Page 162 - classs 6 a_Neat

EXAMPIE:7 lf x = 1, y = -2 and 2 = 3.find the value of SOLUTION Column method: We write the expressions so that the like terms are in a column as
(i) x + y + z _ 3xyz (ii) 3xy - 15x y + 4z shown below and add columnwise.
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SOLUTION (i) Substituting x = 1, y = - 2 and z = 3 in the given expression, we get: 5x +7y-6z 2
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x + y + z - 3xyz = (1) + (-2) + (3) -3 × 1 × (-2) × 3 +3x +4y
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= 1 - 8 + 27 + 18 = 38. + 9x - 9y + 2z 2
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(ii) Substituting x = 1, y = - 2 and z = 3 in the given expression, we get: - 2x + 2y
3xy -15x y + 4z = 3 × 1 × (-2) - 15 × (1) × (-2) + 4 × 3 15x + 4y - 4z 2
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=3 × 1 × 16 + 30 + 12 = 48 + 30 + 12 = 90. Alternative method The sum of the given expressions
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EXAMPIE:8 Identify monomials, binomials and trinomialsjrom the following expressions: = (5x + 7y - 6z ) + (4y + 3x ) + (9x + 2z - 9y) + (2y - 2x )
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(i)-3xyz (ii) 4x yz + 9 - 5x 3 (iii)-7 = (5x + 3x + 9x - 2x ) + (7y + 4y - 9y + 2y) + (-6z + 2z )
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(iv) x + y +z -p 2 (v) x + 5 (vi) 6a b = (5 + 3 + 9 - 2)x + (7 + 4 - 9 + 2)y + (-6 + 2)z 2 = 15x + 4y - 4z .
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SOLUTION Clearly, each of the expressions given in (i), (iii) and (vi) contains only one term. So,
each one of them is a monomial. SUBTRACTION OF ALGEBRAIC EXPRESSIONS
The expression given in (v) contains two terms, and therefore, it is a binomial. RULE Change the sign of each term of the expression to be subtracted and add it to the expression from which
The expression given in (ii) conlains three terms, and therefore, it is a trinomial. subtraction ls to be made.
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The expression given in (iv) contains four terms, so it is none of the given type. EXAMPIE:13 Subtract 6xy= 4x - y - 2from x - 3xy + 7y +5.
EXAMPIE:9 Write down the coefficient of SOLUTION Arranging the like terms columnWise, changing the sign of each term of the
(t) x in 9xy (ii) a in -7abc (iii) xyz in -xyz (iv) bin -abc expression to be subtracted and then adding, we get:
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SOLUTION (i)The coefficient of x in 9xy is 9y. x -3xy + 7y + 5
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(ii)The coefficient of a in -7abc is -7bc. - 4x + 6xy - y -2 (change the sign of each term and add)
(iii)The coefficient of xyz in -xyz is -1. + - + -
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(iv)The coefficient of b in -abc is -ac. 5x - 9.xy + 8y +7
EXAMPIE:14 From the sum of6x 4-3x + 7x - 5x + 1 and-3x + 5x -9x + 7x-2 subtract
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OPERATIONS ON ALGEBRAIC EXPRESSIONS 2x - 5x + 2x - 6x-8.
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SOLUTION We have:
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ADDITION OF ALGEBRAIC EXPRESSIONS 6x - 3x + 7x - 5x +1
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The basic principle of addition of algebraic expressions is that only like terms can be added. Plus -3x + 5 - 9x + 7x - 2
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We know that 5 tables + 7 tables = 12 tables. 3x + 2x - 2x + 2x -1
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But, '5 tables + 7 chairs' does not form 12 tables or 12 chairs. Minus 2x - 5x + 2x - 6x -8
Thus, the sum of two unlike terms can only be indicated. - + - + +
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RULE OF ADDITION The sum of several like terms ts another like term whose coefficient ts the x + 7 x - 4x + 8x+7
sum of the coefficients of the like terms. USE OF GROUPING SYMBOLS
EXAMPIE:10 Add 6xy , -4xy2, xy , 5xy . When we make operations on two or more algebraic expressions. we separate them by the symbols of group-
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SOLUTION The required sum = 6xy + ( -4 )xy + xy + 5xy 2 ings, namely. parentheses ( ). braces { } and brackets [ ].
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= (6 -4 + 1 + 5 )xy = Bxy . In simplifying such expressions, we first remove the grouping symbols, using the laws given below:
EXAMPIE:11 Collect the 2 like terms and simplify: (i) If a '+ · sign precedes a symbol of grouping, the grouping symbol may be removed without any
5x -2x + 7 -9 + 7 x -3x + 4x - x + 1 change in the sign of the terms.
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SOLUTION Rearranging and collecting the like terms, we get: (ii) If a ·-· sign precedes a symbol of grouping. the grouping symbol may be removed and the sign of
5x -2x + 7 -9 + 7 x -3x + 4x - x + 1 each term is changed.
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= 5x -3x + 4x -2x + 7 x - x + 7 -9 + 1 (iii) If more than one grouping symbol is present in an expression, we remove the innermost
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= (5 -3 + 4)x + (-2 + 7 -l) x + (7 -9 + 1) grouping symbol first and combine the like terms, if any. We continue the process outwards until all
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= 6x + 4x -1. the grouping symbols have been removed.
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COLUMN METHOD In this method, each expression ts written in a separate row such that their like terms are EXAMPIE:15 stmplify: (a -8ab - 5) + (3ab - 4a + 8)
arranged one below the other in a column. Then, addition or subtraction of the terms ts done columnwise. SOLUTION Clearly, a ·+· sign precedes the second parenthesis, so we remove it without changing
the signs of the terms within it.
EXAMPIE:12 Add the following expressions: (a 2 - 8ab - 5) + (3ab - 4a 2 + 8)
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5x + 7y - 6z , 4y + 3x , 9x + 2z - 9y and 2y - 2x 2 = a - 8ab - 5 + 3ab - 4a + 8
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= (1 - 4)a + (-8 + 3)ab + (-5 + 8)
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=3a - 5ab + 3.
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