**Question 1. Add the following algebraic expressions:**

**(i) 3a**^{2}b, -4a^{2}b, 9a^{2}b

^{2}b, -4a

^{2}b, 9a

^{2}b

**Solution:**

3a

^{2}b, -4a^{2}b, 9a^{2}b

Now we have add the given expression

= 3a^{2}b + (-4a^{2}b) + 9a^{2}b

= 3a^{2}b – 4a^{2}b + 9a^{2}b

= 8a^{2}b

**(ii) 2/3a, 3/5a, -6/5a**

**Solution:**

We have to add the given expression

2/3a + 3/5a + (-6/5a)

2/3a + 3/5a – 6/5a

Now take LCM for 3 and 5 which will be 15

= (2×5)/(3×5)a + (3×3)/(5×3)a – (6×3)/(5×3)a

= 10/15a + 9/15a – 18/15a

= (10a+9a-18a)/15

= a/15

**(iii) 4xy**^{2} – 7x^{2}y, 12x^{2}y -6xy^{2}, -3x^{2}y + 5xy^{2}

^{2}– 7x

^{2}y, 12x

^{2}y -6xy

^{2}, -3x

^{2}y + 5xy

^{2}

**Solution:**

We have to add the given expression

4xy^{2}– 7x^{2}y + 12x^{2}y – 6xy^{2}– 3x^{2}y + 5xy^{2}

Now rearrange the expression:

12x^{2}y – 3x^{2}y – 7x^{2}y – 6xy^{2}+ 5xy^{2}+ 4xy

3xy^{2}+ 2x^{2}y

**(iv) 3/2a – 5/4b + 2/5c, 2/3a – 7/2b + 7/2c, 5/3a + 5/2b – 5/4c**

**Solution:**

3/2a – 5/4b + 2/5c, 2/3a – 7/2b + 7/2c, 5/3a + 5/2b – 5/4c

Now add the given expression

3/2a – 5/4b + 2/5c + 2/3a – 7/2b + 7/2c + 5/3a + 5/2b – 5/4c

rearrange

3/2a + 2/3a + 5/3a – 5/4b – 7/2b + 5/2b + 2/5c + 7/2c – 5/4c

Now take LCM of (2 and 3 is 6), (4 and 2 is 4), (5,2 and 4 is 20)

(9a+4a+10a)/6 + (-5b-14b+10b)/4 + (8c+70c-25c)/20

23a/6 – 9b/4 + 53c/20

**(v) 11/2xy + 12/5y + 13/7x, -11/2y – 12/5x – 137xy**

**Solution:**

11/2xy + 12/5y + 13/7x, -11/2y – 12/5x – 13/7xy

Now add the given expression

11/2xy + 12/5y + 13/7x + -11/2y – 12/5x – 13/7xy

Now rearrange11/2xy – 13/7xy + 13/7x – 12/5x + 12/5y -11/2y

Now take LCM for (2 and 7 is 14), (7 and 5 is 35), (5 and 2 is 10)

(11xy-12xy)/14 + (65x-84x)/35 + (24y-55y)/10

51xy/14 – 19x/35 – 31y/10

**(vi) 7/2x**^{3} – 1/2x^{3} + 5/3, 3/2x^{3} + 7/4x^{2} – x + 1/3, 3/2x^{2} -5/2x -2

^{3}– 1/2x

^{3}+ 5/3, 3/2x

^{3}+ 7/4x

^{2}– x + 1/3, 3/2x

^{2}-5/2x -2

**Solution:**

Now add the given expression

7/2x^{3}– 1/2x^{2}+ 5/3 + 3/2x^{3}+ 7/4x^{2}– x + 1/3 + 3/2x^{2}-5/2x – 2

Now rearrange

=7/2x^{3}+ 3/2x^{3}– 1/2x^{2}+ 7/4x^{2}+ 3/2x^{2}– x – 5/2x + 5/3 + 1/3 – 2

=10/2x^{3}+ 11/4x^{2}– 7/2x + 0/6

=5x^{3}+ 11/4x^{2}-7/2x

**Question 2. Subtract:**

**(i) -5xy from 12xy**

**Solution:**

Subtract the given expression

= 12xy – (- 5xy)

= 5xy + 12xy

= 17xy

**(ii) 2a**^{2} from -7a^{2}

^{2}from -7a

^{2}

**Solution:**

Subtract the given expression

= (-7a

^{2}) – 2a^{2}

= -7a^{2}– 2a^{2}

= -9a^{2}

**(iii) 2a-b from 3a-5b**

**Solution:**

Subtract the given expression

=(3a – 5b) – (2a – b)

= 3a – 5b – 2a + b

= a – 4b

### (**iv) 2x**^{3} – 4x^{2} + 3x + 5 from 4x^{3} + x^{2} + x + 6

^{3}– 4x

^{2}+ 3x + 5 from 4x

^{3}+ x

^{2}+ x + 6

**Solution:**

Subtract the given expression

(4x

^{3}+ x^{2}+ x + 6) – (2x^{3}– 4x^{2}+ 3x + 5)

4x^{3}+ x^{2}+ x + 6 – 2x^{3}+ 4x^{2}– 3x – 52x

^{3}+ 5x^{2}– 2x + 1

**(v) 2/3y**^{3} – 2/7y^{2} – 5 from 1/3y^{3} + 5/7y^{2} + y – 2

^{3}– 2/7y

^{2}– 5 from 1/3y

^{3}+ 5/7y

^{2}+ y – 2

**Solution:**

Subtract the given expression

1/3y^{3}+ 5/7y^{2}+ y – 2 – 2/3y^{3}+ 2/7y^{2}+ 5

On rearranging,

1/3y^{3}– 2/3y^{3}+ 5/7y^{2}+ 2/7y^{2}+ y – 2 + 5

We will group similar expression:

= -1/3y^{3}+ 7/7y^{2}+ y + 3

= -1/3y^{3}+ y^{2}+ y + 3

**(vi) 3/2x – 5/4y – 7/2z from 2/3x + 3/2y – 4/3z**

**Solution:**

Subtract the given expression

2/3x + 3/2y – 4/3z – (3/2x – 5/4y – 7/2z)

On rearranging,

2/3x – 3/2x + 3/2y + 5/4y – 4/3z + 7/2z

We will group similar expression:

LCM of (3 and 2 is 6), (2 and 4 is 4), (3 and 2 is 6)

=(4x-9x)/6 + (6y+5y)/4 + (-8z+21z)/6

= -5x/6 + 11y/4 + 13z/6

**(vii) x**^{2}y – 4/5xy^{2} + 4/3xy from 2/3x2y + 3/2xy^{2} – 1/3xy

^{2}y – 4/5xy

^{2}+ 4/3xy from 2/3x2y + 3/2xy

^{2}– 1/3xy

**Solution:**

Subtract the given expression

2/3x^{2}y + 3/2xy^{2}– 1/3xy – (x^{2}y – 4/5xy^{2}+ 4/3xy)

on rearrange

2/3x^{2}y – x^{2}y + 3/2xy^{2}+ 4/5xy^{2}– 1/3xy – 4/3xy

We will group similar expression:

LCM of (3 and 1 is 3), (2 and 5 is 10), (3 and 3 is 3)

-1/3x^{2}y + 23/10xy^{2}– 5/3xy

**(viii) ab/7 – 35/3bc + 6/5ac from 3/5bc – 4/5ac**

**Solution:**

Subtract the given expression

3/5bc – 4/5ac – (ab/7 – 35/3bc + 6/5ac)

On rearrange

3/5bc + 35/3bc – 4/5ac – 6/5ac – ab/7

We will group similar expression:

LCM of (5 and 3 is 15), (5 and 5 is 5)

(9bc+175bc)/15 + (-4ac-6ac)/5 – ab/7

184bc/15 + -10ac/5 – ab/7

– ab/7 + 184bc/15 – 2ac

**Question 3. Take away:**

**(i) 6/5x**^{2} – 4/5x^{3} + 5/6 + 3/2x from x^{3}/3 – 5/2x^{2} + 3/5x + 1/4

^{2}– 4/5x

^{3}+ 5/6 + 3/2x from x

^{3}/3 – 5/2x

^{2}+ 3/5x + 1/4

**Solution:**

Subtract the given expression

1/3x^{3}– 5/2x^{2}+ 3/5x + 1/4 – (6/5x^{2}– 4/5x^{3}+ 5/6 + 3/2x)

On rearrange

1/3x^{3}+ 4/5x^{3}– 5/2x^{2}– 6/5x^{2}+ 3/5x – 3/2x + 1/4 – 5/6

By grouping similar expressions we get,

LCM of (3 and 5 is 15), (2 and 5 is 10), (5 and 2 is 10), (4 and 6 is 24)

17/15x^{3}– 37/10x^{2}– 9/10x – 14/24

17/15x^{3}– 37/10x^{2}– 9/10x – 7/12

**(ii) 5a**^{2}/2 + 3a^{3}/2 + a/3 – 6/5 from 1/3a^{3} – 3/4a^{2} – 5/2

^{2}/2 + 3a

^{3}/2 + a/3 – 6/5 from 1/3a

^{3}– 3/4a

^{2}– 5/2

**Solution:**

Subtract the given expression

1/3a^{3}– 3/4a^{2}– 5/2 – (5/2a^{2}+ 3/2a^{3}+ a/3 – 6/5)

On rearrange

1/3a^{5}– 3/2a^{3}– 3/4a^{2}– 5/2a^{2}– a/3 – 5/2 + 6/5

By grouping similar expressions we get,

LCM of (3 and 2 is 6), (4 and 2 is 4), (2 and 5 is 10)

= (2a^{3}– 9a^{3})/6 – (3a^{2}+ 10a^{2})/4 – a/3 + (-25+12)/10

= -7/6a^{3}– 13/4a^{2}– a/3 – 13/10

**(iii) 7/4x**^{3} + 3/5x^{2} + 1/2x + 9/2 from 7/2 – x/3 – x^{2}/5

^{3}+ 3/5x

^{2}+ 1/2x + 9/2 from 7/2 – x/3 – x

^{2}/5

**Solution:**

Subtract the given expression

7/2 – x/3 – 1/5x

^{2}– (7/4x^{3}+ 3/5x^{2}+ 1/2x + 9/2)On rearranging,

-7/4x

^{3}– 1/5x^{2}– 3/5x^{2}– x/3 – x/2 + 7/2 – 9/2By grouping similar expressions we get,

LCM of (3 and 2 is 6)

-7/4x

^{3}– 4/5x^{2}– (2x-3x)/6 + (7-9)/2-7/4x

^{3}– 4/5x^{2}– 5/6x – 1

**(iv) y**^{3}/3 + 7/3y^{2} + 1/2y + 1/2 from 1/3 – 5/3y^{2}

^{3}/3 + 7/3y

^{2}+ 1/2y + 1/2 from 1/3 – 5/3y

^{2}

**Solution:**

Subtract the given expression

1/3 – 5/3y^{2}– (1/3y^{2}+ 7/3y^{2}+ 1/2y + 1/2)

On rearrange

-1/3y^{3}– 5/3y^{2}– 7/3y^{2}– 1/2y + 1/3 – 1/2

By grouping similar expressions we get,

LCM of (3 and 3 is 3), (3 and 2 is 6)

-1/3y^{3}+ (-5y^{2}– 7y^{2})/3 – 1/2y + (2-3)/6

-1/3y^{3}– 12/3y^{2}– 1/2y – 1/6

**(v) 2/3ac – 5/7ab + 2/3bc from 3/2ab -7/4ac – 5/6bc**

**Solution:**

Subtract the given expression

3/2ab – 7/4ac – 5/6bc – (2/3ac – 5/7ab + 2/3bc)

On rearrange

3/2ab + 5/7ab – 7/4ac – 2/3ac – 5/6bc – 2/3bc

By grouping similar expressions we get,

LCM of (2 and 7 is 14), (4 and 3 is 12), (6 and 3 is 6)

(21ab+10ab)/14 – (21ac-8ac)/12 – (5bc-4bc)/6

31/14ab – 29/12ac – 3/2bc

**Question 4. Subtract 3x – 4y – 7z from the sum of x – 3y + 2z and -4x + 9y – 11z. **

**Solution:**

First we will find the sum:

The sum of x – 3y + 2z and -4x + 9y – 11z is

(x – 3y + 2z) + (-4x + 9y – 11z)

On rearrange

x – 4x – 3y + 9y + 2z – 11z

= -3x + 6y – 9z

Now Let’s subtract it from -3x + 6y – 9z

(-3x + 6y – 9z) – (3x – 4y – 7z)

On rearranging again

= -3x – 3x + 6y + 4y – 9z + 7z

= -6x + 10y – 2z

**Question 5. Subtract the sum of 3l – 4m – 7n**^{2} and 2l + 3m – 4n^{2} from the sum of 9l + 2m – 3n^{2} and -3l + m + 4n^{2}.

^{2}and 2l + 3m – 4n

^{2}from the sum of 9l + 2m – 3n

^{2}and -3l + m + 4n

^{2}.

**Solution:**

Sum of 3l – 4m – 7n

^{2}and 2l + 5m – 4n^{2}

3l – 4m – 7n^{2}+ 2l + 3m – 4n^{2}

On rearrange

3l + 2l – 4m + 3m – 7n^{2}– 4n^{2}

5l – m – 11n^{2}……………………..eq. (1)

Sum of 9l + 2m – 3n^{2}and -3l + m + 4n^{2}

9l + 2m – 3n^{2}+ (-3l + m + 4n^{2})

On rearrange

9l – 3l + 2m + m – 3n^{2}+ 4n^{2}

6l + 3m + n^{2}……………………….eq. (2)

Let us subtract equ (i) from (ii), we get

6l + 3m + n^{2}– (5l – m – 11n^{2})

On rearrange

6l – 5l + 3m + m + n2 + 11n2

l + 4m + 12n^{2}

**Question 6. Subtract the sum of 2x – x**^{2} + 5 and -4x – 3 + 7x^{2} from 5.

^{2}+ 5 and -4x – 3 + 7x

^{2}from 5.

**Solution:**

Sum of 2x – x

^{2}+ 5 and -4x – 3 + 7x^{2}is

2x – x^{2}+ 5 + (-4x – 3 + 7x^{2})

2x – x^{2}+ 5 – 4x – 3 + 7x^{2}

On rearrange

– x^{2}+ 7x^{2}+ 2x – 4x + 5 – 3

6x^{2}-2x + 2 …………eq (i)

Let subtract eq (i) from 5 we will get,

5 – (6x^{2}-2x + 2)

5 – 6x^{2}+ 2x – 2

3 + 2x – 6x^{2}

**Question 7. Simplify each of the following:**

**(i) x**^{2} – 3x + 5 – 1/2(3x^{2} – 5x + 7)

^{2}– 3x + 5 – 1/2(3x

^{2}– 5x + 7)

**Solution:**

x

^{2}– 3x + 5 – 1/2(3x^{2}– 5x + 7)

On rearrange

x^{2}– 3/2x^{2}– 3x + 5/2x + 5 – 7/2

We will group similar expression:

LCM of (1 and 2 is 2)

= (2x^{2}– 3x^{2})/2 – (6x + 5x)/2 + (10-7)/2

= -1/2x^{2}– 1/2x + 3/2

**(ii) [5 – 3x + 2y – (2x – y)] – (3x – 7y + 9)**

**Solution:**

5 – 3x + 2y – 2x + y – 3x + 7y – 9

On rearrange

= – 3x – 2x – 3x + 2y + y + 7y + 5 – 9

We will group similar expression:

= -8x + 10y – 4

**(iii) 11/2x**^{2}y – 9/4xy^{2} + 1/4xy – 1/14y^{2}x + 1/15yx^{2} + 1/2xy

^{2}y – 9/4xy

^{2}+ 1/4xy – 1/14y

^{2}x + 1/15yx

^{2}+ 1/2xy

**Solution:**

On rearrange

11/2x^{2}y + 1/15x^{2}y – 9/4xy^{2}– 1/14xy^{2}+ 1/4xy + 1/2xy

We will group similar expression:

LCM of (2 and 15 is 30), (4 and 14 is 56), (4 and 2 is 4)

= (165x^{2}y + 2x^{2}y)/30 + (-126xy^{2}– 4xy^{2})/56 + (xy + 2xy)/4

= 167/30x^{2}y – 130/56xy^{2}+ 3/4xy

= 167/30x^{2}y – 65/28xy^{2}+ 3/4xy

**(iv) (1/3y**^{2} – 4/7y + 11) – (1/7y – 3 + 2y^{2}) – (2/7y – 2/3y^{2} + 2)

^{2}– 4/7y + 11) – (1/7y – 3 + 2y

^{2}) – (2/7y – 2/3y

^{2}+ 2)

**Solution:**

On rearrange

1/3y^{2}– 2y^{2}– 2/3y^{2}– 4/7y – 1/7y – 2/7y + 11 + 3 – 2

We will group similar expression:

LCM of (3, 1 and 3 is 3), (7, 7 and 7 is 7)

= (y^{2}– 6y^{2}+ 2y^{2})/3 – (4y – y – 2y)/7 + 12

= -3/3y^{2}– 7/7y + 12

= -y^{2}– y + 12

**(v) -1/2a**^{2}b^{2}c + 1/3ab^{2}c – 1/4abc^{2} – 1/5cb^{2}a^{2} + 1/6cb^{2}a – 1/7c^{2}ab + 1/8ca^{2}b

^{2}b

^{2}c + 1/3ab

^{2}c – 1/4abc

^{2}– 1/5cb

^{2}a

^{2}+ 1/6cb

^{2}a – 1/7c

^{2}ab + 1/8ca

^{2}b

**Solution:**

On rearrange

-1/2a^{2}b^{2}c – 1/5a^{2}b^{2}c + 1/3ab^{2}c + 1/6ab^{2}c – 1/4abc^{2}– 1/7abc^{2}+ 1/8a^{2}bc

We will group similar expression:

LCM of (2 and 5 is 10), (3 and 6 is 6), (4 and 7 is 28)

-7/10a^{2}b^{2}c + 1/2ab^{2}c – 11/28abc^{2}+ 1/8a^{2}bc

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