3x4 2y-7=0 9x-10y-14=0

Comparing with a,x + b,y+ ¢,=0 Comparing with a,x + bay + c= 0
3 5 “a, =9,b,=-10, c,= -14

“ay=5, by= 5,05 -7

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Ex 3.1,3 CAEmOOeOm
On comparing the ratios “ , & 7 , find out whether the
following pair of linear equations are consistent, or inconsistent.
(iv) 5x — 3y = 11, -10x + 6y = -22
5x-3y-11 =0 (1)
-10x + 6y +22 =0 (2)
5x-3y-11 =0 -10x + 6y +22=0
Comparing with a,x + b,y + c, = 0 | Comparing with a,x + b,y+c,=0
~a,=5,b,=—-3,c,=-11 “a,=-10,b,=6,c,=22

teachoo.com
Ex 3.1, 3
On comparing the ratios “ , 7 & 7 , find out whether the
following pair of linear equations are consistent, or inconsistent.
(v) 5x4 2y=8; 2x4 3y=12
4
3Xx+2y-8=0 (1)
2x +3y-12=0 (2)
5x+2y-8=0 2x +3y-12=0
Comparing with a,x + b,y +c, =0 Comparing with a,x + boy + ¢,=0
4 “a, =2,b)=3,c,=-12
“aes, b,=2,c,=-8

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Ex 3.1, 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
x+y=5 (1)

2x+2y=10 _ ...(2)
x+y=5 2x + 2y =10
xt+y-5=0 2x+2y-10=0
Comparing with a,x + b,y + c,=0 | Comparing with a,x + b,y+c,=0
“a,=1,b,=1,¢,=-5 a, =2,b,=2,¢,=-10

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Ex 3.1, 4
Which of the following pairs of linear equations are consistent/
inconsistent? If consistent, obtain the solution graphically
(ii) x-y =8, 3x-3y =16
x-y=8 (1)
3x — 3y = 16 (2)
x-y=8 3x - 3y =16
x-y-8=0 3x —3y-16=0
Comparing with a,x +b,y+c,=0 | Comparing with a,x + b,y+¢,=0
“a,=1,b,=-1,c,=-8 “a, =3,b,=-3,¢,=-16

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Ex 3.1, 4
Which of the following pairs of linear equations are consistent/
inconsistent? If consistent, obtain the solution graphically
(iii) 2x + y-6=0, 4x-2y-4=0
2x+y-6=0 .-(1}
4x-2y-4=0 ...(2)
2x+y-6=0 4x-2y-4=0
Comparing with a,x + b,y + c, = 0 | Comparing with a,x + b,y+c,=0
a, =2,b6,=1,¢,=-6 “a, =4,b,=-2,¢,=-4

teachoo.
Ex 3.1, 4 ‘eachoo.com
Which of the following pairs of linear equations are consistent/
inconsistent? If consistent, obtain the solution graphically
(iv) 2x - 2y-4=0, 4x -4y-5=0
2x-2y-4=0 (1)
4x-4y-5=0 ...(2)
2x—-2y-4=0 4x-4y-5=0
Comparing with a,x + b,y + c, = 0 | Comparing with a,x + b,y+c,=0
“a, =2,b,=-2,¢,=-4 “a =4,b).=-4,c,=-5

Ex 3.1,5 teachoo.com

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.
po
—<——$_____——+

x
Let Length of Rectangular Garden be x meters
& Breadth of Rectangular Garden be y meters
Given
Half perimeter of rectangular garden is 36m
; x 2(Length + Breadth)= 36
x+y=36 .(1)

Ex 3.1, 6 teachoo.com
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 lines (ii) parallel lines (iii) coincident lines
Given equation

2x+3y-8=0_..(1)
Therefore,

a,=2,b,=3,c,=-8

Ex 3.1, 7 teachoo.com
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 x-axis, and shade the triangular region.
Our equations are

x-y=-l .(1)

3x + 2y=12 ...(2)

teachoo.com
Ex3.2,1
Solve the following pair of linear equations by the substitution
method.
(i) xt+y=14
x-y=4

x +y=14 (1)

x-y=4 (2)
From equation (1)

xty=14

x=14-y

Ex 3.2,1 teachoo.com
Solve the following pair of linear equations by the substitution
method.
(ii}s-t=3
Ss t
3 + z =6
s-t=3 (1)
Ss t
gta 6 ...{2)
From (1)
s-t=3
s=3+t

Ex 3.2, 1 teachoo.com
Solve the following pair of linear equations by the substitution
method.
(iii) 3x -y = 3
9x - 3y=9

3x-y=3 (1)

9x-3y=9 ...(2)
Solving (1)

3x-y=3

3Bx=yt+3

yet 3

3

Ex 3.2, 1 teachoo.com
Solve the following pair of linear equations by the substitution
method.
(iv) 0.2x + 0.3y = 1.3
0.4x + O.5y = 2.3

0.2x + 0.3y = 1.3 (1)
0.4x + 0.5y = 2.3 (2)
From (1)

0.2x + 0.3y = 1.3
Multiplying both side by 10

(0.2x + 0.3y} x 10 = 1.3 x 10

2x + 3y = 13

2x = 13 -3y

Ex 3.2, 1 teachoo.com
Solve the following pair of linear equations by the substitution
method.
(v) V2x + V3y = 0
v3x — vV8y =0

V2x+v3y=0 wa(1)
v3x—V8 y = 0 (2)
From (1)

V2x+vV3y=0

V2x=-v3y

x23

~ Vf2

Ex 3.2,1 teachoo.com
Solve the following pair of linear equations by the substitution
method.
ay 3X Sy _
(vi) 7 _ 3 = 2
x ye 13
312%
Removing fractions from both equations
3x Sy x ,y_13
273? 372 = %
Multiplying both equations by 6 | Multiplying both equations by 6
3x oy x y_ 13
6x —-6x==6x-2 6x —-+6x-=6x —
2 3 3 2 6
9x—10y =- 12 ...{1) 2x+3y=13 (2)

Ex 3.2, 2 teachoo.com
Solve 2x + 3y= 11 and 2x - 4y = — 24 and hence find the value of ‘m’
for which y = mx + 3.

2x+3y=11 (1)

2x — 4y =—-24 (2)
From (1)

2x+3y=11

2x=11-3y

11 -—3y
x=———
2

Ex 3.2, 3 teachoo.com
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.
Let Larger Number be x
& Smaller Number be y
Given that
Difference between two numbers = 26
x-y=26 (1)
Also, one number is 3 times other
x= 3y w(2}

Ex 3.2, 3 teachoo.com
Form the pair of linear equations for the following problems and
find their solution by substitution method.
(ii) The larger of two supplementary angles exceeds the smaller
by 18 degrees. Find them.
Let Larger angle be x
& Smaller angle be y
Given that
Larger angle exceeds Smaller angle by 18°
x-y=18 (1)
Also both angles are supplementary
Sum of both angles = 180°

Ex 3.2, 3 teachoo.com
Form the pair of linear equations for the following problems and
find their solution by substitution method.
(iii) The coach of a cricket team buys 7 bats and 6 balls for Rs 3800.
Later, she buys 3 bats and 5 balls for Rs 1750. Find the cost of each
bat and each ball.
Let Cost of one bat = Rs x
& Cost of one ball = Rs. y

Given that
Coach buys 7 bats and 6 balls for Rs 3800

7 x (Cost of one bat ) + 6 x (Cost of one ball) = 3800

7x + 6y = 3800 ...(1)

Ex 3.2, 3 teachoo.com
Form the pair of linear equations for the following problems and
find their solution by substitution method.
(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 Rs 105 and for a journey of 15 km, the charge paid is
Rs 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?
Let Fixed charge = Rs x

& Charge per km = Rs y
Given that Charge paid for 10 km is Rs 105

Fixed charge + 10 X (Charge per km) = Rs 105

x+10y=105 ...(1)

Ex 3.2, 3 teachoo.com
Form the pair of linear equations for the following problems and
find their solution by substitution method.
(v) A fraction becomes = , if 2 is added to both the numerator and
the denominator. If, 3 is added to both the numerator and the
denominator it becomes = . Find the fraction.
Let Numerator be x
& Denominator be y
So, fraction is 2
y

Given that
If 2 is added to both the numerator and the denominator, fraction
becomes—

11

Ex 3.2, 3 teachoo.com
Form the pair of linear equations for the following problems and
find their solution by substitution method.
(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?
Let Present age of Jacob = x years
& Present age of Jacob’s son = y years

Five years hence (later),

Jacob’s Age =x +5

Jacob son’s Age =y +5

teachoo.com
Ex 3.3, 1 (Elimination)
Solve the following pair of linear equations by the elimination
method and the substitution method :
(i)x+y=5 and 2x-3y=4
xty=5 (1)
2x -—3y=4 (2)
Multiplying equation (1) by 2
Axty)=2%5
2x +2y=10 (3)

Solve the following pair of linear equations by the elimination method and the substitution method :
(ii) 3x + 4y = 10 and 2x – 2y = 2

Solve the following pair of linear equations by the elimination method and the substitution method :
(iii) 3x – 5y – 4 = 0 and 9x = 2y + 7

Solve the following pair of linear equations by the elimination method and the substitution method :
(iv) 𝑥/2+2𝑦/3=−1 𝑎𝑛𝑑 𝑥−𝑦/3=3

Ex 3.3, 2 teachoo.com
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 is the fraction?

Let Numerator be x

and Denominator be y

So, Fraction is

Ex 3.3, 2 teachoo.com
Form the pair of linear equations in the following problems, and
find their solutions (if they exist) by the elimination method :
(ii) 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?
Let Present age of Nuri = x years
& Present age of Sonu = y years

Five years ago ,

Nuri’s age = x — 5 years

Sonu’s age = y — 5 years

Ex 3.3, 2 teachoo.com
Form the pair of linear equations in the following problems, and
find their solutions (if they exist) by the elimination method :

(iii) 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.

Number is of the form

x Y
Tens Units
place place
Let Digit at Units place = y
& Digit at Tens place = x

Ex 3.3, 2 teachoo.com
Form the pair of linear equations in the following problems, and
find their solutions (if they exist) by the elimination method :

(iv) 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 Rs 50 and Rs 100 she received.
Let Number of Rs. 50 notes = x

& Number of Rs. 100 notes = y
Given that
Total notes is 25
(Number of Rs 50 notes) + (Number of Rs 100 notes) = 25

xty=25 (1)

Ex 3.3, 2 teachoo.com
Form the pair of linear equations in the following problems, and find
their solutions (if they exist) by the elimination method
(v} A lending library has a fixed charge for the first three days and an
additional charge for each day thereafter. Saritha paid Rs 27 fora
book kept for seven days, while Susy paid Rs 21 for the book she kept
for five days. Find the fixed charge and the charge for each extra day.
Let Fixed charge for first 3 days = Rs x
& Additional charge after 3 days = Rs y per day

Given that

Saritha paid Rs 27 for a book kept for seven days

Fixed charge for first three days = Rs 27

+ (Number of additional days kept) x (Additional charges per day)

teachoo.com
Example 1
Check graphically, whether the pair of equations x + 3y = 6 and
2x — 3y= 12 is consistent. If so, solve them graphically.
Let equations be
x+3y=6 (1)

2x —3y=12 (2)

Let’s draw their graphs

teachoo.com
Example 2
Graphically, find whether the following pair of equations has no
solution, unique solution or infinitely many solutions:
24 3
Sx-8y+1=0&3x- —y+—=0
5 5
Given equations are
5x -8y+1=0
3. 24 + 3.9
x SYS
Simplifying them
...(1}
5x - 8y=-1
15x — 24y = -3 ...(2}

Example 3 teachoo.com
Champa went to a ‘Sale’ to purchase some pants and skirts. When her
friends asked her how many of each she had bought, she answered,
“The number of skirts is two less than twice the number of pants
purchased. Also, the number of skirts is four less than four times the
number of pants purchased”. Help her friends to find how many pants
and skirts Champa bought.
Let Number of pants purchased by Champa be x
& Number of skirts purchased by Champa be y
Given that
Number of skirts is two less than twice the number of pants purchased
(Number of skirts purchased) = 2 x (Number of pants purchased) - 2
y =2(x)- 2
2x-y= 2 ...(1)

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Example 4
Solve the following pair of equations by substitution method:
7X—15y=2
Xt2y=3

7x— 15y =2 (1)

x+2y=3 ...(2)
From (1)

7x —15y =2

7x =2+15y

2+ 15y
a

Example 5 teachoo.com
Aftab tells his daughter, “Seven years ago, | was seven times as old as
you were then. Also, three years from now, | shall be three times as
old as you will be.” (Isn’t this interesting?) Solve by the method of
substitution..
Let the Current age of Aftab be x years.
& let Current age of Aftab’s daughter be y years.

Given that
“Seven years ago, | was seven times as old as you were then”
Seven years ago,

Age of Aftab = x - 7

Age of Aftab’s daughter = y - 7

Example 6 teachoo.com
Romila went to a stationery shop and purchased 2 pencils & 3
erasers for Rs 9. Her friend Sonali saw the new variety of pencils
and erasers with Romila, and she also bought 4 pencils and 6
erasers of the same kind for Rs 18. Find the cost of each pencil and
each eraser.
Let the Cost of Pencil be Rs x
& Let the Cost of Eraser be Rs y
Given that
Romila purchased 2 pencils & 3 erasers for Rs 9

2 x (Cost of pencil) + 3 x (Cost of eraser) =9

2x+3y=9 (1)

Example 7 teachoo.com
Two rails are represented by the equations x + 2y— 4 = O and
2x + 4y — 12 = 0. Will the rails cross each other?
Given equations are
x+2y=4 (1)
2x + 4y = 12 (2)
We need to check if the rails cross each other
i.e. if both equations form a unique solution
From (1)
x+2y-4=0
x=4-2y

Example 8 teachoo.com
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, find their monthly incomes.
Given that
Ratio of income of two persons is 9: 7
Let Income of 1° person be 9x
& Income for 2 person be 7x
Similarly,
Ratio of expenditures of two persons is 4: 3
Let Expenditure of 1t person be 4y
& Expenditure of 2" person be 3y

Example 9 teachoo.com
Use elimination method to find all possible solutions of the following
pair of linear equations :

2x+3y=8

Ax + 6y=7

2x + 3y=8 (1)

4x + 6y=7 (2)
We multiply (1) by 2

2(2x + 3y)=2*8

4x + 6y = 16 (3)

teachoo.com
Example 10
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?
Number is of the form
x YY

Tens Units

place place
Let Digit at Units place = y

& Digit at Tens place = x

Example 1 teachoo.com
Akhila went to a fair in her village. She wanted to enjoy rides on
the Giant Wheel and play Hoopla (a game in which you throw a
ring on the items kept in a stall, and if the ring covers any object
completely, you get it). The number of times she played Hoopla is
half the number of rides she had on the Giant Wheel. If each ride
costs Rs 3, and a game of Hoopla costs Rs 4, how would you find
out the number of rides she had and how many times she played
Hoopla, provided she spent Rs 20. Represent the situation
algebraically and graphically.

Let the number of rides Akhila has on the giant wheel be x.

Let the number of times Akhila played Hoopla be y.

teachoo.com

Example 2
Romila went to a stationery shop and purchased 2 pencils & 3 erasers
for Rs 9. Her friend Sonali saw the new variety of pencils and erasers
with Romila, and she also bought 4 pencils and 6 erasers of the same
kind for Rs 18. Represent this situation algebraically and graphically.
Let the Cost of Pencil be Rs x
& Let the Cost of Eraser be Rs y
Given that
Romila purchased 2 pencils & 3 erasers for Rs 9

2 x (Cost of pencil} + 3 x (Cost of eraser) = 9

2x+3y=9 (1)

teachoo.com
Example 3
Two rails are represented by the equations x + 2y-— 4 =O and
2x + 4y — 12 = 0. Represent this situation geometrically.
Let equations be
xt2y=4 (1)
2x + 4y = 12 ..{2)

teachoo.com

Example 14
From a bus stand in Bangalore , if we buy 2 tickets to 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 Malleswaram, and to
Yeshwanthpur.
Let the Price of 1 ticket to Malleswaram = Rs x

& Price of 1 ticket to Yeshwanthpur = Rs y
Given that,
If we buy 2 tickets to Malleswaram and 3 tickets to Yeshwanthpur,
the total costis Rs.46

Example 15 teachoo.com
For which values of p does the pair of equations given below has
unique solution?

4x+py+8=0

2x+2y+2=0
4x + py+8=0 (1)
2x+2y+2=0 ..{2)
4x + py+8=0 2x + 2y+2=0
Comparing with a,x + b,y + c, = 0 | Comparing with a,x + b,y+c,=0
~a,=4,b,=p,c,=8 “a =2,b),=2,c),=2

Example 16 teachoo.com
For what values of k will the following pair of linear equations
have infinitely many solutions?

kx + 3y— (k-3) =0

12x+ky-k=0

kx + 3y—(k—3) =0 (1)

12x+ky-k=0 (2)
kx + 3y -(k-3)=0 12x+ky—-k=0
Comparing with a,x + b,y + c, = 0 | Comparing with a,x + b,y+c,=0
“a, =k,b,=3,c,=—(k—3) “a, =12,b,=k,c)=-k

Example 17 teachoo.com
Solve the pair of equations:

243243

x oy

S_4__9

x oy

2 3 _ 1

,t5=13 (4) } let== u

1

S42 —=V

yoy 2 ...(2) y
So, our equations become

2u+3v =13 _...(3) | 5u-4v=-2 ..{4)

Example 18 teachoo.com
Solve the following pair of equations by reducing them to a pair of
linear equations :

> 4,14 <9

x-1l y-2

6 3 14

x-1l y-2

5 1 1
—— +—— = AL —=
goityn2 2 (1) let—— = u
to.

6 3 =1 (2) y-2

x-1 y-2
So, our equations become

Sutv =2 (3) | 6u-3v=1 {4}

Example 19 teachoo.com
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.
Let the speed of boat in still water be x km/hr
& let the speed of stream be y km/hr
SN
<= .

Now, Speed of strea

Speed downstream=x+y

Speed upstream = x -—y

A test consists of ‘True’ or ‘False’ questions. One mark is awarded for every correct answer while ¼ mark is deducted for every wrong answer. A student knew answers to some of the questions. Rest of the questions he attempted by guessing. He answered 120 questions and got 90 marks.
Type of Question
Marks given for correct

Amit is planning to buy a house and the layout is given below. The design and the measurement has been made such that areas of two bedrooms and kitchen together is 95 sq. m.
Based on the above information, answer the following questions:
Question 1
Form the pair of linear equations in two variables from this situation.
Question 2
Find the length of the outer boundary of the layout.
Question 3
Find the area of each bedroom and kitchen in the layout.
Question 4
Find the area of living room in the layout.
Question 5
Find the cost of laying tiles in kitchen at the rate of Rs. 50 per sq.m

It is common that Governments revise travel fares from time to time based on various factors such as inflation ( a general increase in prices and fall in the purchasing value of money) on different types of vehicles like auto, Rickshaws, taxis, Radio cab etc. The auto charges in a city comprise of a fixed charge together with the charge for the distance covered. Study the following situations
Name of the city
Distance travelled (Km)
Amount paid (Rs.)
City A
10
75
15
110
City B
8
91
14
145
Situation 1:
In city A, for a journey of 10 km, the charge paid is
Rs 75 and for a journey of 15 km, the charge paid is Rs 110.
Situation 2:
In a city B, for a journey of 8km, the charge paid is
Rs 91 and for a journey of 14 km, the charge paid is Rs 145.
Refer Situation 1:
In city A, for a journey of 10 km, the charge paid is
Rs 75 and for a journey of 15 km, the charge paid is Rs 110.
Question 1
If the fixed charges of auto rickshaw be Rs x and the running charges be Rs y km/hr, the pair of linear equations representing situation is
(a) x + 10y = 110, x + 15y = 75
(b) x + 10y = 75, x + 15y = 110
(c) 10x + y = 110, 15x + y = 75
(d) 10x + y = 75, 15x + y =110
Refer Situation 1:
In city A, for a journey of 10 km, the charge paid is
Rs 75 and for a journey of 15 km, the charge paid is Rs 110.
Question 2
A person travels a distance of 50 km. The amount he has to pay is
(a) Rs 155
(b) Rs 255
(c) Rs 355
(d) Rs 455
Refer Situation 2:
In a city B, for a journey of 8km, the
charge paid is Rs 91 and for a journey of 14 km, the charge paid is Rs 145
Question 3
What will a person have to pay for travelling a distance of 30 km?
(a) Rs 185
(b) Rs 289
(c) Rs 275
(d) Rs 305
Question 4
The graph of lines representing the conditions are: (situation 2)

Question 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.
Question 1 Assuming that the speed of first car and second car be u km/h and v km/h respectively. What is the relative speed of both cars while they are travelling in the same direction? (a) (u + v) km/hr (b) (u – v) km/hr (c) (u/v) km/hr (d) (uv) km/hr

Question John and Jivanti are playing with the marbles in the playground. They together have 45 marbles and John has 15 marbles more than Jivanti.
Question 1 The number of marbles Jivanti had: (a) 15 (b) 30 (c) 40 (d) 5
Let Number of marbles with John = x
Number of marbles with Jivanti = y

The pair of equations 5x – 15y = 8 and 3x – 9y = 24/5 has
(A)one solution (B) two solutions
(C) infinitely many solutions (D) no solution

The sum of the digits of a two-digit number is 9. If 27 is added
to it, the digits of the number get reversed. The number is
(A) 25 (B) 72 (C) 63 (D) 36

Graphically, the pair of equations
6x – 3y + 10 = 0
2x – y + 9 = 0 represents two lines which are
(a)intersecting at exactly one point
(b) intersecting at exactly two points
(c) coincident
(d) parallel

The pair of equations x + 2y + 5 = 0 and –3x – 6y + 1 = 0 have:
(a) a unique solution (b) exactly two solutions
(c) infinitely many solutions (d) no solution

If a pair of linear equations is consistent, then the lines will be:
(a)parallel
(b) always coincident
(c) intersecting or coincident
(d) always intersecting

The pair of equations y = 0 and y = –7 has:
(a)one solution (b) two solutions
(c) infinitely many solutions (d) no solution

The pair of equations x = a and y = b graphically represents lines which are:
(a)parallel
(b) intersecting at (b, a)
(c) coincident
(d) intersecting at (a, b)

For what value of k, do the equations 3x – y + 8 = 0 and 6x – ky = –16 represent coincident lines?
(A) 1/2 (B) − 1/2 (C) 2 (D) −2

If the lines given by 3x + 2ky = 2 and 2x + 5y + 1 = 0 are parallel, then the value of k is
(A) (-5)/4 (B) 2/5 (C) 15/4 (D) 3/2

The value of c for which the pair of equations cx – y = 2 and 6x – 2y = 3 will have infinitely many solutions is
(A) 3 (B) – 3 (C) –12 (D) no value

One equation of a pair of dependent linear equations is –5x + 7y = 2. The second equation can be:
(a)10x + 14y + 4 = 0 (b) –10x – 14y + 4 = 0
(c) –10x + 14y + 4 = 0 (d) 10x – 14y = − 4

A pair of linear equations which has a unique solution x = 2, y = –3 is:
(a)x + y = –1
(b) 2x + 5y = –11 2x – 3y = –5 4x + 10y = –22
(c) 2x – y = 1
(d) x − 4y -14 = 0 3x + 2y = 0 5x - y − 13 = 0

If x = a, y = b is the solution of the equations x – y = 2 and x + y = 4, then the values of a and b are, respectively
(A) 3 and 5
(B) 5 and 3
(C) 3 and 1
(D) –1 and –3

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

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

Ex3.1,1 teachoo.com
Aftab tells his daughter, “Seven years ago, | was seven times as old as
you were then. Also, three years from now, | shall be three times as
old as you will be.” Represent this situation algebraically & graphically.
Let the Current age of Aftab be x years.
& let Current age of Aftab’s daughter be y years.

Given that
“Seven years ago, | was seven times as old as you were then”
Seven years ago,

Age of Aftab =x - 7

Age of Aftab’s daughter = y - 7

Ex 3.1, 2 teachoo.com
The coach of a cricket team buys 3 bats and 6 balls for Rs 3900.
Later, she buys another bat and 3 more balls of the same kind for
Rs 1300. Represent this situation algebraically and geometrically.
Let the Cost of one bat be Rs x
& let Cost of one ball be Rs y
Given that
3 bats and 6 balls cost Rs 3900
3 x Cost of one bat + 6 x Cost of one ball = 3900

3x + 6y = 3900

3(x + 2y) = 3 x 1300

x + 2y = 1300 (1)

Ex 3.1, 3 teachoo.com
The cost of 2 kg of apples and 1kg of grapes on a day was found
to be Rs 160. After a month, the cost of 4 kg of apples and 2 kg of
grapes is Rs 300. Represent the situation algebraically and
geometrically.
Let the Cost of Apples per kg be Rs x

& let Cost of grapes per kg be Rs y
Given that
2 kg apples and 1 kg grapes cost Rs 160

2 x Cost per kg of apples + 1 x Cost per kg of grapes = 160
2x +y = 160 (1)

Ex3.5,1 teachoo.com
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- 3y-3=0
3x- 9y-2=0

x-3y-3=0 (1)

3x-9y-2=0 (2)
x-3y-3=0 3x -9y-2=0
Comparing with a,x + b,y + c,=0 | Comparing with a,x + b,y+c,=0
“a, =1,b,=-3, ¢,=-3 “a, =3,b,=-9, c, =-2

teachoo.com
Ex3.5,1
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
(ii) 2x+y=5
3x+2y=8

ax+ty=5 (1)

3x+2y=8 (2)
2x+y=5 3x+2y=8
2x+1ly-5=0 3x+2y—-8=0
Comparing with a,x + b,y + c, = 0} Comparing with a,x + b,y +c, =0
“a, =2,b,=1,¢,=-5 “a, =3,b,=2,c,=-8

Ex3.5,1 teachoo.com
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
(iii) 3x —5y = 20
6x — 10y = 40
3x —5y = 20 (1)
6x —10y = 40 (2)
3x—-5y = 20 6x — 10y = 40
3x-5y—-20=0 6x — 10y — 40 =0
Comparing with a,x + b,y + c, = 0 | Comparing with a,x + boy + c, =0
“a, = 3,b,=-5,c,=-20 “a, = 6,b, =-10,c, =—40

teachoo.

Ex3.5,1 ‘eachoo.com
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
(iv) x-3y-7=0

3x-3y-15=0
x-3y-7=0 (1)
3x-3y -15=0 (2)
x-3y-7=0 3x-—3y-15=0
Comparing with a,x + b,y + c,=0 | Comparing with a,x + b,y + c, =0
“a, =1,b,=-3,c,=-7 “ay = 3, b,=-3, c,=—-15

Ex 3.5, 2 teachoo.com
(i} For which values of a and b does the following pair of linear
equations have an infinite number of solutions?

2xt+ 3y =7

(a—b}x+(a+b) y= 3a+b-2
2x + 3y=7 (1)
(a—b)x + (a+b) y=3at+b-2 (2)
2x + 3y =7 {a—b)x + (a+ b)y =3a+b-2
2x+3y-7=0 (a—-b)x + (a+ bjy- (3a+b-2)=0
Comparing with a,x + by + ¢,= 0] Comparing with ax + b,y +c, =0
* a, = 2,b, = 3,¢,=—-7 +a, = (a—b),b, = (a +b),

c,=— (3a + b — 2)

Ex3.5,2 teachoo.com
(ii) For which value of k will the following pair of linear equations
have no solution?

3xty=1

(2k-1)x + (k-—1) y=2k+1
3xty =1 (1)
(2k—1)x + (k-1)y=2k +1 ...(2)
3x+y=1 (2k —1)x + (k—-1)jy=2k+1
3x+y-1=0 (2k — 1)x + (k-1)y —(2k + 1)=0
Comparing with a,x + b,y + c, = 0 | Comparing with a,x + b,y+c,=0
+a, =3,b,=1,¢,=-1 a, = (2k— 1), b= (k—-1),

c,=—(2k+1)

Ex 3.5, 3 (Substitution method) feachoo.com
Solve the following pair of linear equations by the substitution and
cross-multiplication methods :

8x+5y=9

3x+2y=4

8x+Sy=9 (1)

3x+2y=4 ...(2)
From (1)

8x+5y=9

8x =9—5Sy

_@-5y)
yee
8

Ex 3.5, 4 teachoo.com
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.
Let Fixed charge = Rs x

& Charge per day taken food = Rs y per day
Given that charge paid by student A for 20 days is Rs 1000
Fixed charge + 20 x (Charge per day) = 1000

x + 20y = 1000 {1}

Ex 3.5, 4 teachoo.com
Form the pair of linear equations in the following problems and find
their solutions (if they exist) by any algebraic method :
{ii) A fraction becomes = when 1is subtracted from the numerator &
it becomes = when 8 is added to its denominator. Find the fraction
Let Numerator be x
& Denominator be y

So, Fraction is

y
Given that,
If 1 is subtracted from numerator fraction becomes :

Ex 3.5,4 teachoo.com
Form the pair of linear equations in the following problems and find
their solutions (if they exist) by any algebraic method :
(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?
Let Number of Right answers be x

& Number of Wrong answers be y

Ex 3.5,4 teachoo.com
Form the pair of linear equations in the following problems and find
their solutions (if they exist) by any algebraic method :
(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?
er
A 100 km B
Let Speed of first car be x km/hr
& Speed of second car be y km/hr

Ex 3.5,4 teachoo.com
Form the pair of linear equations in the following problems and find
their solutions (if they exist) by any algebraic method :
(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.
Let Length of rectangle be x units y
& Breadth of rectangle be y units
x

Hence,

Area = Length x Breadth

Area = xy

Ex 3.6,1 teachoo.com
Solve the following pairs of equations by reducing them to a pair of
linear equations:
ay 1 1
(i) in + coil 2
1 1 13
+,2-2
3x 2y 6
1 1 a
12,4 2) (1) Leto = u
2x By
1
2,4-8 (2) == V
3x 2y 6 y
So, our equations become
tuyaty =2 tyaty -8
2 37 guts
3ut2v_, 2u+3v 13
2x3 2x3 6
3u+2v=12 ...{3) 2u¢3v=13— --(4)

Ex 3.6, 1 teachoo.com
Solve the following pairs of equations by reducing them to a pair of
linear equations:
way 4
(iii) x +3y =14

3 4y = 23

x y ~

4

—+3y =14 ...(1) }

x 1

Let-=u

3 x

x 4y=23 — ...(2)
So, our equations become

4u+3y=14_ ...(3) | 3u-4y=23 ...(4)

Ex 3.6,1 teachoo.com
Solve the following pairs of equations by reducing them to a pair of
linear equations:
7x —2
(v) es
xy
Bx +7
ext 245
xy
Given
7x —2 8x +7
Tx-2y . 8+ 79 45
xy xy
ars Das
xy xy xy xy -
7 2 5
TTF 8 7
y * —+—=15
y x
—2 7
a tye
we 7 8
~ % —4+—=15 (2)
x y

Ex 3.6, 1 teachoo.com
Solve the following pairs of equations by reducing them to a pair of
linear equations:
(vii) 22+ =4
x+y x-y
15 5
x+y x-y
10, 2 24 1 Let—— =u
1
15 5 &—F=Vv
yay ey =-2 ...(2) x-y
So, our equations become
10u + 2v=4 (3) | 15u-5v =-2 (4)

Ex 3.6,2 teachoo.com
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 & speed of the current.
Let the speed of boat in still water be x km/hr
& let the speed of current be y km/hr
~
<= .
Speed of strea

Now,

Speed downstream=x+y

Speed upstream = x —y

Ex 3.6, 2 (Introduction) teachoo.com
Formulate the following problems as a pair of equations, and hence
find their solutions:

(ii) 2 women & 5 mencan together finish an embroidery work in 4
days, while 3 women & 6 men can finish it in 3 days. Find the time
taken by 1 woman alone to finish the work, & also that taken by 1
man alone.

A person completes work in 2 days

Work completed in 1 day =>

A person completes work in 3 days

Work completed in 1 day =5

A person completes work in x days

Work completed in 1 day =<

teachoo.com
Ex 3.6, 2
Formulate the following problems as a pair of equations, and
hence find their solutions:
(iii) Roohi travels 300 km to her home partly 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.
Let speed of train be x km/hr
& speed of bus be y km/hr

Ex 3.7,1 teachoo.com
The ages of two friends Ani and Biju differ by 3 years. Ani’s father
Dharam is twice as old as Ani and Biju is twice as old as his sister
Cathy. The ages of Cathy and Dharam differ by 30 years. Find the
ages of Aniand Biju.
Let the Present age of Ani = x years
and the Present age of Biju = y years.
Given that the ages of Ani and Biju differ by 3
x-y=3 ..(1)

Given that
Ani’s father Dharam is twice as old as Ani

Present Age of Dharam = 2x

Ex 3.7, 2 teachoo.com
One says, “Give me a hundred, friend! | shall then become twice as
rich as you”. The other replies, “If you give me ten, | 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)].
Let Amount of money with first person = Rs x

& Amount of money with second person = Rs y
When the first Person takes 100 from the second Person
Money with first Person = x + 100
Money with Second Person = y - 100

Ex 3.7, 3 teachoo.com
A train covered a certain distance at a uniform speed. If the train
would have been 10 km/h 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.
Let Speed of train = x km/h
& Time taken = y hours.

We know that,

Speed = Distance

Time
Distance = Speed x Time
Distance = xy (1)

Ex 3.7, 4 (Introduction) feachoo.com
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 ina
row, there would be 2 rows more. Find the number of students in
the class.
Introduction
Let 2 Students stand in 1 row and there are 10 rows in total
e e
Total the number of Students ° °
e es
= Number of Students in 1 row x Number of rows, .
= 2 x 10 e e
e e
= 20 Students ° °
e e
e e
e e

Ex 3.7, 5 teachoo.com
Ina A ABC, 2C=32ZB=2(ZA+ ZB). Find the three angles.
A
Let ZA=x
&ZB=y

Given that

2C=3 2B

2C=3y B c
Also,

Z2C=2 (ZA+ ZB)

2C=2 (x+y)

Ex 3.7, 6 teachoo.com
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.
Given equations

Sx-y=5 (1)

3x-y=3 (2)

Ex 3.7,7 teachoo.com
Solve the following pair of linear equations:
(i) px+qy=p-q
qx+py=pt+q
Given
px+qy=p-—q gx-py=pt+q
Multiplying both sides by q Multiplying both sides by p
a(px + qy) = a(p — q) p(qx + py) = p(p + q)
pqx+q’y=pq-q’_ -(1) paxt p’y=p?+pq_ ...(2)
Hence, our equations are
paxtqy=pq-q _ ...(1)
pax+p’y=p? +pq — ...{2)

Ex 3.7, 7 teachoo.com
Solve the following pair of linear equations:
(ii) ax + by =c
bxtay=1+c
Solving equations
ax + by=c bx + ay= 1+¢
Multiplying both sides by a Multiplying both sides by b
a (ax + by) = ac b (bx + ay) =b (1+)
a’x + aby = ac ...(1} b’x+aby=b+be ...(2}
Hence, the equations are
a°x + aby = ac ...(1)
b’x+aby=b+be _ ...(2)

Ex 3.7, 7 teachoo.com
Solve the following pair of linear equations:
wee xy
(iii) a5 =0

ax + by =a? + b?

x Yy _

ab 0 (1)

ax + by = a? + b? ..(2)
Solving Equation (1)

x Yy _

a 7 b- 0

bx -— ay _

ab 0
bx - ay =0 ...(3)

Ex3.7,7 teachoo.com
Solve the following pair of linear equations:
(iv) (a — b)x + (a+ b)y = a? — 2ab — b?
(a + b) (x+y) =a? + b?
{a-b) x + (a+b) y =a? - 2ab - b? (1)
(a+b) (x+y) =a? +b? ...(2)
Solving equation (2)
(a+b) (x+y) =a?+ b?
(a+ b)x + (a+ b)y = a? + b?
{at+b) y=a2+b?-(atb)x
_a? +b? -(atb)x
y= (a+b)

Ex 3.7, 7 teachoo.com
Solve the following pair of linear equations:
(v) 152x — 378y =- 74
—378x + 152y =- 604
152x - 378y = -74 ...(1)
-378x + 152y = -604 ..(2)
From equation (1)
152x — 378y = -74
378y = 152x + 74
v= 152x+74
378
2(76x + 37)
y=" 3489)

Ex 3.7, 8 teachoo.com
ABCD is a cyclic quadrilateral (see Fig. 3.7). Find the angles of the
cyclic quadrilateral.
Given that c
ZA = 4y +20 B <a
V 3y-5
2B = 3y-
vos sf
ha Sy
2C = -4x a §&
LA
Z2D=-7x+5 A D
We know that in a cyclic quadrilateral,
Sum of the opposite angles is 180°
Therefore,
ZA + ZC = 180°
& 2B + ZD = 180°

Chapter 3 Class 10 Linear Equations In Two Variables
- MCQ Worksheet 1
by teachoo

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Chapter 3 Class 10 - Linear Equations in Two Variables - MCQ Questions - Worksheet 2 - Teachoo.pdf
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Chapter 3 Class 10 Linear Equations In Two Variables
- Mix Questions Worksheet 1
by teachoo

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Chapter 3 Class 10 Linear Equations In Two Variables
- Assertion and Reasoning
Worksheet 1
by teachoo

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Chapter 3 Class 10 Linear Equations In Two Variables
- Case Based Question
Worksheet 1
by teachoo

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