Download Grade 10 Linear Equations in Two Variables

ID : eu-10-Linear-Equations-in-Two-Variables [1]
Grade 10
Linear Equations in Two Variables
For more such worksheets visit www.edugain.com
Answer t he quest ions
(1)
Misha and Maciej have some bananas. Misha says to Maciej, "f f you give me 16 of your
bananas, I will have twice the number of bananas lef t with you". Maciej replies, "if you give me 16
of your bananas I will have the same number of bananas lef t with you". Find the the number of
bananas with Misha and T atiana separately.
(2)
Which of the f ollowing equations has a unique solution?
3x + 5y + 9 = 0, 12x + 20y + 36 = 0
3x + 5y + 9 = 0, 12x + 20y + 37 = 0
3x + 5y + 38 = 0, 12x + 7y + 33 = 0
3x + 5y + 9 = 0, 15x + 25y + 38 = 0
(3)
Vladimir has only €5 and €10 notes with him. If the total number of notes that he has is 16 and
the amount of money with him is €135, then the number of €5 and €10 notes are, respectively
(4) T here are two numbers. If three times the larger of two numbers is divided by the smaller one,
we get 5 as quotient and 13 as remainder. If f ive times the smaller of two numbers is divided by
the larger one, we get 2 as quotient and 23 as remainder. Find the numbers.
(5)
T he f ather's age is 4 times his son's age. Af ter 15 years, the age of the f ather will be 13/7 times
his son's age. Find the present ages of the son and the f ather.
(6) Ania is three times as old as her son. Af ter 18 years, Ania will be two times as hold as her son.
Find the current age of Ania and her son.
(7) Find value of k f or which equations -p + kq + 3 = 0 and 2p - 2q - 6 = 0, will have inf initely many
solutions.
(8)
If twice the daughter's age in years is added to mother's age, the sum is 50. If twice the mother's
age is added to the daughter's age, the sum is 76. Find the age of mother and daughter.
(9) T hree years ago Filip was f our times older than his daughter. Af ter three years, Filip will be 4
years more than two times the age of his daughter. Find the present age of Filip and his
daughter.
(10) In a two digit number, the ten's digit is twice the units's digit. If 27 is added to the number, the
digit interchange their places. Find the number.
Choose correct answer(s) f rom given choice
(11) A two digit number is 3 more than 4 times the sum of its digits. If 18 is added to the number, the
digit interchange their places. Find the number.
a. 44
b. 35
c. 53
d. 26
Copyright 2016 www.edugain.com
Personal use only. Commercial use is strictly prohibited.
ID : eu-10-Linear-Equations-in-Two-Variables [2]
(12) T he pair of equations 3x + y - 2 = 0 and -3x - y + 2 = 0 have
a. no solution
b. exactly two solutions
c. inf initely many solutions
d. a unique solution
(13) Which of the f ollowing equations has an inf inite number of solutions?
a. 3x + 4y + 9 = 0, 12x + 6y + 6 = 0
b. 3x + 4y + 2 = 0, 12x + 16y + 9 = 0
c. 3x + 4y + 2 = 0, 15x + 20y + 9 = 0
d. 3x + 4y + 2 = 0, 12x + 16y + 8 = 0
(14) Find value of k such that equations -2x + ky + 3 = 0 and 2x + 2y - 2 = 0, represents parallel
lines.
a. -3
b. -2
c. 0
d. -4
(15) T he pair of equations y = -3 and y = -7 has
a. no solution
b. inf initely mainy solution
c. two solutions
d. an unique solution
© 2016 Edugain (www.edugain.com).
All Rights Reserved
Copyright 2016 www.edugain.com
Many more such worksheets can be
generated at www.edugain.com
Personal use only. Commercial use is strictly prohibited.
ID : eu-10-Linear-Equations-in-Two-Variables [3]
Answers
(1)
112 and 80 items
Step 1
Let the number of bananas with Misha and Maciej be x and y. It is given that
x + 16 = 2 (y - 16)
x - 16 = y + 16
Step 2
On subtracting second equation f rom f irst equation
2 × 16 = 2 (y - 16) - (y + 16)
32 = y - 48
y = 5 × 16
y = 80
Step 3
Substitute value of y in f irst equation,
x + 16 = 2 (80 - 16)
x = 112
(2)
3x + 5y + 38 = 0, 12x + 7y + 33 = 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 3x + 5y + 38 = 0,
12x + 7y + 33 = 0, since
3
≠
12
5
7
Step 3
T heref ore answer is 3x + 5y + 38 = 0, 12x + 7y + 33 = 0
Copyright 2016 www.edugain.com
Personal use only. Commercial use is strictly prohibited.
ID : eu-10-Linear-Equations-in-Two-Variables [4]
(3)
5 and 11
Step 1
Lets assume
number €5 notes = x
number €10 notes = y
Step 2
Is is given that total number of notes is 16,
x + y = 16 ....................... (1)
Step 3
It is also given that total amount of money is €135,
(5)x + (10)y = 135 ....................... (2)
Step 4
Now multiply Eq. (1) by 5 and subtract is f rom Eq (2),
⇒ (5 - 5)x + (10 - 5)y = (135 - 5 × 16)
⇒ (5)y = 55
⇒ y = (55)/(5)
⇒ y = 11
Step 5
Now substiture value of y in Eq. (1),
x + 11 = 16
⇒ x = 16 - 11
⇒x=5
Step 6
T heref ore number of €5 notes = 5, and number of €10 notes = 11
(4) 36 and 19
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 = 5y + 13 => 3x - 5y - 13 = 0
5y = 2x + 23 => 2x - 5y + 23 = 0
Step 3
On solving these two equations we get x = 36 and y = 19
Copyright 2016 www.edugain.com
Personal use only. Commercial use is strictly prohibited.
ID : eu-10-Linear-Equations-in-Two-Variables [5]
(5)
6 years and 24 year
Step 1
Let's assume, the present of the son be 'x'.
It is given that the f ather's age is 4 times his son's age, theref ore the present age of the
f ather is 4x.
Step 2
Af ter 15 years, the age of the son = x + 15,
the age of the f ather = 4x + 15
Step 3
It is also given that af ter 15 years, the age of the f ather will be 13/7 times his son's age.
T heref ore,
4x + 15 = 13/7(x + 15)
⇒ 4x - 13/7(x) = 13/7(15) - 15
⇒x=6
Step 4
T heref ore, the present age of the son = 6 years,
the present age of the f ather = 4x = 4 × 6 = 24 years.
(6) 54 and 18 years
Step 1
Let age of Ania be x years and age of be y years
Step 2
It is given that
x = 3y => 3y - x = 0
(x + 18) = 2(y + 18) => x - 2y = 18
Step 3
On adding above two equations
(3y - x) + (x - 2y) = 18
y = 18
Step 4
Hence their ages are x = 3y = 54 years and y = 18 years
Copyright 2016 www.edugain.com
Personal use only. Commercial use is strictly prohibited.
ID : eu-10-Linear-Equations-in-Two-Variables [6]
(7) 1
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
On substituting coef f icients in above condition,
-1
=
2
k
=
-2
3
-6
Step 3
T href ore k = 1
(8)
34 and 8 years
Step 1
Let the age of mother be x and that of daughter be y. It is given that
x + 2y = 50
2x + y = 76
Step 2
On solving above two equations we get x = 34 and y = 8 years
Copyright 2016 www.edugain.com
Personal use only. Commercial use is strictly prohibited.
ID : eu-10-Linear-Equations-in-Two-Variables [7]
(9) 23 years and 8 years
Step 1
Let x and y be the current age of Filip and his daughter respectively.
It is given that, T hree years ago Filip was f our times older than his daughter
(x - 3) = 4 (y - 3)
x - 4y = -9 ____________________(1)
Step 2
It is also given that, af ter three years Filip will be 4 years more than two times the age of
his daughter
(x + 3) = 2 (y + 3) + 4
x - 2y = 7 ____________________(2)
Step 3
On subtracting eq. 1 f rom 2
2y = 16
y = 8 years
Step 4
On substituting value of y in eq. 1
x - 4(8) = -9
x = 23 years
Step 5
T href ore,
present age of Filip = x = 23 years
present age of his daughter = y = 8 years
(10) 36
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) + 27 = 10y + x
9x - 9y = -27
x - y = -3 ________________________(2)
Step 2
On adding two equations,
y - 2x + x - y = -3
-x = -3
x=3
Step 3
y = 2x = 2×3 = 6
T heref ore number is 36
Copyright 2016 www.edugain.com
Personal use only. Commercial use is strictly prohibited.
ID : eu-10-Linear-Equations-in-Two-Variables [8]
(11) b. 35
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) + 3
6x - 3y = 3 ________________________(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 = 3 and y = 5
T heref ore number is 35
(12) c. inf initely many solutions
Step 1
Lets calculate ratio of coef f icients of these equations
a1
=
a2
b1
-3
=
b2
c1
3
1
-1
=
c2
-2
2
Step 2
On comparing these ratios we observe
a1
=
a2
b1
b2
=
c1
c2
Step 3
T heref ore these equations have inf initely many solutions
Copyright 2016 www.edugain.com
Personal use only. Commercial use is strictly prohibited.
ID : eu-10-Linear-Equations-in-Two-Variables [9]
(13) d. 3x + 4y + 2 = 0, 12x + 16y + 8 = 0
Step 1
Equations a1x + b1y + c1 = 0 and a1x + b1y + c1 = 0 have inf initely many solutions if ,
a1
=
a2
b1
c1
=
b2
c2
Step 2
If we try this condition on given f our set of equations, it is only satisf ied by pair 3x + 4y + 2
= 0, 12x + 16y + 8 = 0, since
3
=
12
4
2
=
16
8
Step 3
T heref ore answer is 3x + 4y + 2 = 0, 12x + 16y + 8 = 0
(14) b. -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
3
≠
-2
Step 3
T href ore k = -2
(15) a. no solution
Step 1
T he conditions given in two equations are conf licting (i.e. they cannot be satisf ied by a
single value of y simultaneously.
If value of y is -3, x ≠ -7
Similarly, if value of y is -7, x ≠ -3
Step 2
T heref ore this pair of equations have no solution.
Copyright 2016 www.edugain.com
Personal use only. Commercial use is strictly prohibited.