Download Grade 10 Linear Equations in Two Variables

ID : ph-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)
Find value of k f or which equations -2x + 6y + 1 = 0 and kx - 3y - 2 = 0, will have no solution.
(2)
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.
(3)
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 6 as remainder. If f our times the smaller of two numbers is divided by
the larger one, we get 1 as quotient and 22 as remainder. Find the numbers.
(4) Find value of k such that equations -3x + ky - 2 = 0 and -9x - 3y - 6 = 0, represents coincident
lines.
Choose correct answer(s) f rom given choice
(5)
T wo numbers are in ratio 3:4. If 4 is added to both the numbers, ratio becomes 4:5. Find the
numbers.
a. 18 and 24
b. 9 and 12
c. 15 and 20
d. 12 and 16
(6) T hree years ago Jejomar was f ive times older than his son. Af ter three years, Jejomar will be 9
years more than two times the age of his son. Find the present age of Jejomar and his son.
a. 28 and 8 years
b. 27 and 7 years
c. 30 and 10 years
d. 26 and 6 years
(7) If f ollowing pair of equations have inf initely many solutions, f ind the value of a and b.
-24x - 30y = -18
(3a + 2b)x + (3a + b)y = -9
(8)
a. a = -5, b = 3
b. a = -6, b = 3
c. a = -7, b = 3
d. a = -7, b = 2
A two digit number is 3 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. 38
b. 56
c. 74
d. 47
(9) Which of the f ollowing equations has an inf inite number of solutions?
a. 5x + 4y + 3 = 0, 20x + 16y + 11 = 0
b. 5x + 4y + 3 = 0, 15x + 12y + 9 = 0
c. 5x + 4y + 10 = 0, 15x + 5y + 8 = 0
d. 5x + 4y + 3 = 0, 15x + 12y + 11 = 0
(C) 2016 Edugain (www.Edugain.com)
Personal use only, commercial use is strictly prohibited
ID : ph-10-Linear-Equations-in-Two-Variables [2]
(10) Which of the f ollowing equations has a unique solution?
a. 3x + 5y + 9 = 0, 15x + 25y + 45 = 0
b. 3x + 5y + 9 = 0, 18x + 30y + 47 = 0
c. 3x + 5y + 9 = 0, 15x + 25y + 47 = 0
d. 3x + 5y + 46 = 0, 15x + 6y + 42 = 0
(11) T he pair of equations 3p - q + 1 = 0 and -9p - 5 = 0 have
a. exactly two solutions
b. a unique solution
c. inf initely many solutions
d. no solution
(12) 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. 0.5
d. -1
(13) If a pair of linear equations is consistent, then the lines will be
a. always intersecting
b. always coincident
c. intersecting or coincident
d. parallel
(14) 9 chairs and 6 tables cost ₱4350 and 6 chairs and 8 tables cost ₱4300. Find the cost of 2 chairs
and 2 tables.
a. ₱1450
b. ₱850
c. ₱1200
d. ₱950
(15) Kidlat is three times as old as his son. Af ter 10 years, Kidlat will be two times as hold as his son.
Find the current age of Kidlat and his son.
a. 36 and 12 years
b. 30 and 10 years
c. 33 and 11 years
d. 27 and 9 years
© 2016 Edugain (www.edugain.com).
All Rights Reserved
(C) 2016 Edugain (www.Edugain.com)
Many more such worksheets can be
generated at www.edugain.com
Personal use only, commercial use is strictly prohibited
ID : ph-10-Linear-Equations-in-Two-Variables [3]
Answers
(1)
1
Step 1
Equations a1x + b1y + c1 = 0 and a1x + b1y + c1 = 0 have no solution if ,
a1
=
a2
b1
≠
b2
c1
c2
Step 2
On substituting coef f icients in above condition,
k
=
-2
-3
≠
6
-2
1
Step 3
T href ore k = 1
(2)
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
(C) 2016 Edugain (www.Edugain.com)
Personal use only, commercial use is strictly prohibited
ID : ph-10-Linear-Equations-in-Two-Variables [4]
(3)
26 and 12
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 + 6 => 3x - 6y - 6 = 0
4y = 1x + 22 => 1x - 4y + 22 = 0
Step 3
On solving these two equations we get x = 26 and y = 12
(4) -1
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
=
-9
k
=
-3
-2
-6
Step 3
T href ore k = -1
(C) 2016 Edugain (www.Edugain.com)
Personal use only, commercial use is strictly prohibited
ID : ph-10-Linear-Equations-in-Two-Variables [5]
(5)
d. 12 and 16
Step 1
Since numbers are in ratio 3:4, lets assume numbers are 3x and 4x
Step 2
When 4 is added to both numbers, numbers become (3x + 4) and (4x + 4)
Step 3
Since new ratio is 4:5
(3x + 4)/(4x + 4) = 4/5
⇒ 5 (3x + 4) = 4 (4x + 4)
⇒ 15x + 20 = 16x + 16
⇒ 15x - 16x = + 16 - 20
⇒ -1x = -4
⇒ x = -4/-1
⇒x=4
Step 4
T heref ore numbers are,
3x = 3 × 4 = 12
4x = 4 × 4 = 16
(C) 2016 Edugain (www.Edugain.com)
Personal use only, commercial use is strictly prohibited
ID : ph-10-Linear-Equations-in-Two-Variables [6]
(6) a. 28 and 8 years
Step 1
Let x and y be the current age of Jejomar and his son respectively.
It is given that, T hree years ago Jejomar was f ive times older than his son
(x - 3) = 5 (y - 3)
x - 5y = -12 ____________________(1)
Step 2
It is also given that, af ter three years Jejomar will be 9 years more than two times the age
of his son
(x + 3) = 2 (y + 3) + 9
x - 2y = 12 ____________________(2)
Step 3
On subtracting eq. 1 f rom 2
3y = 24
y = 8 years
Step 4
On substituting value of y in eq. 1
x - 5(8) = -12
x = 28 years
Step 5
T href ore,
present age of Jejomar = x = 28 years
present age of his son = y = 8 years
(C) 2016 Edugain (www.Edugain.com)
Personal use only, commercial use is strictly prohibited
ID : ph-10-Linear-Equations-in-Two-Variables [7]
(7) b. a = -6, b = 3
Step 1
From the equations, -24x - 30y = -18 and (3a + 2b)x + (3a + b)y = -9, we notice that,
a1 = -24, b1 = -30, c1 = -18
and a2 = 3a + 2b, b2 = 3a + b, c2 = -9
Step 2
It is given that, the equations -24x - 30y = -18 and (3a + 2b)x + (3a + b)y = -9 have inf initely
many solutions.
a1
T heref ore,
=
a2
-24
⇒
=
b2
-30
=
3a + 2b
b1
3a + b
c1
c2
=
-18
-9
Step 3
On comparing f irst two terms,
-24
⇒
-30
=
3a + 2b
3a + b
By cross multiplying both sides, we get,
-24(3a + b) = -30(3a + 2b)
⇒ -72a -24b = -90a -60b
⇒ 18a = -36b
⇒
a
=
-36
b
18
⇒ a = -2 b
Step 4
Now, lets compare f irst and last term,
-24
3a + 2b
=
-18
-9
By cross multiplying both sides, we get,
-24(-9) = -18(3a + 2b)
⇒ 216 = -54a -36b
On substituting a = -2 b,
⇒ 216 = -54(-2 b) -36b
⇒b=3
Step 5
a = -2 b
⇒ -2 × 3
⇒ -6
Step 6
T heref ore, the value of a = -6,
and the value of b = 3.
(C) 2016 Edugain (www.Edugain.com)
Personal use only, commercial use is strictly prohibited
ID : ph-10-Linear-Equations-in-Two-Variables [8]
(8)
d. 47
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) + 27 = 10y + x
9x - 9y = -27
x - y = -3 ________________________(2)
Step 4
On solving equations (1) and (4), we get x = 4 and y = 7
T heref ore number is 47
(C) 2016 Edugain (www.Edugain.com)
Personal use only, commercial use is strictly prohibited
ID : ph-10-Linear-Equations-in-Two-Variables [9]
(10) d. 3x + 5y + 46 = 0, 15x + 6y + 42 = 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 + 46 = 0,
15x + 6y + 42 = 0, since
3
≠
15
5
6
Step 3
T heref ore answer is 3x + 5y + 46 = 0, 15x + 6y + 42 = 0
(11) b. a unique solution
Step 1
Lets calculate ratio of coef f icients of these equations
a1
=
a2
b1
-9
=
b2
c1
3
-1
0
=
c2
1
-5
Step 2
On comparing these ratios we observe
a1
≠
a2
b1
b2
Step 3
T heref ore these equations have a unique solution
(12) a. 1
(13) c. intersecting or coincident
(C) 2016 Edugain (www.Edugain.com)
Personal use only, commercial use is strictly prohibited
ID : ph-10-Linear-Equations-in-Two-Variables [10]
(14) c. ₱1200
Step 1
Let the cost of a chair and table be ₱ x and ₱ y respectively. It is given that
9 x + 6 y = 4350
6 x + 8 y = 4300
Step 2
On multiplying f irst equation by 4 and second equation by 3
36 x + 24 y = 17400
18 x + 24 y = 12900
Step 3
Subtract second equation f rom f irst equation 1
=> 18 x = 4500
=> x = (4500)/(18) = ₱250
Step 4
Substitute this value in f irst equation
=> 9 × 250 + 6 y = 4350
=> 6 y = 4350 - 9 × 250 = 2100
=> y = (2100)/(6) = ₱350
Step 5
Price of 2 chairs and 2 tables
=> 2x + 2y = 2 × 250 + 2 × 350
=> = ₱1200
(15) b. 30 and 10 years
Step 1
Let age of Kidlat be x years and age of be y years
Step 2
It is given that
x = 3y => 3y - x = 0
(x + 10) = 2(y + 10) => x - 2y = 10
Step 3
On adding above two equations
(3y - x) + (x - 2y) = 10
y = 10
Step 4
Hence their ages are x = 3y = 30 years and y = 10 years
(C) 2016 Edugain (www.Edugain.com)
Personal use only, commercial use is strictly prohibited