Algebraic Expressions
Exercise 12.2
1). Simplify combining like terms:
(i) 21b – 32 + 7b – 20b
Solution:
21b – 32 + 7b – 20b = 21b + 7b – 20b – 32
= ( 21 + 7 – 20 ) b – 32
= 8b – 32
Ans: 21b – 32 + 7b – 20b = 8b – 32
(ii) – z2 + 13z2 – 5z + 7z3 – 15z
Solution:
– z2 + 13z2 – 5z + 7z3 – 15z = 7z3 – z2 + 13z2 – 5z – 15z
= 7z3 +12z2 – 20z
Ans: – z2 + 13z2 – 5z + 7z3 – 15z =7z3 +12z2 – 20z
(iii) p – (p – q) – q – (q – p)
Solution:
p – (p – q) – q – (q – p) = p – p + q – q – q + p
= p – p + p + q – q – q
= p – q
Ans: p – (p – q) – q – (q – p) = = p – q
(iv) 3a – 2b – ab – (a – b + ab) + 3ab + b – a
Solution:
3a – 2b – ab – (a – b + ab) + 3ab + b – a
= 3a – 2b – ab – a + b – ab + 3ab + b – a
= 3a – a – a – ab – ab + 3ab – 2b + b + b
= 2a – a –2ab + 3ab – b + b
= a + ab
Ans: 3a – 2b – ab – (a – b + ab) + 3ab + b – a = a + ab
(v) 5x2y – 5x2 + 3yx2 – 3y2 + x2 – y2 + 8xy2 – 3y2
Solution:
= 5x2y – 5x2 + 3yx2 – 3y2 + x2 – y2 + 8xy2 – 3y2
= 5x2y + 3yx2 – 5x2 + x2 + 8xy2 – 3y2 – y2– 3y2
= 5x2y + 3yx2 – 4x2 + 8xy2 – 7y2
= 8x2y – 4x2 + 8xy2 – 7y2
Ans:5x2y –5x2 +3yx2 –3y2 +x2 –y2 +8xy2 –3y2 = 8x2y – 4x2 +8xy2 –7y2
(vi) (3y2 + 5y – 4) – (8y – y2 – 4)
Solution:
(3y2 + 5y – 4) – (8y – y2 – 4)
= 3y2 + 5y – 4 – 8y + y2 + 4
= 3y2 + y2 + 5y – 8y + 4 – 4
= 4y2 – 3y
Ans: (3y2 + 5y – 4) – (8y – y2 – 4) = 4y2 – 3y
2). Add:
(i) 3mn, – 5mn, 8mn, – 4mn
Solution:
(3mn) + (– 5mn) + (8mn) + (– 4mn)
= 3mn – 5mn + 8mn – 4mn
= ( 3 – 5 + 8 – 4)mn
= 2mn
Ans: (3mn) + (– 5mn) + (8mn) + (– 4mn) = 2mn
(ii) t – 8tz, 3tz – z, z – t
Solution:
(t – 8tz) + (3tz – z) + (z – t)
= t – 8tz + 3tz – z + z – t
= t – t – 8tz + 3tz – z + z
= – 5tz
Ans: (t – 8tz ) + (3tz – z) + (z – t) = – 5tz
Solution:
(iii) – 7mn + 5, 12mn + 2, 9mn – 8, – 2mn – 3
Solution:
(– 7mn + 5 ) + (12mn + 2) + (9mn – 8) + (– 2mn – 3)
= – 7mn + 5 + 12mn + 2 + 9mn – 8 – 2mn – 3
= – 7mn + 12mn + 9mn – 2mn + 5 + 2 – 8 – 3
= (– 7 + 12 + 9 – 2)mn + 5 + 2 – 8 – 3
= 12mn – 4
Ans: (– 7mn + 5) + (12mn + 2) + (9mn – 8) + (– 2mn – 3) = 12mn – 4
(iv) a + b – 3, b – a + 3, a – b + 3
Solution:
(a + b – 3) + (b – a + 3) + (a – b + 3)
= a + b – 3 + b – a + 3 + a – b + 3
= a – a + a + b + b – b – 3 + 3 + 3
= a + b + 3
Ans: (a + b – 3) + (b – a + 3) + (a – b + 3) = a + b + 3
(v) 14x + 10y – 12xy – 13, 18 – 7x – 10y + 8xy, 4xy
Solution:
(14x + 10y – 12xy – 13) + (18 – 7x – 10y + 8xy) + (4xy)
= 14x + 10y – 12xy – 13 + 18 – 7x – 10y + 8xy + 4xy
= 14x – 7x – 12xy + 8xy + 4xy + 10y – 10y – 13 + 18
= 7x + 0xy +0y + 5
= 7x + 5
Ans: (14x+10y –12xy –13)+(18–7x–10y+8xy)+(4xy) = 7x + 5
(vi) 5m – 7n, 3n – 4m + 2, 2m – 3mn – 5
Solution:
(5m – 7n) + (3n – 4m + 2) + (2m – 3mn – 5)
= 5m – 7n + 3n – 4m + 2 + 2m – 3mn – 5
= 5m – 4m + 2m – 7n + 3n – 3mn + 2– 5
= 3m – 4n – 3mn – 3
Ans: (5m –7n)+(3n–4m+2)+(2m–3mn–5) = 3m – 4n – 3mn – 3
(vii) 4x2y, – 3xy2, –5xy2, 5x2y
Solution:
(4x2y) + (– 3xy2) + (–5xy2) + (5x2y)
= 4x2y – 3xy2 –5xy2 + 5x2y
= 4x2y+ 5x2y – 3xy2 –5xy2
= 9x2y – 8xy2
Ans: (4x2y) + (– 3xy2) + (–5xy2) + (5x2y) = 9x2y – 8xy2
(viii) 3p2q2 – 4pq + 5, – 10 p2q2, 15 + 9pq + 7p2q2
Solution:
(3p2q2 – 4pq + 5) + (– 10 p2q2) + (15 + 9pq + 7p2q2)
= 3p2q2 – 4pq + 5 – 10p2q2 +15 + 9pq + 7p2q2
= 3p2q2 – 10p2q2 + 7p2q2 – 4pq + 9pq + 5 +15
= 0p2q2 + 5pq +20
= 5pq +20
Ans: (3p2q2–pq+5)+(–10p2q2)+(15+9pq+7p2q2) = 5pq + 20
(ix) ab – 4a, 4b – ab, 4a – 4b
Solution:
(ab – 4a)+ (4b – ab) + (4a – 4b)
= ab – 4a + 4b – ab + 4a – 4b
= ab – ab – 4a + 4a + 4b – 4b
= 0ab +0a + 0b
= 0
Ans: (ab – 4a)+ (4b – ab) + (4a – 4b) = 0
(x) x2 – y2 – 1, y2 – 1 – x2, 1 – x2 – y2
Solution:
(x2 – y2 – 1) + (y2 – 1 – x2)+ (1 – x2 – y2)
= x2 – y2 – 1 + y2 – 1 – x2 + 1 – x2 – y2
= x2 – x2 – x2 – y2 + y2 – y2 – 1 – 1 + 1
= – x2 – y2 – 1
Ans: (x2 – y2 – 1) + (y2 – 1 – x2)+ (1 – x2 – y2) = – x2 – y2 – 1
3). Subtract:
(i) –5y2 from y2
Solution:
y2 – (– 5y2 )
= y2 + 5y2
= 6y2
Ans: y2 – (– 5y2 ) = 6y2
(ii) 6xy from –12xy
Solution:
–12xy – (6xy )
= –12xy – 6xy
= –18xy
Ans: –12xy – (6xy ) = –18xy
(iii) (a – b) from (a + b)
Solution:
(a + b) – (a – b)
= a + b – a + b
= a – a + b + b
= 2b
Ans: (a + b) – (a – b) = 2b
(iv) a (b – 5) from b (5 – a)
Solution:
b (5 – a) – a (b – 5)
= 5b – ab – ab + 5a
= 5b – 2ab + 5a
Ans: b (5 – a) – a (b – 5) = 5b – 2ab + 5a
(v) –m2 + 5mn from 4m2 – 3mn + 8
Solution:
4m2 – 3mn + 8 – (–m2 + 5mn )
= 4m2 – 3mn + 8 + m2 – 5mn
= 4m2 + m2 – 3mn – 5mn + 8
= 5m2 – 8mn + 8
Ans: 4m2 – 3mn + 8 – (–m2 + 5mn ) = 5m2 – 8mn + 8
(vi) – x2 + 10x – 5 from 5x – 10
Solution:
5x – 10 – (– x2 + 10x – 5 )
= 5x – 10 + x2 – 10x + 5
= x2 + 5x – 10x – 10 + 5
= x2 – 5x – 5
Ans: 5x – 10 – (– x2 + 10x – 5 ) = x2 – 5x – 5
(vii) 5a2 – 7ab + 5b2 from 3ab – 2a2 – 2b2
Solution:
3ab – 2a2 – 2b2 – (5a2 – 7ab + 5b2 )
= 3ab – 2a2 – 2b2 –5a2 + 7ab – 5b2
= 3ab + 7ab – 2a2 –5a2 – 2b2 – 5b2
= 10ab – 7a2 – 7b2
Ans: 3ab – 2a2 – 2b2 – (5a2 – 7ab + 5b2 ) = 10ab – 7a2 – 7b2
(viii) 4pq – 5q2 – 3p2 from 5p2 + 3q2 – pq
Solution:
5p2 + 3q2 – pq – (4pq – 5q2 – 3p2 )
= 5p2 + 3q2 – pq – 4pq + 5q2 + 3p2
= 5p2 + 3p2 – pq – 4pq + 3q2+ 5q2
= 8p2 – 5pq + 8q2
Ans: 5p2 + 3q2 – pq – (4pq – 5q2 – 3p2 ) = 8p2 – 5pq + 8q2
4). (a) What should be added to x2 + xy + y2 to obtain 2x2 + 3xy?
Let the expression to be added be A
x2 + xy + y2 + A = 2x2 + 3xy
→ A = 2x2 + 3xy – ( x2 + xy + y2)
= 2x2 + 3xy – x2 – xy – y2
= 2x2 – x2 + 3xy – xy – y2
= x2 + 2xy – y2
Ans: the required expression is x2 + 2xy – y2
(b) What should be subtracted from 2a + 8b + 10 to get – 3a + 7b + 16?
Let the expression to be subtracted be A
2a + 8b + 10 – A = – 3a + 7b + 16
→ 2a + 8b + 10 – ( – 3a + 7b + 16 ) = A
→ A = 2a + 8b + 10 – ( – 3a + 7b + 16 )
= 2a + 8b + 10 + 3a – 7b – 16
= 2a + 3a + 8b – 7b + 10 – 16
= 5a + b – 6
Ans: the required expression is 5a + b – 6
5). What should be taken away from 3x2 – 4y2 + 5xy + 20 to obtain
– x2 – y2 + 6xy + 20?
Let the expression to be taken away be A
3x2 – 4y2 + 5xy + 20 – A = – x2 – y2 + 6xy + 20
→ 3x2 – 4y2 + 5xy + 20 – (– x2 – y2 + 6xy + 20 ) = A
→ A = 3x2 – 4y2 + 5xy + 20 – ( – x2 – y2 + 6xy + 20 )
= 3x2 – 4y2 + 5xy + 20 + x2 + y2 – 6xy – 20
= 3x2 + x2 + 5xy – 6xy – 4y2 + y2 + 20 – 20
= 4x2 – xy – 3y2
Ans: the required expression is 4x2 – xy – 3y2
6). (a) From the sum of 3x – y + 11 and – y – 11, subtract 3x – y – 11.
[ (3x – y + 11) + ( – y – 11)] – ( 3x – y – 11)
= [3x – y + 11 – y – 11] – ( 3x – y – 11)
= (3x – y – y + 11 – 11) – ( 3x – y – 11)
= (3x – 2y ) – ( 3x – y – 11)
= 3x – 2y – 3x + y + 11
= 3x – 3x – 2y + y + 11
= 0x – y + 11
= – y + 11
Ans: the required expression is – y + 11
(b) From the sum of 4 + 3x and 5 – 4x + 2x2, subtract the sum of 3x2 – 5x and –x2 + 2x + 5.
[ (4 + 3x) + ( 5 – 4x + 2x2)] – [(3x2 – 5x) + (–x2 + 2x + 5)]
= [ 4 + 3x + 5 – 4x + 2x2] – [3x2 – 5x –x2 + 2x + 5]
= (4 + 3x + 5 – 4x + 2x2) – (3x2 – 5x –x2 + 2x + 5)
= ( 2x2 + 3x – 4x +4 + 5) – (3x2 –x2 – 5x + 2x + 5)
= (2x2 – x + 9) – (2x2 – 3x + 5)
= 2x2 – x + 9 – 2x2 + 3x – 5
= 2x2 – 2x2 – x + 3x + 9 – 5
= 0x2 +2x + 4
= 2x + 4
Ans: the required expression is 2x + 4
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