GRADE: VIII SUBJECT: Opt. Maths READING SECTION: ALGEBRA Ordered Pairs. Two real numbers (x,y) written in certain or specific order is called an ordered pair. Step 1: Consider two numbers 4 and 7. We usually write ordered pair as (4,7) where 4 is the first component and 7 is the second component of the ordered pair. The pair of numbers (a, b) where ‘a’ is the first component and ‘b’ is the second element, is also called an ordered pair. The first and second component of an ordered pair is also known as ‘x’ component and ‘y’ component respectively i.e. (x, y) in the coordinate plane. Example: (2, 3), (5,2), (6, 7) etc. are ordered pairs. It is used to represent the location of certain object or point on coordinate plane. Ordered pairs representation Figure: 1 Figure: 2 In figure: 2, the ordered pairs are (-3, 4), (3, 4), (2, 0), (0, -3), (5, -4) and (-4, -2) Equality of Ordered Pairs The ordered pairs (a,b) and (c,d) are equal ordered pair If and only if a=c and b=d. Lets see worked out examples: Which of the following ordered pairs are equal? a) (3,2) and (3,3) b) (5,3) and (5,3) Solution: a Solution: b The first component of both the The first component of both the ordered pairs are equal; i.e. 3=3, ordered pairs are equal; i.e. 5=5, The second component of these The second component of these ordered pairs are not equal; i.e. ordered pairs are also equal; i.e. 3=3 2≠3. Therefore, (5,3) = (5,3). Therefore, (3,2) ≠ (3,3). This is equal ordered pair This is not equal ordered pair EXERCISE: 1 1. Identify which of the following pairs are equal. a. (2,3) and (3,2) b. (1,3) and (1,3) c. (4,4) and (5,5) d. (1,a) and (1,b) e. (2,4) and (2,4) f. (3,7) and (3,7) Example 2: If (x, y) = (a, b) Example 3: If (x, y) = (3, -5) By equality of ordered pairs , By equality of ordered pairs, x = a and y = b x = 3 and y = -5 EXERCISE: 2 2. Find the value of x and y if: a. (x, 4) = (2, y) b. (x, y) = (-3, 1) c. (4, -2) = (x, y) d. (x, y) = (-9, -7) e. (-5, -6) = (x, y) Example 4: If ( x+1, y-2) = (4, 3) Example 5: If (2x+1, 3y-2) = (9, 4) By equality of ordered pairs, By equality of ordered pairs, x+1 = 4 y-2 = 3 2x+1 = 9 3y-2 = 4 x = 4-1 y = 3-2 2x = 9-1 3y = 4-2 x = 3 y = 1 2x = 8 3y = 2 3 x = 4 y = Example 6: 2 퐴 = 1, 2 푠 푎 푠푒푡. 퐹푟표푚 푡ℎ푠 푠푒푡 푚푎푘푒 푎푙푙 푝표푠푠푏푙푒 표푟푑푒푟푒푑 푝푎푟푠. Solution: 2 3 Here, the possible ordered pairs from the given set Example 7: If (푥 +3, 푦 − 5) = (7, 3) are shown by the following scheme. By equality of ordered pairs, 2 3 푥 +3 = 7 푦 − 5 = 3 푥2 = 7 − 3 푦3 = 3 + 5 2 3 푥 = 4 푦 = 8 x = ±2 y = 2 Thus, the ordered pairs are: (1, 1), (1, 2), (2, 1) 푎푛푑 (2, 2) EXERCISE: 3 3. Find the value of x and y if: a. (x+y,2) = (6,x-y) b. (2x+y,2) = (6,2x-y) c. (2x-1,-3) = (3,y+3) d. (x+y,1) = (2,2x-y) Miscellaneous Exercise: 1. In each of the following ordered pairs write down the first component and second component. 푎) (4, 5) b) (2, 3) c) (8, 5) d) (푎, 푏 + 2) e) (푎 + 5, 6) 2. Which of the following ordered pairs are equal? a) (6, 2) 푎푛푑 2, 6 b) (8, 7) 푎푛푑 (7, 8) c) (12 − 3, 5 + 3) 푎푛푑 (9, 8) 12 15 35 35 d) (4, 5) 푎푛푑 (5 − 1, 6 − 1) e) (4, 5) 푎푛푑 , f) (5, 7) 푎푛푑 , 3 3 7 5 3. Find a and b, if a) (a-1, b)= (2, 3) b) (a+3, b-4) = (0, 5) c) (2a-3, 4) = (4a, b+5) 4. Following ordered pairs are equal. Find the values of x and y. a) 4, 푦 푎푛푑 푥, 7 b) 6, 푦 푎푛푑 푥, 5 c) 2, 3 푎푛푑 푥 − 1, 푦 d) 푥, 푦 − 1 푎푛푑 (4, 5) e) 푦, 푦 + 1 푎푛푑 (푥 + 2, 5) f) 푥 + 3, 4 푎푛푑 (6, 푦) g) 3푥 + 2, 2푦 + 1 푎푛푑 (5, 3) h) 3푥 + 1, 5 푎푛푑 4, 3푦 − 1 5. From each of the following sets make all possible ordered pairs. a) 퐴 = *푎, 푏+ b) 푃 = *푥, 푦) c) 푋 = *1, 2, 3+ Contact Person Name: Mr. Manoj Kunwar Phone No.: 9843383236 Email: [email protected] .