Algebraic Expressions

Exercise 12.2

Question 1: Simplify combining like terms:

(a) `21b – 32 + 7b – 20b`

Answer: `21b – 32 + 7b – 20b`
`= 21b + 7b – 20b – 32`
`= 28b – 20b – 32 = 8b – 32`

(b) `– z^2 + 13z^2 – 5z + 7z^3 – 15z`

Answer: `- z^2 + 13z^2 – 5z + 7z^3 – 15z`
`= - z^2 + 13z^2 – 5z – 15z + 7z^3`
`= 12z^2 – 20z + 7z^3`

(c) `p – (p – q) – q – (q – p)`

Answer: `p – (p – q) – q – (q – p)`
`= p – p + q – q – q + p`
`= p – p + p + q – q – q`
`= p – q`

(d) `3a – 2b – ab – (a – b + ab) + 3ab + b – a`

Answer: `3a – 2b – ab – a + b – ab + 3ab + b – a`
`= 3a – a – a – 2b + b + b – ab – ab + 3ab`
`= 3a – 2a – 2b + 2b – 2ab + 3ab`
`= a + ab`


(e) `5x^2y – 5x^2 + 3yx^2 – 3y^2 + x^2 – y^2 + 8xy^2 – 3y^2`

Answer: `5x^2y – 5x^2 + 3yx^2 – 3y^2 + x^2 – y^2 + 8xy^2 – 3y^2`
`= 5x^2y + 3x^2y – 5x^2 + x^2 – 3y^2 – y^2 + 8xy^2`
`= 8x^2y – 4x^2 – 4y^2 + 8xy^2`

(f) `(3y^2 + 5y – 4) – (8y – y^2 – 4)`

Answer: `(3y^2 + 5y – 4) – (8y – y^2 – 4)`
`= 3y^2 + 5y – 4 – 8y + y^2 + 4`
`= 3y^2 + y^2 + 5y – 8y – 4 + 4`
`= 4y^2 – 3y`

Question 2: Add the following:

(a) `3mn`, `- 5mn`, `8mn`, `- 4mn`

Answer: `3mn – 5mn + 8mn – 4mn`
`= 3mn + 8mn – 5mn – 4mn`
`= 11mn – 9mn = 8mn`

(b) `t – 8tz`, `3tz – z`, `z – t`

Answer: `t – 8tz + 3tz – z + z – t`
`= t – t – 8tz + 3tz – z + z`
`= - 5tz`

(c) `– 7mn + 5`, `12mn + 2`, `9mn – 8`, `- 2mn – 3`

Answer: `– 7mn + 5 + 12mn + 2 + 9mn – 8 - 2mn – 3`
`= - 7mn + 12mn + 9mn – 2mn + 5 + 2 – 8 – 3`
`= 12mn + 7 – 11`
`= 12mn – 4`

(d) `a + b` `– 3`, `b – a + 3`, `a – b + 3`

Answer: `a + b – 3 + b – a + 3 + a – b + 3`
`= a – a + a + b + b – b – 3 + 3 + 3`
`= a + b + 3`

(e) `14x + 10y – 12xy – 13`, `18 – 7x – 19y + 8xy`, `4xy`

Answer: `14x + 10y – 12xy – 13 + 18 – 7x – 19y + 8xy + 4xy`
`= 14x – 7x – 10y – 19y – 12xy + 7xy + 4xy – 13 + 18`
`= 7x – 29y – xy + 5`

(f) `5m – 7n`, `3n – 4m + 2`, `2m - 3mn – 5`

Answer: `5m – 7n + 3n – 4m + 2 + 2m - 3mn – 5`
`= 5m – 4m + 2m – 7n + 3n – 3mn + 2 – 5`
`= 3m – 4n – 3mn – 3`

(g) `4x^2y`, `- 3xy^2`, `- 5xy^2`, `5x^2y`

Answer: `4x^2y - 3xy^2 - 5xy^2 + 5x^2y`
`= 4x^2y + 5x^2y – 3xy^2 – 5xy^2`
`= 9x^2y – 8xy^2`

(h) `3p^2q^2 – 4pq + 5`, `- 10p^2q^2`, `15 + 9pq + 7p^2q^2`

Answer: `3p^2q^2 – 4pq + 5 - 10p^2q^2 + 15 + 9pq + 7p^2q^2`
`= 3p^2q^2 – 10p^2q^2 + 7p^2q^2 – 4pq + 9pq + 5 + 15`
`= 10p^2q^2 – 10p^2q^2 + 5pq + 20`
`= 5pq + 20`

(i) `ab – 4a`, `4b – ab`, `4a – 4b`

Answer: `ab – 4a + 4b – ab + 4a – 4b`
`= ab – ab – 4a + 4a + 4b – 4b = 0`

(j) `x^2 – y^2 – 1`, `y^2 – 1 – x^2`, 1 – x^2 – y^2`

Answer: `x^2 – y^2 – 1 + y^2 – 1 – x^2 + 1 – x^2 – y^2`
`= x^2 – x^2 – x^2 – y^2 + y^2 – y^2 – 1 – 1 + 1`
`= - x^2 – y^2 – 1`


Question 3: Subtract the following:

(a) `0.5y^2` from `y^2`

Answer: `y^2 – 0.5y^2 = 0.5y^2`

(b) `6xy` from `– 12xy`

Answer: `- 12xy – 6xy = - 18xy`

(c) `(a – b)` from `(a + b)`

Answer: `a + b – (a – b) = a + b – a + b = 2b`

(d) `a(b – 5)` from `b(5 – a)`

Answer: `b(5 – a) – a(b – 5)`
`= 5b – ab – ab + 5a = 5a + 5b – 2ab`

(e) `– m^2 + 5mn` from `4m^2 – 3mn + 8`

Answer: `4m^2 – 3mn + 8 – ( - m^2 + 5mn)`
`= 4m^2 – 3mn + 8 + m^2 – 5mn`
`= 4m^2 + m^2 – 3mn – 5mn + 8`
`= 5m^2 – 8mn + 8`

(f) `– x^2 + 10x – 5` from `5x – 10`

Answer: `5x – 10 – ( - x^2 + 10x – 5)`
`= 5x – 10 + x^2 – 10x + 5`
`= 5x – 10x + x^2 – 10 + 5`
`= - 5x + x^2 – 5`

(g) `5a^2 – 7ab + 5b^2` from `3ab – 2a^2 – 2b^2`

Answer: `3ab – 2a^2 – 2b^2 – (5a^2 – 7ab + 5b^2)`
`= 3ab – 2a^2 – 2b^2 – 5a^2 + 7ab – 5b^2`
`= 3ab + 7ab – 2a^2 – 5a^2 – 2b^2 – 5b^2`
`= 10ab – 7a^2 – 7b^2`

(h) `4pq – 5q^2 – 3p^2` from `5p^2 + 3q^2 – pq`

Answer: `5p^2 + 3q^2 – pq – (4pq – 5q^2 – 3p^2)`
`= 5p^2 + 3q^2 – pq – 4pq + 5q^2 + 3p^2`
`= 5p^2 + 3p^2 + 3q^2 + 5q^2 – pq – 4pq`
`= 8p^2 + 8q^2 – 5pq`

Question 4: What should be added to `x^2 + xy + y^2` to obtain `2x^2 + 3xy?`

Answer: For this, you need to subtract the first expression from the second expression:

`= 2x^2 + 3xy – (x^2 + xy + y^2)`
`= 2x^2 + 3xy – x^2 – xy – y^2`
`= 2x^2 – x^2 + 3xy – xy – y^2`
`= x^2 + 2xy – y^2`


Question 5: What should be subtracted from `2a + 8b + 10` to get `– 3a + 7b + 16?`

Answer: For this, you need to subtract the second expression from first expression:

`= 2a + 8b + 10 – ( - 3a + 7b + 16)`
`= 2a + 8b + 10 + 3a – 7b – 16`
`= 2a + 3a + 8b – 7b + 10 – 16`
`= 5a + b – 6`

Question 6: What should be taken away from `3x^2 – 4y^2 + 5xy + 20` to obtain `– x^2 – y^2 + 6xy + 20?`

Answer: For this, you need to subtract the second term from first term:

`= 3x^2 – 4y^2 + 5xy + 20 – (– x^2 – y^2 + 6xy + 20)`
`= 3x^2 – 4y^2 + 5xy + 20 + x^2 + y^2 – 6xy – 20`
`= 3x^2 + x^2 – 4y^2 + y^2 + 5xy – 6xy + 20 – 20`
`= 4x^2 – 3y^2 – xy`

Question 7: From the sum of `3x – y + 11` and `– y – 11`, subtract `3x – y – 11`.

Answer: `3x – y + 11 + (– y – 11) – (3x – y – 11)`
`= 3x – y + 11 – y – 11 – 3x + y + 11`
`= 3x – 3x – y – y + y + 11 – 11 + 11`
`= - y + 11`

Question 8: From the sum of `4 + 3x` and `5 – 4x + 2x^2`, subtract the sum of `3x^2 – 5x` and `– x^2 + 2x + 5`.

Answer: Sum of first pair of expressions:

`4 + 3x + 5 – 4x + 2x^2`
`= 4 + 5 + 3x – 4x + 2x^2 ``= 9 – x + 2x^2`

Sum of second pair of expressions:
`3x^2 – 5x – x^2 + 2x + 5`
`= 3x^2 – x^2 – 5x + 2x + 5`
`= 2x^2 – 3x + 5`

Subtraction: `9 – x + 2x^2 – (2x^2 – 3x + 5)`
`= 9 – x + 2x^2 – 2x^2 + 3x – 5`
`= 2x^2 – 2x^2 – x + 3x + 9 – 5`
`= 2x + 4`



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