Download Class 10 Linear Equations in Two Variables

ID : in-10-Linear-Equations-in-Two-Variables [1]
Class 10
Linear Equations in Two Variables
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Answer t he quest ions
(1)
T here are two numbers. If f our times the larger of two numbers is divided by the smaller one, we
get 8 as quotient and 0 as remainder. If six times the smaller of two numbers is divided by the
larger one, we get 3 as quotient and 0 as remainder. Find the numbers.
(2)
Neha and Sunil have some apples. Neha says to Sunil, "f f you give me 12 of your apples, I will
have twice the number of apples lef t with you". Sunil replies, "if you give me 12 of your apples I
will have the same number of apples lef t with you". Find the the number of apples with Neha and
Archana separately.
(3)
Bala is three times as old as his son. Af ter 16 years, Bala will be two times as hold as his son.
Find the current age of Bala and his son.
(4) In a two digit number, sum of its digits is 13. If 27 is added to the number, the digit interchange
their places. Find the number.
(5)
Priyanka has only Rs.2 and Rs.1 notes with her. If the total number of notes that she has is 19
and the amount of money with her is Rs.26, then the number of Rs.2 and Rs.1 notes are,
respectively
(6) Five years ago T ina was six times older than her son. Af ter f ive years, T ina will be 6 years more
than two times the age of her son. Find the present age of T ina and her son.
(7) A two digit number is 9 more than 4 times the sum of its digits. If 18 is added to the number, the
digit interchange their places. Find the number.
Choose correct answer(s) f rom given choice
(8)
Find value of k f or which equations 3x - y + 4 = 0 and kx + y - 1 = 0, will have no solution.
a. -5
b. -4
c. -1
d. -3
(9) If a pair of linear equations is consistent, then the lines will be
a. intersecting or coincident
b. always intersecting
c. always coincident
d. parallel
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ID : in-10-Linear-Equations-in-Two-Variables [2]
(10) Which of the f ollowing conditions is true if the system of equations below is shown to be
inconsistent
a1x + b 1y + c1 = 0, a2x + b 2y + c2 = 0
a.
c.
a1
≠
b1
≠
c1
b.
a1
a2
=
=
c1
a2
b2
c2
b1
b2
c2
a1
b1
c1
a1
b1
c1
a2
=
b2
≠
c2
d.
=
a2
b2
=
c2
(11) Find value of k such that equations 2p + kq - 2 = 0 and -2p + 2q - 1 = 0, represents parallel
lines.
a. 2
b. -3
c. -2
d. 0
(12) T wo numbers are in ratio 2:3. If 5 is subtracted f rom both the numbers, ratio becomes 1:2. Find
the numbers.
a. 10 and 15
b. 8 and 12
c. 14 and 21
d. 12 and 18
(13) Which of the f ollowing equations has a unique solution?
a. 4x + 5y + 8 = 0, 16x + 20y + 25 = 0
b. 4x + 5y + 8 = 0, 12x + 15y + 24 = 0
c. 4x + 5y + 8 = 0, 12x + 15y + 25 = 0
d. 4x + 5y + 26 = 0, 12x + 6y + 22 = 0
(14) In a two digit number, the ten's digit is twice the units's digit. If 36 is added to the number, the
digit interchange their places. Find the number.
a. 48
b. 36
c. 63
d. 84
(15) For which value of a, f ollowing pair of linear equations have no solution.
a x + y = a2 and x + a y = 1
a. -1
b. 0
c. 2
d. 1
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ID : in-10-Linear-Equations-in-Two-Variables [3]
Answers
(1)
40 and 20
Step 1
Let the larger number be x and smaller number be y
Step 2
We know that Dividend = (Divisor × Quotient) + Remainder
T heref ore we can write relationship provided as,
4x = 8y + 0 => 4x - 8y - 0 = 0
6y = 3x + 0 => 3x - 6y + 0 = 0
Step 3
On solving these two equations we get x = 40 and y = 20
(2)
84 and 60 items
Step 1
Let the number of apples with Neha and Sunil be x and y. It is given that
x + 12 = 2 (y - 12)
x - 12 = y + 12
Step 2
On subtracting second equation f rom f irst equation
2 × 12 = 2 (y - 12) - (y + 12)
24 = y - 36
y = 5 × 12
y = 60
Step 3
Substitute value of y in f irst equation,
x + 12 = 2 (60 - 12)
x = 84
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ID : in-10-Linear-Equations-in-Two-Variables [4]
(3)
48 and 16 years
Step 1
Let age of Bala be x years and age of be y years
Step 2
It is given that
x = 3y => 3y - x = 0
(x + 16) = 2(y + 16) => x - 2y = 16
Step 3
On adding above two equations
(3y - x) + (x - 2y) = 16
y = 16
Step 4
Hence their ages are x = 3y = 48 years and y = 16 years
(4) 58
Step 1
Let the ten's digit be x and unit digits be y, hence number of 10x + y
Now it is given that
x + y = 13 ________________________(1) and
(10x + y) + 27 = 10y + x
9x - 9y = -27
x - y = -3 ________________________(2)
Step 2
On adding two equations,
2x + y - y = 13 - 3
2x = 10
x=5
Step 3
y = 13 - x = 13 - 5 = 8
T heref ore number is 58
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ID : in-10-Linear-Equations-in-Two-Variables [5]
(5)
7 and 12
Step 1
Lets assume
number Rs.2 notes = x
number Rs.1 notes = y
Step 2
Is is given that total number of notes is 19,
x + y = 19 ....................... (1)
Step 3
It is also given that total amount of money is Rs.26,
(2)x + (1)y = 26 ....................... (2)
Step 4
Now multiply Eq. (1) by 2 and subtract is f rom Eq (2),
⇒ (2 - 2)x + (1 - 2)y = (26 - 2 × 19)
⇒ (-1)y = -12
⇒ y = (-12)/(-1)
⇒ y = 12
Step 5
Now substiture value of y in Eq. (1),
x + 12 = 19
⇒ x = 19 - 12
⇒x=7
Step 6
T heref ore number of Rs.2 notes = 7, and number of Rs.1 notes = 12
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ID : in-10-Linear-Equations-in-Two-Variables [6]
(6) 29 years and 9 years
Step 1
Let x and y be the current age of T ina and her son respectively.
It is given that, Five years ago T ina was six times older than her son
(x - 5) = 6 (y - 5)
x - 6y = -25 ____________________(1)
Step 2
It is also given that, af ter f ive years T ina will be 6 years more than two times the age of her
son
(x + 5) = 2 (y + 5) + 6
x - 2y = 11 ____________________(2)
Step 3
On subtracting eq. 1 f rom 2
4y = 36
y = 9 years
Step 4
On substituting value of y in eq. 1
x - 6(9) = -25
x = 29 years
Step 5
T href ore,
present age of T ina = x = 29 years
present age of her son = y = 9 years
(7) 57
Step 1
Let the ten's digit be x and unit digits be y, hence number of 10x + y
Step 2
It is given that
10x + y = 4(x + y) + 9
6x - 3y = 9 ________________________(1)
Step 3
It is also given that
(10x + y) + 18 = 10y + x
9x - 9y = -18
x - y = -2 ________________________(2)
Step 4
On solving equations (1) and (4), we get x = 5 and y = 7
T heref ore number is 57
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ID : in-10-Linear-Equations-in-Two-Variables [7]
(8)
d. -3
Step 1
Equations a1x + b1y + c1 = 0 and a1x + b1y + c1 = 0 have no solution if ,
a1
=
a2
b1
c1
≠
b2
c2
Step 2
On substituting coef f icients in above condition,
k
1
=
3
-1
≠
-1
4
Step 3
T href ore k = -3
(9) a. intersecting or coincident
(10)
c.
a1
=
a2
b1
≠
b2
c1
c2
Step 1
T wo equations are inconsistent if no solution exists which can satisf y both the equations.
T his situation would arise when ratio of coef f icients of variables are same, while ratio of
constants is dif f erent, i.e.
a1
=
a2
b1
b2
≠
c1
,
c2
where a1, b1 and a2, b2 are the coef f icients of the variables and c1 and c2 are the
constants of the equations respectively.
Step 2
Hence, the true condition f or the given equations is
a1
a2
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=
b1
b2
≠
c1
.
c2
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ID : in-10-Linear-Equations-in-Two-Variables [8]
(11) c. -2
Step 1
Equations a1x + b1y + c1 = 0 and a1x + b1y + c1 = 0 represent parallel lines if ,
a1
=
a2
b1
c1
≠
b2
c2
Step 2
On substituting coef f icients in above condition,
2
=
-2
k
≠
2
-2
-1
Step 3
T href ore k = -2
(12) a. 10 and 15
Step 1
Since numbers are in ratio 2:3, lets assume numbers are 2x and 3x
Step 2
When 5 is subtracted f rom both numbers, numbers become (2x - 5) and (3x - 5)
Step 3
Since new ratio is 1:2
(2x - 5)/(3x - 5) = 1/2
⇒ 2 (2x - 5) = 1 (3x - 5)
⇒ 4x - 10 = 3x - 5
⇒ 4x - 3x = - 5 + 10
⇒ 1x = 5
⇒ x = 5/1
⇒x=5
Step 4
T heref ore numbers are,
2x = 2 × 5 = 10
3x = 3 × 5 = 15
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ID : in-10-Linear-Equations-in-Two-Variables [9]
(13) d. 4x + 5y + 26 = 0, 12x + 6y + 22 = 0
Step 1
Equations a1x + b1y + c1 = 0 and a1x + b1y + c1 = 0 have unique solution if ,
a1
≠
a2
b1
b2
Step 2
If we try this condition on given f our equations, it is only satisf ied by pair 4x + 5y + 26 = 0,
12x + 6y + 22 = 0, since
4
≠
12
5
6
Step 3
T heref ore answer is 4x + 5y + 26 = 0, 12x + 6y + 22 = 0
(14) a. 48
Step 1
Let the ten's digit be x and unit digits be y, hence number of 10x + y
Now it is given that
y - 2x = 0 ________________________(1) and
(10x + y) + 36 = 10y + x
9x - 9y = -36
x - y = -4 ________________________(2)
Step 2
On adding two equations,
y - 2x + x - y = -4
-x = -4
x=4
Step 3
y = 2x = 2×4 = 8
T heref ore number is 48
(15) a. -1
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