20 exercises of merging similar items of rational numbers

20 exercises of merging similar items of rational numbers

Example 1. Merging similar items
（1）(3x-5y)-(6x+7y)+(9x-2y)
（2）2a-[3b-5a-(3a-5b)]
（3）(6m2n-5mn2)-6(m2n-mn2)
（1）(3x-5y)-(6x+7y)+(9x-2y)
=3x-5y-6x-7y + 9x-2y (remove brackets correctly)
=(3-6 + 9) x + (- 5-7-2) y (merge congeners)
=6x-14y
(2) 2A - [3b-5a - (3a-5b)]
=2A - [3b-5a-3a + 5B] (remove parentheses first)
=2A - [- 8A + 8b] (merge similar items in time)
=2A + 8a-8b (without brackets)
=10a-8b
(3) (6m2n-5mn2) - 6 (m2n-mn2) (note the factor 6 before the second bracket)
=6m2n-5mn2-2m2n + 3mn2
=(6-2) m2N + (- 5 + 3) Mn2
=4m2n-2mn2
Example 2: a = 3x2-4xy + 2Y2, B = x2 + 2xy-5y2
Find: (1) a + B (2) A-B (3) if 2a-b + C = 0, find C
(1）A+B=(3x2-4xy+2y2)+(x2+2xy-5y2)
=3x2-4xy + 2Y2 + x2 + 2xy-5y2 (without brackets)
=(3 + 1) x2 + (- 4 + 2) XY + (2-5) Y2 (merge congeners)
=4x2-2xy-3y2 (in descending order of x)
（2）A-B=(3x2-4xy+2y2)-(x2+2xy-5y2)
=3x2-4xy + 2y2-x2-2xy + 5y2 (without brackets)
=(3-1) x2 + (- 4-2) XY + (2 + 5) Y2 (merge congeners)
=2x2-6xy + 7y2 (in descending order of x)
（3）∵2A-B+C=0
∴C=-2A+B
=-2(3x2-4xy+2y2)+(x2+2xy-5y2)
=-6x2 + 8xy-4y2 + x2 + 2xy-5y2
=(- 6 + 1) x2 + (8 + 2) XY + (- 4-5) Y2 (merge congeners)
=-5x2 + 10xy-9y2 (in descending order of x)
Example 3. Calculation:
（1）m2+(-mn)-n2+(-m2)-(-0.5n2)
（2）2(4an+2-an)-3an+(an+1-2an+1)-(8an+2+3an)
(3) Simplification: (X-Y) 2 - (X-Y) 2 - [(X-Y) 2 - (X-Y) 2]
（1）m2+(-mn)-n2+(-m2)-(-0.5n2)
=M2-mn-n2-m2 + N2 (without brackets)
=(-) m2 Mn + (-) N2
=-M2-mn-n2 (in descending order of M)
（2）2(4an+2-an)-3an+(an+1-2an+1)-(8an+2+3an)
=8An + 2-2an-3an-an + 1-8an + 2-3an (without brackets)
=0 + (- 2-3-3) An-an + 1 (merge congeners)
=-an+1-8an
(3) (X-Y) 2 - (X-Y) 2 - [(X-Y) 2 - (X-Y) 2] [take (X-Y) 2 as a whole]
=(X-Y) 2 - (X-Y) 2 - (X-Y) 2 + (X-Y) 2 (remove brackets)
=(1 -- +) (X-Y) 2 ("merge congeners")
=(x-y)2
Example 4 find the value of 3x2-2 {X-5 [x-3 (x-2x2) - 3 (x2-2x)] - (x-1)}, where x = 2
Analysis: because the given formula is more complex, in general, we should simplify the integral first, and then substitute the given value x = - 2, pay attention to the sign when removing the brackets, and merge the similar terms in time to make the operation simple
The original formula = 3x2-2 {X-5 [x-3x + 6x2-3x2 + 6x] - x + 1} (without parentheses)
=3x2-2 {X-5 [3x2 + 4x] - x + 1} (merge similar items in time)
=3x2-2 {x-15x2-20x-x + 1} (without brackets)
=3x2-2 {- 15x2-20x + 1} (simplifying the formula in braces)
=3x2 + 30x2 + 40x-2 (without braces)
=33x2+40x-2
When x = - 2, the original formula is 33 × (- 2) 2 + 40 × (- 2) - 2 = 132-80-2 = 50
Example 5. If 16x3m-1y5 and - x5y2n + 1 are of the same kind, find the value of 3M + 2n
∵ 16x3m-1y5 and - x5y2n + 1 are similar terms
The times corresponding to X and y should be equal respectively
Ψ 3m-1 = 5 and 2n + 1 = 5
Ψ M = 2 and N = 2
∴3m+2n=6+4=10
This topic examines our understanding of the concept of similar terms
Example 6. Given x + y = 6, xy = - 4, find the value of (5x-4y-3xy) - (8x-y + 2XY)
(5x-4y-3xy)-(8x-y+2xy)
=5x-4y-3xy-8x+y-2xy
=-3x-3y-5xy
=-3(x+y)-5xy
∵x+y=6,xy=-4
The original formula = - 3 × 6-5 × (- 4) = - 18 + 20 = 2
Note: after the simplification of this problem, it is found that the result can be written in the form of - 3 (x + y) - 5xy, so the value of X + y, XY can be substituted into the original formula to get the final result, and there is no need to find out the value of X, Y. this thinking method is called overall substitution. I hope students can pay attention to it in the process of learning
3、 Practice
（1） Calculation:
（1）a-(a-3b+4c)+3(-c+2b)
（2）(3x2-2xy+7)-(-4x2+5xy+6)
（3）2x2-{-3x+6+[4x2-(2x2-3x+2)]}
（2） Simplification
（1）a>0,b