Linear Equation with two variables
Q1. If 2x + 3y = 29 and y = x +3, what is the value of x ?
(a) 2
(b) 4
(c) 6
(d) 8
Answer: (b) 4
2x + 3y = 29 ----> (1)
y = x + 3
⇒ 2y = 2x + 6
⇒ 2x - 2y = -6 ------>(2)
(2)-(1) ⇒ 2x + 3y - 2x + 2y = 29 + 6
⇒ 5y = 35
⇒ y = 7
(1) ⇒ 2x + 3 X 7 = 29
⇒ 2x = 29 - 21 = 8
⇒ x = 4
Q2. If (2, k) is a solution of 3x + y - 5 = 0, then k = ?
(a) -1
(b) -2
(c) -4
(d) -5
Answer: (a) -1
3x + y - 5 = 0
⇒ 3.2 + k - 5 = 0
⇒ 6 + k - 5 = 0
⇒ k = -1
Q3. If 3x + 7y = 72 and 5x - 5y = 20, then what is x + y
(a) 12
(b) 15
(c) 16
(d) 18
Answer: (c) 16
3x + 7y = 72 ----->(1)
5x - 5y = 20
⇒ x - y = 4
⇒ 7x - 7y = 28 -----> (2)
(1)+(2) ⇒ 10x = 100
⇒ x = 10
(1)⇒ 3.10 + 7y = 72
⇒ 7y = 72 - 30
⇒ y = 42/7 = 6
∴ x + y = 10 + 6 = 16
Q4. If x + 1/y = 5, 2x + 3/y = 13, then (2x - 3y) = ?
(a) 1
(b) 3
(c) 6
(d) 7
Answer: (b) 3
x + 1/y = 5
⇒ 2x + 2/y = 10 ------> (1)
2x + 3/y = 13 ----> (2)
(1) - (2) ⇒ 2/y - 3/y = 10 - 13
⇒ (2-3)/y = -3
⇒ 1/y = 3
⇒ y = 1/3
| (1) ⇒ 2x + | 2 | = 10 |
| 1/3 |
⇒ 2x + 6 = 10
⇒ 2x = 4
⇒ x = 2
∴ 2x - 3y = 2 X 2 - 3 X 1/3 = 4 - 1 = 3
Q5. If 2x - 3y = 5 and (x+y)/x = 3/7 then x + y = ?
(a) 102/27
(b) 113/16
(c) 121/29
(d) 135/26
Answer: (d) 135/26
2x - 3y = 5
⇒ 4x - 6y = 10 ------> (1)
(x+y)/x = 3/7
⇒ 7x + 7y = 3x
⇒ 4x + 7y = 0 -----> (2)
(1)-(2)⇒ -13y = 10
⇒ y = -10/13
| (1) ⇒ 4x - 6 X | -10 | = 10 |
| 13 |
| ⇒ 4x + | 60 | = 10 |
| 13 |
| ⇒ 4x = 10 - | 60 | = | 130 - 60 | = | 70 |
| 13 | 13 | 13 |
⇒ x = 70/(13X4) = 70/52
| ∴ x + y = | 70 | - | 10 | = | 280 - 10 |
| 52 | 13 | 52 |
= 270/52 = 135/26
Q6. The system of equations kx - y = 2 and 6x - 2y = 3 has a unique solution when
(a) 2
(b) 3
(c) 4
(d) 5
Answer: (b) 3
For a unique solution
k/6 ≠ (-1)/(-2)
⇒ k ≠ (1/2) X 6 = 3
Q7. The cost of 2 sarees and 4 shirts is Rs. 16000 while cost of one saree is equal to cost of 6 shirts. The cost of 10 shirts is
(a) 5,000
(b) 8,000
(c) 10,000
(d) 12,000
Answer: (c) 10,000
Let the cost of 1 saree = Rs. x
Let the cost of 1 shirt = Rs. y
A/Q 2x + 4y = 16000 -------- (1)
And x = 6y -------- (2)
(1) ⇒ 2. 6 y + 4y = 16000
⇒ 12y + 4y = 16000
⇒ 16y = 16000
⇒ y = 1000
∴ Cost of 10 shirts = 10 X 1000 = Rs. 10,000
Q8. For what value of k, the system of equations kx - 3y + 2 = 0 and 6x - 2y + 3 = 0 has a unique solution ?
(a) 9
(b) 11
(c) 15
(d) 16
Answer: (a) 9
| For unique solution, | a1 | ≠ | b1 |
| a2 | b2 |
| ⇒ | k | ≠ | -3 |
| 6 | -2 |
| ⇒ | k | ≠ | 3 |
| 6 | 2 |
| ⇒ k ≠ | 3 | X 6 |
| 2 |
⇒ k ≠ 9
Q9. If x/4 + y/3 = 5/12 and x/2 + y = 1 then the value of 2x + y is
(a) 1/3
(b) 5/2
(c) 5/3
(d) 6/7
Answer: (b) 5/2
| x | + | y | = | 5 | |
| 4 | 3 | 12 |
| ⇒ | 3x + 4y | = | 5 |
| 12 | 12 |
⇒ 3x + 4y = 5 ------- (1)
x/2 + y = 1
⇒ x + 2y = 2
⇒ 3x + 6y = 6 ------- (2)
(2) - (1) ⇒ 2y = 1
⇒ y = 1/2
(1) ⇒ 3x + 4y = 5
⇒ 3x + 4.(1/2) = 5
⇒ 3x + 2 = 5
⇒ 3x = 3
⇒ x = 1
∴ 2x + y = 2. 1 + ½ = 2 + ½ = 5/2
Q10. If 2a + 3b = 17 and 2a+2 - 3b+1 = 5 then a = ?, b = ?
(a) a = 3, b = 2
(b) a = 1, b = 3
(c) a = 2, b = 5
(d) a = 3, b = 5
Answer: (a) a = 3, b = 2
2a + 3b = 17 ------(1)
2a+2 - 3b+1 = 5
⇒ 2a.22 - 3b.3 = 5
⇒ 4.2a - 3.3b = 5 --------(2)
putting 2a = x , 3b = y
(1)⇒ x + y = 17
⇒ 4x + 4y = 68 ------- (3)
(2)⇒ 4x - 3y = 5 ------- (4)
(3) - (4) ⇒ 7y = 63
⇒ y = 9
⇒ 3b = 9
⇒ 3b = 32
⇒ b = 2
(1)⇒ 2a + 32 = 17
2a = 17 - 9 = 8 = 23
⇒ a = 3
∴ a = 3, b = 2
Practice Test Exam