Question 10 (the simpler the month, the better) merge the similar items in seventh grade mathematics.

Example 1, merging similar projects.

( 1)(3x-5y)-(6x+7y)+(9x-2y)

(2)2a-[3b-5a-(3a-5b)]

(3)(6m2n-5mn2)-6(m2n-mn2)

Solution: (1) (3x-5y)-(6x+7y)+(9x-2y)

=3x-5y-6x-7y+9x-2y (remove brackets correctly)

=(3-6+9)x+(-5-7-2)y (merging similar items)

=6x- 14y

(2)2a-[3b-5a-(3a-5b)] (brackets should be removed step by step in the order of brackets, brackets and braces)

=2a-[3b-5a-3a+5b] (remove the brackets first)

=2a-[-8a+8b] (merging similar items by time)

=2a+8a-8b (remove brackets)

= 10a-8b

(3)(6m2n-5mn2)-6(m2n-mn2) (note that there is a factor 6 before the second bracket)

=6m2n-5mn2-2m2n+3mn2 (bracket removal and distribution rules are carried out at the same time)

=(6-2)m2n+(-5+3)mn2 (merger of similar projects)

=4m2n-2mn2

Example 2. It is known that A=3x2-4xy+2y2 and B=x2+2xy-5y2.

Find: (1)A+B (2)A-B (3) If 2A-B+C=0, find C.

Solution: (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 (merging similar items)

=4x2-2xy-3y2 (in descending order of X)

(2)A-B =(3 x2-4xy+2 y2)-(x2+2xy-5 y2)

=3x2-4xy+2y2-x2-2xy+5y2 (without brackets)

= (3-1) x2+(-4-2) xy+(2+5) y2 (merging similar items)

=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 (without brackets, pay attention to the distribution law)

= (-6+1) x2+(8+2) xy+(-4-5) y2 (merging similar items)

=-5x2+ 10xy-9y2 (in descending order of X)

Example 3. Calculation:

( 1)m2+(-Mn)-N2+(-m2)-(-0.5 N2)

(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]

Solution: (1) m2+(-Mn)-N2+(-m2)-(-0.5n2)

=m2-mn-n2-m2+n2 (remove brackets)

=(-)m2-Mn+(-)N2 (merger of similar projects)

=-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 (brackets removed)

=0+(-2-3-3)an-an+ 1 (merging similar items)

=-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 (without brackets)

= (1-+) (x-y) 2 ("merging similar projects")

=(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 known to be complicated, in general, the algebraic expression should be simplified first, and then substituted into the given value x=-2. Pay attention to symbols when removing brackets, and merge similar terms in time to make the operation simple.

Solution: The original formula = 3x2-2 {x-5 [x-3x+6x2-3x2+6x]-x+1} (brackets removed).

=3x2-2{x-5[3x2+4x]-x+ 1} (merging similar items by time)

= 3x2-2 {x-15x2-20x-x+1} (brackets removed)

= 3x2-2 {-15x2-20x+1} (simplified formula in braces)

=3x2+30x2+40x-2 (braces removed)

=33x2+40x-2

When x=-2, the original formula = 33× (-2) 2+40× (-2)-2 =132-80-2 = 50.

Example 5. If 16x3m- 1y5 and-x5x2n+1are similar terms, find the value of 3m+2n.

Solution: ∫ 16x3m- 1y 5 and -x5y2n+ 1 are similar terms.

The time corresponding to x and y should be equal respectively.

∴3m- 1=5 and 2n+ 1=5.

∴m=2 and n=2

∴3m+2n=6+4= 10

This topic examines our understanding of the concept of similar items.

Example 6. Given x+y=6 and xy=-4, find the value of (5x-4y-3xy)-(8x-y+2xy).

Solution: (5x-4y-3xy)-(8x-y+2xy)

=5x-4y-3xy-8x+y-2xy

=-3x-3y-5xy

=-3(x+y)-5xy

∫x+y = 6，xy=-4

∴ Original formula =-3×6-5×(-4)=- 18+20=2

Note: after simplifying this question, it is found that the result can be written in the form of -3(x+y)-5xy, so you can get the final result by substituting the values of x+y and xy into the original formula, and you don't need to ask for the values of x and y. This idea is called whole substitution, so I hope students will pay attention to it during their study.

Third, practice.

(1) calculation:

( 1)a-(a-3b+4c)+3(-c+2b)

(2)(3x2-2xy+7)-(-4x2+5xy+6)

(3)2 x2-{-3x+6+[4x 2-(2 x2-3x+2)]]

(2) simplification

( 1)a & gt； 0，b & lt0，|6-5b|-|3a-2b|-|6b- 1|

(2) 1 & lt； a & lt3、| 1-a|+|3-a|+|a-5|

(3) When a= 1, b=-3 and c= 1, find the value of the algebraic formula a2b-[a2b-(5abc-a2c)]-5abc.

(4) When the algebraic expression -(3x+6)2+2 obtains the maximum value, find the value of the algebraic expression 5x-[-x2-(x+2)].

(5) x2-3xy=-5, xy+y2=3, and find the value of x2-2xy+y2.

Practice reference answer: