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LINEAR EQUATIONS IN ONE VARIABLE - Q&A

EXERCISE 14(A)

1. Solve the following equations:
(i) 20 = 6 + 2x
(ii) 15 + x = 5x + 3
(iii) (3x + 2) / (x - 6) = -7
(iv) 3a - 4 = 2(4 - a)
(v) 3(b - 4) = 2(4 - b)
(vi) (x + 2) / 9 = (x + 4) / 11
(vii) (x - 8) / 5 = (x - 12) / 9
(viii) 5(8x + 3) = 9(4x + 7)
(ix) 3(x + 1) = 12 + 4(x - 1)
(x) 3x / 4 - 1 / 4(x - 20) = x / 4 + 32
(xi) 3a / 5 - 2 / 3(a - 30) + a / 20 = 40
(xii) x - 30% of x = 35
Solution:
(i) 20 = 6 + 2x
20 - 6 = 2x
14 = 2x
x = 14 / 2 = 7

(ii) 15 + x = 5x + 3
15 - 3 = 5x - x
12 = 4x
x = 12 / 4 = 3

(iii) (3x + 2) / (x - 6) = -7
3x + 2 = -7(x - 6)
3x + 2 = -7x + 42
3x + 7x = 42 - 2
10x = 40
x = 40 / 10 = 4

(iv) 3a - 4 = 2(4 - a)
3a - 4 = 8 - 2a
3a + 2a = 8 + 4
5a = 12
a = 12 / 5 = 2.4

(v) 3(b - 4) = 2(4 - b)
3b - 12 = 8 - 2b
3b + 2b = 8 + 12
5b = 20
b = 20 / 5 = 4

(vi) (x + 2) / 9 = (x + 4) / 11
11(x + 2) = 9(x + 4)
11x + 22 = 9x + 36
11x - 9x = 36 - 22
2x = 14
x = 14 / 2 = 7

(vii) (x - 8) / 5 = (x - 12) / 9
9(x - 8) = 5(x - 12)
9x - 72 = 5x - 60
9x - 5x = -60 + 72
4x = 12
x = 12 / 4 = 3

(viii) 5(8x + 3) = 9(4x + 7)
40x + 15 = 36x + 63
40x - 36x = 63 - 15
4x = 48
x = 48 / 4 = 12

(ix) 3(x + 1) = 12 + 4(x - 1)
3x + 3 = 12 + 4x - 4
3x + 3 = 8 + 4x
3 - 8 = 4x - 3x
x = -5

(x) 3x / 4 - 1 / 4(x - 20) = x / 4 + 32
Multiply entire equation by 4:
3x - 1(x - 20) = x + 128
3x - x + 20 = x + 128
2x + 20 = x + 128
2x - x = 128 - 20
x = 108

(xi) 3a / 5 - 2 / 3(a - 30) + a / 20 = 40
Multiply by LCM (60):
12(3a) - 40(a - 30) + 3(a) = 2400
36a - 40a + 1200 + 3a = 2400
-a + 1200 = 2400
-a = 1200
a = -1200

(xii) x - 30% of x = 35
x - 0.3x = 35
0.7x = 35
x = 35 / 0.7 = 50

2. Solve:
(i) (x + 1)(x + 2) = (x + 11)(x - 2)
(ii) 3x / 4 - (x - 1) / 2 = (x - 2) / 3
(iii) (5x - 4) / 8 - (x - 3) / 5 = (x + 6) / 4
Solution:
(i) (x + 1)(x + 2) = (x + 11)(x - 2)
x2 + 3x + 2 = x2 + 9x - 22
3x + 2 = 9x - 22
24 = 6x
x = 4

(ii) 3x / 4 - (x - 1) / 2 = (x - 2) / 3
Multiply by 12:
9x - 6(x - 1) = 4(x - 2)
9x - 6x + 6 = 4x - 8
3x + 6 = 4x - 8
14 = x

(iii) (5x - 4) / 8 - (x - 3) / 5 = (x + 6) / 4
Multiply by 40:
5(5x - 4) - 8(x - 3) = 10(x + 6)
25x - 20 - 8x + 24 = 10x + 60
17x + 4 = 10x + 60
7x = 56
x = 8

3. If (2x + 1) / (3x - 2) = 9/10, find the value of x.
Solution:
10(2x + 1) = 9(3x - 2)
20x + 10 = 27x - 18
28 = 7x
x = 4

4. If (x - 1) / (x + 1) = (2x - 5) / (2x - 11), find the value of x.
Solution:
(x - 1)(2x - 11) = (2x - 5)(x + 1)
2x2 - 13x + 11 = 2x2 - 3x - 5
-13x + 11 = -3x - 5
16 = 10x
x = 1.6

5. Find x, if (4x + 3) / (2x - 5) = 2
Solution:
4x + 3 = 2(2x - 5)
4x + 3 = 4x - 10
3 = -10 (False)
No solution.

6. Solve:
(i) (x2 - 9) / (5 + x2) = -5 / 9
(ii) (y2 + 4) / (3y2 + 7) = 1 / 2
Solution:
(i) 9(x2 - 9) = -5(5 + x2)
9x2 - 81 = -25 - 5x2
14x2 = 56
x2 = 4 => x = ±2

(ii) 2(y2 + 4) = 1(3y2 + 7)
2y2 + 8 = 3y2 + 7
1 = y2 => y = ±1

7. Solve: 1/(2x - 3) + 1/(x - 5) = 1 1/9
Solution:
[(x - 5) + (2x - 3)] / [(2x - 3)(x - 5)] = 10/9
(3x - 8) / (2x2 - 13x + 15) = 10/9
9(3x - 8) = 10(2x2 - 13x + 15)
27x - 72 = 20x2 - 130x + 150
20x2 - 157x + 222 = 0
x = 6 or x = 1.85

8. Solve: x/2 - 1/5 = x/3 + 1/4
Solution:
x/2 - x/3 = 1/4 + 1/5
x/6 = 9/20
x = 54/20 = 2.7

9. Solve: (2x - 3) / (2x + 1) = (3x + 1) / (3x - 1)
Solution:
(2x - 3)(3x - 1) = (3x + 1)(2x + 1)
6x2 - 11x + 3 = 6x2 + 5x + 1
2 = 16x
x = 1/8


EXERCISE 14(B)

1. 30 less than 4 times a number is 10 more than twice the number. Find the number.
Solution:
4x - 30 = 2x + 10
2x = 40
x = 20

2. A number is as much greater than 35 as is less than 53. Find the number.
Solution:
x - 35 = 53 - x
2x = 88
x = 44

3. The sum of two numbers is 25. One of the numbers exceeds the other by 9. Find the numbers.
Solution:
x + (x + 9) = 25
2x = 16 => x = 8
Numbers are 8 and 17.

4. A number is multiplied by 5 and 25 is subtracted from the product. The result is equal to four times the number plus 10. Find the number.
Solution:
5x - 25 = 4x + 10
x = 35

5. The sum of two numbers is 4500. If 10% of one number is 12.5% of the other, find the numbers.
Solution:
0.1x = 0.125(4500 - x)
0.1x = 562.5 - 0.125x
0.225x = 562.5
x = 2500
Numbers are 2500 and 2000.

6. The sum of two numbers is 405 and their ratio is 8:7. Find the numbers.
Solution:
8x + 7x = 405
15x = 405 => x = 27
Numbers: 216 and 189.

7. The ages of A and B are in the ratio 7:5. Ten years hence, the ratio of their ages will be 9:7. Find their present ages.
Solution:
(7x + 10)/(5x + 10) = 9/7
49x + 70 = 45x + 90
4x = 20 => x = 5
Ages: 35 and 25.

8. Find the number whose double is 45 greater than its half.
Solution:
2x = x/2 + 45
4x = x + 90
3x = 90 => x = 30

9. The difference between the squares of two consecutive numbers is 31. Find the numbers.
Solution:
(x + 1)2 - x2 = 31
2x + 1 = 31
2x = 30 => x = 15
Numbers: 15 and 16.

10. Find a number such that when 5 is subtracted from 5 times the number, the result is 4 more than twice the number.
Solution:
5x - 5 = 2x + 4
3x = 9 => x = 3

11. The numerator of a fraction is 5 less than its denominator. If 3 is added to both, the fraction becomes 4/5. Find the original fraction.
Solution:
(x - 5 + 3) / (x + 3) = 4/5
5(x - 2) = 4(x + 3)
5x - 10 = 4x + 12
x = 22
Fraction: 17/22

12. The difference between two numbers is 36. The quotient when one number is divided by other is 4. Find the two numbers.
Solution:
x - y = 36; x = 4y
4y - y = 36
3y = 36 => y = 12
Numbers: 48 and 12.

13. Five years ago, Mohit was thrice of Manish. 10 years later, Mohit will be twice as old as Manish. Find their present ages.
Solution:
M - 5 = 3(m - 5) => M = 3m - 10
M + 10 = 2(m + 10) => M = 2m + 10
3m - 10 = 2m + 10 => m = 20, M = 50
Ages: 50 and 20.

14. The denominator of a fraction is 3 more than its numerator. If numerator is increased by 7 and denominator decreased by 2, fraction is 2. Find the sum of numerator and denominator.
Solution:
(x + 7) / (x + 3 - 2) = 2
x + 7 = 2(x + 1)
x = 5
Fraction: 5/8. Sum = 13.

15. The digits of a two-digit number differ by 3. If digits are interchanged and added to original, result is 121. Find the number.
Solution:
11(x + y) = 121 => x + y = 11
x - y = 3 => x=7, y=4 (74)
y - x = 3 => y=7, x=4 (47)
Numbers: 47 or 74.


Test yourself

1. Multiple Choice Type:
(i) If (x - 2) / 3 = (x + 3) / 2, then value of x is:
(a) -13 (b) 13 (c) 5 (d) 1/13
Answer: (a)
2(x - 2) = 3(x + 3) => 2x - 4 = 3x + 9 => x = -13

(ii) If 2/3 x + 10 = x/4 - 5, then value of x is:
(a) 12 (b) -36 (c) -12 (d) 36
Answer: (b)
5x/12 = -15 => x = -36

(iii) The sum of three consecutive odd numbers is 57. The middle number is:
(a) 17 (b) 19 (c) 21 (d) 23
Answer: (b)
3x = 57 => x = 19

(iv) 3a = 4(a - 2), then a is equal to:
(a) 8 (b) -8 (c) 2 (d) -2
Answer: (a)
3a = 4a - 8 => a = 8

(v) x exceeds 6 by 3. Then x is:
(a) 18 (b) 3 (c) 9 (d) 2
Answer: (c)
x - 6 = 3 => x = 9

(vi) Statement 1: If 5 is added to 3 times a number, it becomes 14. This statement in the form of an equation is 3x + 5 = 14.
Statement 2: The roots of the equation (x - 3) / 5 = (x + 1) / 4 is -17.
(a) Both statements are true.
(b) Both the statements are false.
(c) Statement 1 is true, and statement 2 is false.
(d) Statement 1 is false, and statement 2 is true.
Answer: (a)
Both are correct.

(vii) Assertion (A): The equation 4x - 13 = 14 - 5x has the solution x = 3.
Reason (R): For any equation of the form ax + b = cx + d, solution is x = (d - b) / (a - c).
(a) (1) (b) (2) (c) (3) (d) (4)
Answer: (a)
Both true and R explains A.

(viii) Assertion (A): If the sum of two consecutive numbers is 11, then the numbers are 5 and 6.
Reason (R): If two consecutive numbers are x and x + 1, then x + (x + 1) = 11.
(a) (1) (b) (2) (c) (3) (d) (4)
Answer: (a)
Both true and R explains A.

(ix) Assertion (A): If the difference of two numbers is 5 and their sum is 19, then the numbers are 12 and 7.
Reason (R): We can verify the answer by substituting values in the given problem statement.
(a) (1) (b) (2) (c) (3) (d) (4)
Answer: (b)
Both true, but R is a method, not the mathematical derivation.

2. Solve: (2x + 1) / (3x + 2) = 5/7
Solution:
14x + 7 = 15x + 10 => x = -3

3. Solve: (5x - 7) / (3x + 1) = (5x - 1) / (3x + 5)
Solution:
15x2 + 4x - 35 = 15x2 + 2x - 1
2x = 34 => x = 17

4. Solve: (y + 1) / (y + 2) = (y + 11) / (y + 8)
Solution:
y2 + 9y + 8 = y2 + 13y + 22
-14 = 4y => y = -3.5

5. Solve: (x + 3) / 2 - (x - 2) / 2 = 1.6 + x / 4
Solution:
2.5 = 1.6 + 0.25x
0.9 = 0.25x => x = 3.6

6. Find two numbers such that one of them exceeds the other by 9 and their sum is 81.
Solution:
x + x + 9 = 81 => 2x = 72 => x = 36
Numbers: 36, 45

7. The length of a rectangular plot exceeds its breadth by 5 metres. If the perimeter of the plot is 142 metres. Find the dimensions of the plot.
Solution:
2(2x + 5) = 142 => 2x + 5 = 71 => x = 33
Dimensions: 33m and 38m

8. The numerator of a fraction is 4 less than its denominator. If 1 is added to both its numerator and denominator, it becomes 1/2. Find the fraction.
Solution:
(x - 3) / (x + 1) = 1/2
2x - 6 = x + 1 => x = 7
Fraction: 3/7

9. A number consists of two digits whose sum is 9. If 27 is subtracted from the number, its digits are reversed. Find the number.
Solution:
9x + 9 - 27 = 90 - 9x
18x = 108 => x = 6
Number: 63

10. The ages of Amit and Archana are in the ratio 4:5. If Amit is 4 years 8 months old, find the age of Archana.
Solution:
Amit = 14/3 years.
Archana = (14/3) * (5/4) = 35/6 years
Age: 5 years 10 months.

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Quick Review Flashcards - Click to flip and test your knowledge!
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A statement which states that two algebraic expressions are equal.
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An equation involving only one variable with its highest power equal to one.
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