Download Grade 10 Linear Equations in Two Variables

ID : pk-10-Linear-Equations-in-Two-Variables [1]
Grade 10
Linear Equations in Two Variables
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Answer t he quest ions
(1)
T he mother's age is 8 times her son's age. Af ter 14 years, the age of the mother will be 31/10
times her son's age. Find the present ages of the son and the mother.
(2)
Fareeha is three times as old as her son. Af ter 19 years, Fareeha will be two times as hold as
her son. Find the current age of Fareeha and her son.
(3)
In a two digit number, the ten's digit is twice the units's digit. If 18 is added to the number, the
digit interchange their places. Find the number.
(4) Which of the f ollowing equations has an inf inite number of solutions?
2x + 3y + 1 = 0, 8x + 12y + 4 = 0
2x + 3y + 5 = 0, 6x + 4y + 0 = 0
2x + 3y + 1 = 0, 6x + 9y + 6 = 0
2x + 3y + 1 = 0, 6x + 9y + 3 = 0
(5)
6 pens and 3 pencils cost Rs.33 and 8 pens and 10 pencils cost Rs.86. Find the cost of one pen
and one pencil separately.
(6) T here are two numbers. If three times the larger of two numbers is divided by the smaller one,
we get 6 as quotient and 9 as remainder. If f ive times the smaller of two numbers is divided by
the larger one, we get 2 as quotient and 5 as remainder. Find the numbers.
(7) 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
a1
≠
a2
a1
≠
b2
=
b1
c1
c2
≠
c1
a2
b2
c2
a1
a2
c1
=
b1
a1
a2
(8)
b1
=
b2
=
b1
b2
c2
=
c1
c2
T wo numbers are in ratio 3:4. If 10 is subtracted f rom both the numbers, ratio becomes 1:2. Find
the numbers.
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ID : pk-10-Linear-Equations-in-Two-Variables [2]
Choose correct answer(s) f rom given choice
(9) For which value of m, f ollowing pair of linear equations have inf initely many solutions.
mx+8y=m
2x+my=m- 2
a. 2
b. -4
c. 4
d. -2
(10) For which value of p, f ollowing pair of linear equations have inf initely many solutions.
p x + y = p2 and x + p y = 1
a. -1
b. 2
c. 0
d. 1
(11) If a pair of linear equations is consistent, then the lines will be
a. always coincident
b. parallel
c. intersecting or coincident
d. always intersecting
(12) Find value of k such that equations -3x + ky + 3 = 0 and -6x - 4y + 6 = 0, represents coincident
lines.
a. 2
b. 0
c. -4
d. -2
(13) A two digit number is 6 more than 4 times the sum of its digits. If 27 is added to the number, the
digit interchange their places. Find the number.
a. 58
b. 49
c. 85
d. 67
(14) Jaleel has only Rs.10 and Rs.1 notes with him. If the total number of notes that he has is 21 and
the amount of money with him is Rs. 111, then the number of Rs.10 and Rs.1 notes are,
respectively
a. 8 and 13
b. 10 and 11
c. 12 and 9
d. 9 and 12
Check True/False
(15) Equations -x + 3y = 1 and 3x - 9y = c will have an unique solution f or all real values of c.
T rue
False
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ID : pk-10-Linear-Equations-in-Two-Variables [3]
Answers
(1)
6 years and 48 year
Step 1
Let's assume, the present of the son be 'x'.
It is given that the mother's age is 8 times her son's age, theref ore the present age of the
mother is 8x.
Step 2
Af ter 14 years, the age of the son = x + 14,
the age of the mother = 8x + 14
Step 3
It is also given that af ter 14 years, the age of the mother will be 31/10 times her son's age.
T heref ore,
8x + 14 = 31/10(x + 14)
⇒ 8x - 31/10(x) = 31/10(14) - 14
⇒x=6
Step 4
T heref ore, the present age of the son = 6 years,
the present age of the mother = 8x = 8 × 6 = 48 years.
(2)
57 and 19 years
Step 1
Let age of Fareeha be x years and age of be y years
Step 2
It is given that
x = 3y => 3y - x = 0
(x + 19) = 2(y + 19) => x - 2y = 19
Step 3
On adding above two equations
(3y - x) + (x - 2y) = 19
y = 19
Step 4
Hence their ages are x = 3y = 57 years and y = 19 years
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ID : pk-10-Linear-Equations-in-Two-Variables [4]
(3)
24
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) + 18 = 10y + x
9x - 9y = -18
x - y = -2 ________________________(2)
Step 2
On adding two equations,
y - 2x + x - y = -2
-x = -2
x=2
Step 3
y = 2x = 2×2 = 4
T heref ore number is 24
(4) 2x + 3y + 1 = 0, 6x + 9y + 3 = 0
Step 1
Equations a1x + b1y + c1 = 0 and a1x + b1y + c1 = 0 have inf initely many solutions if ,
a1
=
a2
b1
=
b2
c1
c2
Step 2
If we try this condition on given f our set of equations, it is only satisf ied by pair 2x + 3y + 1
= 0, 6x + 9y + 3 = 0, since
2
=
6
3
9
=
1
3
Step 3
T heref ore answer is 2x + 3y + 1 = 0, 6x + 9y + 3 = 0
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ID : pk-10-Linear-Equations-in-Two-Variables [5]
(5)
Rs.2 and Rs.7
Step 1
Let the cost of a pen and pencil be Rs. x and Rs. y respectively. It is given that
6 x + 3 y = 33
8 x + 10 y = 86
Step 2
On multiplying f irst equation by 10 and second equation by 3
60 x + 30 y = 330
24 x + 30 y = 258
Step 3
Subtract second equation f rom f irst equation 1
=> 36 x = 72
=> x = (72)/(36) = Rs.2
Step 4
Substitute this value in f irst equation
=> 6 × 2 + 3 y = 33
=> 3 y = 33 - 6 × 2 = 21
=> y = (21)/(3) = Rs.7
(6) 25 and 11
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,
3x = 6y + 9 => 3x - 6y - 9 = 0
5y = 2x + 5 => 2x - 5y + 5 = 0
Step 3
On solving these two equations we get x = 25 and y = 11
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ID : pk-10-Linear-Equations-in-Two-Variables [6]
(7)
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
(8)
=
b1
≠
b2
c1
.
c2
15 and 20
Step 1
Since numbers are in ratio 3:4, lets assume numbers are 3x and 4x
Step 2
When 10 is subtracted f rom both numbers, numbers become (3x - 10) and (4x - 10)
Step 3
Since new ratio is 1:2
(3x - 10)/(4x - 10) = 1/2
⇒ 2 (3x - 10) = 1 (4x - 10)
⇒ 6x - 20 = 4x - 10
⇒ 6x - 4x = - 10 + 20
⇒ 2x = 10
⇒ x = 10/2
⇒x=5
Step 4
T heref ore numbers are,
3x = 3 × 5 = 15
4x = 4 × 5 = 20
(9) c. 4
(10) d. 1
(11) c. intersecting or coincident
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ID : pk-10-Linear-Equations-in-Two-Variables [7]
(12) d. -2
Step 1
Equations a1x + b1y + c1 = 0 and a1x + b1y + c1 = 0 represent coincident lines if ,
a1
=
a2
b1
=
b2
c1
c2
Step 2
On substituting coef f icients in above condition,
-3
=
-6
k
=
-4
3
6
Step 3
T href ore k = -2
(13) a. 58
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) + 6
6x - 3y = 6 ________________________(1)
Step 3
It is also given that
(10x + y) + 27 = 10y + x
9x - 9y = -27
x - y = -3 ________________________(2)
Step 4
On solving equations (1) and (4), we get x = 5 and y = 8
T heref ore number is 58
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ID : pk-10-Linear-Equations-in-Two-Variables [8]
(14) b. 10 and 11
Step 1
Lets assume
number Rs.10 notes = x
number Rs.1 notes = y
Step 2
Is is given that total number of notes is 21,
x + y = 21 ....................... (1)
Step 3
It is also given that total amount of money is Rs.111,
(10)x + (1)y = 111 ....................... (2)
Step 4
Now multiply Eq. (1) by 10 and subtract is f rom Eq (2),
⇒ (10 - 10)x + (1 - 10)y = (111 - 10 × 21)
⇒ (-9)y = -99
⇒ y = (-99)/(-9)
⇒ y = 11
Step 5
Now substiture value of y in Eq. (1),
x + 11 = 21
⇒ x = 21 - 11
⇒ x = 10
Step 6
T heref ore number of Rs.10 notes = 10, and number of Rs.1 notes = 11
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ID : pk-10-Linear-Equations-in-Two-Variables [9]
(15) False
Step 1
On comparing equations with standard f orm of equations a1x + b 1y + c1 = 0 and a2x + b 2y
+ c2 = 0, we get,
a1 = -1, b1 = 3, c1 = 1,
and a2 = 3, b2 = -9, c2 = c
Step 2
Now,
a1
=
-1
a2
and
b1
b2
3
=
3
-9
Step 3
Since (a1/a2) = (b1/b2) , the two lines represented by the equations -x + 3y = 1 and 3x - 9y
= c are parallel, and these equations will not have an unique solution, f or all values of c.
T heref ore, the answer is False.
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