2. Linear
Question 1. Find the value of a
1
5
1
5
2
a.
a
b.
1
5 15
3
13
7
Question 2. Write the additive inverse of each of the following:
1
2
a. 1
b.
C.
1
7
5
uestion 3. Evaluate:
9
25
1
b.
C.
3x10
3
10
8
3
27
uestion 4. Verify that - (-x) = x for:
Answers
1. Add the following rational numbers:
(i) -5/7 and 3/7
(ii) -15/4 and 7/4
(iii) -8/11 and -4/11
(iv) 6/13 and -9/13
Solution:
Since the denominators are of same positive numbers we can add them directly
(i) -5/7 + 3/7 = (-5+3)/7 = -2/7
(ii) -15/4 + 7/4 = (-15+7)/4 = -8/4
Further dividing by 4 we get,
-8/4 = -2
(iii) -8/11 + -4/11 = (-8 + (-4))/11 = (-8-4)/11 = -12/11
(iv) 6/13 + -9/13 = (6 + (-9))/13 = (6-9)/13 = -3/13
2. Add the following rational numbers:
(i) 3/4 and -5/8
Solution: The denominators are 4 and 8
By taking LCM for 4 and 8 is 8
We rewrite the given fraction in order to get the same denominator
3/4 = (3×2) / (4×2) = 6/8 and
-5/8 = (-5×1) / (8×1) = -5/8
Since the denominators are same we can add them directly
6/8 + -5/8 = (6 + (-5))/8 = (6-5)/8 = 1/8
(ii) 5/-9 and 7/3
Solution: Firstly we need to convert the denominators to positive numbers.
5/-9 = (5 × -1)/ (-9 × -1) = -5/9
The denominators are 9 and 3
By taking LCM for 9 and 3 is 9
We rewrite the given fraction in order to get the same denominator
-5/9 = (-5×1) / (9×1) = -5/9 and
7/3 = (7×3) / (3×3) = 21/9
Since the denominators are same we can add them directly
-5/9 + 21/9 = (-5+21)/9 = 16/9
(iii) -3 and 3/5
Solution: The denominators are 1 and 5
By taking LCM for 1 and 5 is 5
We rewrite the given fraction in order to get the same denominator
-3/1 = (-3×5) / (1×5) = -15/5 and
3/5 = (3×1) / (5×1) = 3/5
Now, the denominators are same we can add them directly
-15/5 + 3/5 = (-15+3)/5 = -12/5
(iv) -7/27 and 11/18
Solution: The denominators are 27 and 18
By taking LCM for 27 and 18 is 54
We rewrite the given fraction in order to get the same denominator
-7/27 = (-7×2) / (27×2) = -14/54 and
11/18 = (11×3) / (18×3) = 33/54
Now, the denominators are same we can add them directly
-14/54 + 33/54 = (-14+33)/54 = 19/54
(v) 31/-4 and -5/8
Solution: Firstly we need to convert the denominators to positive numbers.
31/-4 = (31 × -1)/ (-4 × -1) = -31/4
The denominators are 4 and 8
By taking LCM for 4 and 8 is 8
We rewrite the given fraction in order to get the same denominator
-31/4 = (-31×2) / (4×2) = -62/8 and
-5/8 = (-5×1) / (8×1) = -5/8
Since the denominators are same we can add them directly
-62/8 + (-5)/8 = (-62 + (-5))/8 = (-62-5)/8 = -67/8
(vi) 5/36 and -7/12
Solution: The denominators are 36 and 12
By taking LCM for 36 and 12 is 36
We rewrite the given fraction in order to get the same denominator
5/36 = (5×1) / (36×1) = 5/36 and
-7/12 = (-7×3) / (12×3) = -21/36
Now, the denominators are same we can add them directly
5/36 + -21/36 = (5 + (-21))/36 = 5-21/36 = -16/36 = -4/9
(vii) -5/16 and 7/24
Solution: The denominators are 16 and 24
By taking LCM for 16 and 24 is 48
We rewrite the given fraction in order to get the same denominator
-5/16 = (-5×3) / (16×3) = -15/48 and
7/24 = (7×2) / (24×2) = 14/48
Now, the denominators are same we can add them directly
-15/48 + 14/48 = (-15 + 14)/48 = -1/48
(viii) 7/-18 and 8/27
Solution: Firstly we need to convert the denominators to positive numbers.
7/-18 = (7 × -1)/ (-18 × -1) = -7/18
The denominators are 18 and 27
By taking LCM for 18 and 27 is 54
We rewrite the given fraction in order to get the same denominator
-7/18 = (-7×3) / (18×3) = -21/54 and
8/27 = (8×2) / (27×2) = 16/54
Since the denominators are same we can add them directly
-21/54 + 16/54 = (-21 + 16)/54 = -5/54
3.Simplify:
(i) 8/9 + -11/6
Solution: let us take the LCM for 9 and 6 which is 18
(8×2)/(9×2) + (-11×3)/(6×3)
16/18 + -33/18
Since the denominators are same we can add them directly
(16-33)/18 = -17/18
(ii) 3 + 5/-7
Solution: Firstly convert the denominator to positive number
5/-7 = (5×-1)/(-7×-1) = -5/7
3/1 + -5/7
Now let us take the LCM for 1 and 7 which is 7
(3×7)/(1×7) + (-5×1)/(7×1)
21/7 + -5/7
Since the denominators are same we can add them directly
(21-5)/7 = 16/7
(iii) 1/-12 + 2/-15
Solution: Firstly convert the denominator to positive number
1/-12 = (1×-1)/(-12×-1) = -1/12
2/-15 = (2×-1)/(-15×-1) = -2/15
-1/12 + -2/15
Now let us take the LCM for 12 and 15 which is 60
(-1×5)/(12×5) + (-2×4)/(15×4)
-5/60 + -8/60
Since the denominators are same we can add them directly
(-5-8)/60 = -13/60
(iv) -8/19 + -4/57
Solution: let us take the LCM for 19 and 57 which is 57
(-8×3)/(19×3) + (-4×1)/(57×1)
-24/57 + -4/57
Since the denominators are same we can add them directly
(-24-4)/57 = -28/57
(v) 7/9 + 3/-4
Solution: Firstly convert the denominator to positive number
3/-4 = (3×-1)/(-4×-1) = -3/4
7/9 + -3/4
Now let us take the LCM for 9 and 4 which is 36
(7×4)/(9×4) + (-3×9)/(4×9)
28/36 + -27/36
Since the denominators are same we can add them directly
(28-27)/36 = 1/36
(vi) 5/26 + 11/-39
Solution: Firstly convert the denominator to positive number
11/-39 = (11×-1)/(-39×-1) = -11/39
5/26 + -11/39
Now let us take the LCM for 26 and 39 which is 78
(5×3)/(26×3) + (-11×2)/(39×2)
15/78 + -22/78
Since the denominators are same we can add them