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Chapter 12

Fundamental Concepts of Algebra — Exercise 12(F)

Class - 6 RS Aggarwal Mathematics Solutions

Exercise 12(F)

Question 1(i)

Subtract:

8x8x from 15x15x

Answer

To subtract, we change the sign of the term to be subtracted and then add.

⇒15x−8x⇒(15−8)x⇒7x.\Rightarrow 15x - 8x\\[1em] \Rightarrow (15 - 8)x\\[1em] \Rightarrow 7x.

Hence, the required difference is 7x\bm{7x}.

Question 1(ii)

Subtract:

3ab23ab^2 from 7ab27ab^2

Answer

To subtract, we change the sign of the term to be subtracted and then add.

⇒7ab2−3ab2⇒(7−3)ab2⇒4ab2.\Rightarrow 7ab^2 - 3ab^2\\[1em] \Rightarrow (7 - 3)ab^2\\[1em] \Rightarrow 4ab^2.

Hence, the required difference is 4ab2\bm{4ab^2}.

Question 1(iii)

Subtract:

5b5b from −b-b

Answer

To subtract, we change the sign of the term to be subtracted and then add.

⇒−b−5b⇒(−1−5)b⇒−6b.\Rightarrow -b - 5b\\[1em] \Rightarrow (-1 - 5)b\\[1em] \Rightarrow -6b.

Hence, the required difference is −6b\bm{-6b}.

Question 1(iv)

Subtract:

−4x-4x from 11x11x

Answer

To subtract, we change the sign of the term to be subtracted and then add.

⇒11x−(−4x)⇒11x+4x⇒(11+4)x⇒15x.\Rightarrow 11x - (-4x)\\[1em] \Rightarrow 11x + 4x\\[1em] \Rightarrow (11 + 4)x\\[1em] \Rightarrow 15x.

Hence, the required difference is 15x\bm{15x}.

Question 1(v)

Subtract:

−x-x from xx

Answer

To subtract, we change the sign of the term to be subtracted and then add.

⇒x−(−x)⇒x+x⇒2x.\Rightarrow x - (-x)\\[1em] \Rightarrow x + x\\[1em] \Rightarrow 2x.

Hence, the required difference is 2x\bm{2x}.

Question 1(vi)

Subtract:

2ab2ab from −2ab-2ab

Answer

To subtract, we change the sign of the term to be subtracted and then add.

⇒−2ab−2ab⇒(−2−2)ab⇒−4ab.\Rightarrow -2ab - 2ab\\[1em] \Rightarrow (-2 - 2)ab\\[1em] \Rightarrow -4ab.

Hence, the required difference is −4ab\bm{-4ab}.

Question 1(vii)

Subtract:

−3m2n-3m^2n from −7m2n-7m^2n

Answer

To subtract, we change the sign of the term to be subtracted and then add.

⇒−7m2n−(−3m2n)⇒−7m2n+3m2n⇒(−7+3)m2n⇒−4m2n.\Rightarrow -7m^2n - (-3m^2n)\\[1em] \Rightarrow -7m^2n + 3m^2n\\[1em] \Rightarrow (-7 + 3)m^2n\\[1em] \Rightarrow -4m^2n.

Hence, the required difference is −4m2n\bm{-4m^2n}.

Question 1(viii)

Subtract:

12b\dfrac{1}{2}b from −2b-2b

Answer

To subtract, we change the sign of the term to be subtracted and then add.

⇒−2b−12b⇒(−2−12)b⇒−4−12b⇒−52b\Rightarrow -2b - \dfrac{1}{2}b\\[1em] \Rightarrow \left(-2 - \dfrac{1}{2}\right)b\\[1em] \Rightarrow \dfrac{-4 - 1}{2}b\\[1em] \Rightarrow -\dfrac{5}{2}b

Hence, the required difference is −52b\bm{-\dfrac{5}{2}b}.

Question 1(ix)

Subtract:

−13c-\dfrac{1}{3}c from 3c3c

Answer

To subtract, we change the sign of the term to be subtracted and then add.

⇒3c−(−13c)⇒3c+13c⇒(3+13)c⇒9+13c⇒103c\Rightarrow 3c - \left(-\dfrac{1}{3}c\right)\\[1em] \Rightarrow 3c + \dfrac{1}{3}c\\[1em] \Rightarrow \left(3 + \dfrac{1}{3}\right)c\\[1em] \Rightarrow \dfrac{9 + 1}{3}c\\[1em] \Rightarrow \dfrac{10}{3}c

Hence, the required difference is 103c\bm{\dfrac{10}{3}c}.

Question 2(i)

Subtract:

a+2b−ca + 2b - c from 3a−b+2c3a - b + 2c.

Answer

Required difference =(3a−b+2c)−(a+2b−c)= (3a - b + 2c) - (a + 2b - c)

⇒3a−b+2c−a−2b+c⇒(3a−a)+(−b−2b)+(2c+c)⇒2a−3b+3c.\Rightarrow 3a - b + 2c - a - 2b + c\\[1em] \Rightarrow (3a - a) + (-b - 2b) + (2c + c)\\[1em] \Rightarrow 2a - 3b + 3c.

Hence, the required difference is 2a−3b+3c\bm{2a - 3b + 3c}.

Question 2(ii)

Subtract:

−3a+b+6c-3a + b + 6c from 5a−3b−4c5a - 3b - 4c.

Answer

Required difference =(5a−3b−4c)−(−3a+b+6c)= (5a - 3b - 4c) - (-3a + b + 6c)

⇒5a−3b−4c+3a−b−6c⇒(5a+3a)+(−3b−b)+(−4c−6c)⇒8a−4b−10c.\Rightarrow 5a - 3b - 4c + 3a - b - 6c\\[1em] \Rightarrow (5a + 3a) + (-3b - b) + (-4c - 6c)\\[1em] \Rightarrow 8a - 4b - 10c.

Hence, the required difference is 8a−4b−10c\bm{8a - 4b - 10c}.

Question 2(iii)

Subtract:

2m−3n+p2m - 3n + p from −m+n−2p-m + n - 2p.

Answer

Required difference =(−m+n−2p)−(2m−3n+p)= (-m + n - 2p) - (2m - 3n + p)

⇒−m+n−2p−2m+3n−p⇒(−m−2m)+(n+3n)+(−2p−p)⇒−3m+4n−3p.\Rightarrow -m + n - 2p - 2m + 3n - p\\[1em] \Rightarrow (-m - 2m) + (n + 3n) + (-2p - p)\\[1em] \Rightarrow -3m + 4n - 3p.

Hence, the required difference is −3m+4n−3p\bm{-3m + 4n - 3p}.

Question 2(iv)

Subtract:

−6x−9y+15z-6x - 9y + 15z from x−y−3zx - y - 3z.

Answer

Required difference =(x−y−3z)−(−6x−9y+15z)= (x - y - 3z) - (-6x - 9y + 15z)

⇒x−y−3z+6x+9y−15z⇒(x+6x)+(−y+9y)+(−3z−15z)⇒7x+8y−18z.\Rightarrow x - y - 3z + 6x + 9y - 15z\\[1em] \Rightarrow (x + 6x) + (-y + 9y) + (-3z - 15z)\\[1em] \Rightarrow 7x + 8y - 18z.

Hence, the required difference is 7x+8y−18z\bm{7x + 8y - 18z}.

Question 2(v)

Subtract:

−12p+q−r-\dfrac{1}{2}p + q - r from 12p−13q−32r\dfrac{1}{2}p - \dfrac{1}{3}q - \dfrac{3}{2}r.

Answer

Required difference =(12p−13q−32r)−(−12p+q−r)= \left(\dfrac{1}{2}p - \dfrac{1}{3}q - \dfrac{3}{2}r\right) - \left(-\dfrac{1}{2}p + q - r\right)

⇒12p−13q−32r+12p−q+r⇒(12+12)p+(−13−1)q+(−32+1)r⇒p−43q−12r\Rightarrow \dfrac{1}{2}p - \dfrac{1}{3}q - \dfrac{3}{2}r + \dfrac{1}{2}p - q + r\\[1em] \Rightarrow \left(\dfrac{1}{2} + \dfrac{1}{2}\right)p + \left(-\dfrac{1}{3} - 1\right)q + \left(-\dfrac{3}{2} + 1\right)r\\[1em] \Rightarrow p - \dfrac{4}{3}q - \dfrac{1}{2}r

Hence, the required difference is p−43q−12r\bm{p - \dfrac{4}{3}q - \dfrac{1}{2}r}.

Question 3(i)

Subtract:

3x2−5x+63x^2 - 5x + 6 from 5x2−x−35x^2 - x - 3.

Answer

First we arrange the like terms columnwise. Change the sign of each term to be subtracted and then add.

+5x2−x−3+3x2−5x+6−+−+2x2+4x−9\begin{array}{rrrr} &+5x^2 &- x &- 3 \\ &+3x^2 &- 5x &+ 6 \\ &- &+ &- \\ \hline &+2x^2 &+ 4x &- 9 \end{array}

Hence, the required difference is 2x2+4x−9\bm{2x^2 + 4x - 9}.

Question 3(ii)

Subtract:

2x2+3x−12x^2 + 3x - 1 from x2+x−5x^2 + x - 5.

Answer

First we arrange the like terms columnwise. Change the sign of each term to be subtracted and then add.

+x2+x−5+2x2+3x−1−−+−x2−2x−4\begin{array}{rrrr} &+x^2 &+ x &- 5 \\ &+2x^2 &+ 3x &- 1 \\ &- &- &+ \\ \hline &-x^2 &- 2x &- 4 \end{array}

Hence, the required difference is −x2−2x−4\bm{-x^2 - 2x - 4}.

Question 3(iii)

Subtract:

1−x+x21 - x + x^2 from x2+x−1x^2 + x - 1.

Answer

First we arrange the like terms columnwise. Change the sign of each term to be subtracted and then add.

+x2+x−1+x2−x+1−+−+2x−2\begin{array}{rrrr} &+x^2 &+ x &- 1 \\ &+x^2 &- x &+ 1 \\ &- &+ &- \\ \hline & &+ 2x &- 2 \end{array}

Hence, the required difference is 2x−2\bm{2x - 2}.

Question 3(iv)

Subtract:

a2+a−1a^2 + a - 1 from a−a2a - a^2.

Answer

First we arrange the like terms columnwise. Change the sign of each term to be subtracted and then add.

−a2+a+a2+a−1−−+−2a2+1\begin{array}{rrrr} &-a^2 &+ a & \\ &+a^2 &+ a &- 1 \\ &- &- &+ \\ \hline &-2a^2 & &+ 1 \end{array}

Hence, the required difference is −2a2+1\bm{-2a^2 + 1}.

Question 3(v)

Subtract:

6y2−y−86y^2 - y - 8 from −y2+2y+7-y^2 + 2y + 7.

Answer

First we arrange the like terms columnwise. Change the sign of each term to be subtracted and then add.

−y2+2y+7+6y2−y−8−++−7y2+3y+15\begin{array}{rrrr} &-y^2 &+ 2y &+ 7 \\ &+6y^2 &- y &- 8 \\ &- &+ &+ \\ \hline &-7y^2 &+ 3y &+ 15 \end{array}

Hence, the required difference is −7y2+3y+15\bm{-7y^2 + 3y + 15}.

Question 4(i)

Subtract:

p2−2pq+q2p^2 - 2pq + q^2 from p2+2pq−q2p^2 + 2pq - q^2.

Answer

First we arrange the like terms columnwise. Change the sign of each term to be subtracted and then add.

+p2+2pq−q2+p2−2pq+q2−+−+4pq−2q2\begin{array}{rrrr} &+p^2 &+ 2pq &- q^2 \\ &+p^2 &- 2pq &+ q^2 \\ &- &+ &- \\ \hline & &+ 4pq &- 2q^2 \end{array}

Hence, the required difference is 4pq−2q2\bm{4pq - 2q^2}.

Question 4(ii)

Subtract:

x2−2xy+y2x^2 - 2xy + y^2 from 5xy−2x2−3y25xy - 2x^2 - 3y^2.

Answer

First we arrange the like terms columnwise. Change the sign of each term to be subtracted and then add.

−2x2+5xy−3y2+x2−2xy+y2−+−−3x2+7xy−4y2\begin{array}{rrrr} &-2x^2 &+ 5xy &- 3y^2 \\ &+x^2 &- 2xy &+ y^2 \\ &- &+ &- \\ \hline &-3x^2 &+ 7xy &- 4y^2 \end{array}

Hence, the required difference is −3x2+7xy−4y2\bm{-3x^2 + 7xy - 4y^2}.

Question 4(iii)

Subtract:

5xy−3x2−2y2−35xy - 3x^2 - 2y^2 - 3 from x2−2xy+6y2+8x^2 - 2xy + 6y^2 + 8.

Answer

First we arrange the like terms columnwise. Change the sign of each term to be subtracted and then add.

+x2−2xy+6y2+8−3x2+5xy−2y2−3+−+++4x2−7xy+8y2+11\begin{array}{rrrrr} &+x^2 &- 2xy &+ 6y^2 &+ 8 \\ &-3x^2 &+ 5xy &- 2y^2 &- 3 \\ &+ &- &+ &+ \\ \hline &+4x^2 &- 7xy &+ 8y^2 &+ 11 \end{array}

Hence, the required difference is 4x2−7xy+8y2+11\bm{4x^2 - 7xy + 8y^2 + 11}.

Question 4(iv)

Subtract:

6x3−8x2+4x−26x^3 - 8x^2 + 4x - 2 from 5−4x+6x2−8x35 - 4x + 6x^2 - 8x^3.

Answer

First we arrange the like terms columnwise. Change the sign of each term to be subtracted and then add.

−8x3+6x2−4x+5+6x3−8x2+4x−2−+−+−14x3+14x2−8x+7\begin{array}{rrrrr} &-8x^3 &+ 6x^2 &- 4x &+ 5 \\ &+6x^3 &- 8x^2 &+ 4x &- 2 \\ &- &+ &- &+ \\ \hline &-14x^3 &+ 14x^2 &- 8x &+ 7 \end{array}

Hence, the required difference is −14x3+14x2−8x+7\bm{-14x^3 + 14x^2 - 8x + 7}.

Question 5

How much is a+2ba + 2b greater than b−ab - a?

Answer

Required expression =(a+2b)−(b−a)= (a + 2b) - (b - a)

First we arrange the like terms columnwise. Change the sign of each term to be subtracted and then add.

+a+2b−a+b+−+2a+b\begin{array}{rrr} &+a &+ 2b \\ &-a &+ b \\ &+ &- \\ \hline &+2a &+ b \end{array}

Hence, a+2b\bm{a + 2b} is greater than b−a\bm{b - a} by (2a+b)\bm{(2a + b)}.

Question 6

How much is a−2b+3ca - 2b + 3c greater than 2a+3b−c2a + 3b - c?

Answer

Required expression =(a−2b+3c)−(2a+3b−c)= (a - 2b + 3c) - (2a + 3b - c)

First we arrange the like terms columnwise. Change the sign of each term to be subtracted and then add.

+a−2b+3c+2a+3b−c−−+−a−5b+4c\begin{array}{rrrr} &+a &- 2b &+ 3c \\ &+2a &+ 3b &- c \\ &- &- &+ \\ \hline &-a &- 5b &+ 4c \end{array}

Hence, a−2b+3c\bm{a - 2b + 3c} is greater than 2a+3b−c\bm{2a + 3b - c} by (−a−5b+4c)\bm{(-a - 5b + 4c)}.

Question 7

How much does 3x2−4x+33x^2 - 4x + 3 exceed x3−x2+2x−1x^3 - x^2 + 2x - 1?

Answer

Required expression =(3x2−4x+3)−(x3−x2+2x−1)= (3x^2 - 4x + 3) - (x^3 - x^2 + 2x - 1)

First we arrange the like terms columnwise. Change the sign of each term to be subtracted and then add.

+3x2−4x+3+x3−x2+2x−1−+−+−x3+4x2−6x+4\begin{array}{rrrrr} & &+ 3x^2 &- 4x &+ 3 \\ &+x^3 &- x^2 &+ 2x &- 1 \\ &- &+ &- &+ \\ \hline &-x^3 &+ 4x^2 &- 6x &+ 4 \end{array}

Hence, 3x2−4x+3\bm{3x^2 - 4x + 3} exceeds x3−x2+2x−1\bm{x^3 - x^2 + 2x - 1} by (−x3+4x2−6x+4)\bm{(-x^3 + 4x^2 - 6x + 4)}.

Question 8

How much is p−2q+3rp - 2q + 3r less than 3r+53r + 5?

Answer

Required expression =(3r+5)−(p−2q+3r)= (3r + 5) - (p - 2q + 3r)

First we arrange the like terms columnwise. Change the sign of each term to be subtracted and then add.

+3r+5+p−2q+3r−+−−p+2q+5\begin{array}{rrrrr} & & &+ 3r &+ 5 \\ &+p &- 2q &+ 3r & \\ &- &+ &- & \\ \hline &-p &+ 2q & &+ 5 \end{array}

Hence, p−2q+3r\bm{p - 2q + 3r} is less than 3r+5\bm{3r + 5} by (−p+2q+5)\bm{(-p + 2q + 5)}.

Question 9

What must be subtracted from 3x3x to get 5x+15x + 1?

Answer

Required expression =3x−(5x+1)= 3x - (5x + 1)

First we arrange the like terms columnwise. Change the sign of each term to be subtracted and then add.

+3x+5x+1−−−2x−1\begin{array}{rrr} &+3x & \\ &+5x &+ 1 \\ &- &- \\ \hline &-2x &- 1 \end{array}

Hence, (−2x−1)\bm{(-2x - 1)} must be subtracted from 3x\bm{3x} to get 5x+1\bm{5x + 1}.

Question 10

What must be subtracted from 2x+3y2x + 3y to get y−3xy - 3x?

Answer

Required expression =(2x+3y)−(y−3x)= (2x + 3y) - (y - 3x)

First we arrange the like terms columnwise. Change the sign of each term to be subtracted and then add.

+2x+3y−3x+y+−+5x+2y\begin{array}{rrr} &+2x &+ 3y \\ &-3x &+ y \\ &+ &- \\ \hline &+5x &+ 2y \end{array}

Hence, (5x+2y)\bm{(5x + 2y)} must be subtracted from 2x+3y\bm{2x + 3y} to get y−3x\bm{y - 3x}.

Question 11

What must be subtracted from x2+2x−3x^2 + 2x - 3 to get 2x2−42x^2 - 4?

Answer

Required expression =(x2+2x−3)−(2x2−4)= (x^2 + 2x - 3) - (2x^2 - 4)

First we arrange the like terms columnwise. Change the sign of each term to be subtracted and then add.

+x2+2x−3+2x2−4−+−x2+2x+1\begin{array}{rrrr} &+x^2 &+ 2x &- 3 \\ &+2x^2 & &- 4 \\ &- & &+ \\ \hline &-x^2 &+ 2x &+ 1 \end{array}

Hence, (−x2+2x+1)\bm{(-x^2 + 2x + 1)} must be subtracted from x2+2x−3\bm{x^2 + 2x - 3} to get 2x2−4\bm{2x^2 - 4}.

Question 12

What must be added to 6x6x to get 4x+14x + 1?

Answer

Required expression =(4x+1)−6x= (4x + 1) - 6x

First we arrange the like terms columnwise. Change the sign of each term to be subtracted and then add.

+4x+1+6x−−2x+1\begin{array}{rrr} &+4x &+ 1 \\ &+6x & \\ &- & \\ \hline &-2x &+ 1 \end{array}

Hence, (−2x+1)\bm{(-2x + 1)} must be added to 6x\bm{6x} to get 4x+1\bm{4x + 1}.

Question 13

What must be added to a+2ba + 2b to get b−2ab - 2a?

Answer

Required expression =(b−2a)−(a+2b)= (b - 2a) - (a + 2b)

First we arrange the like terms columnwise. Change the sign of each term to be subtracted and then add.

−2a+b+a+2b−−−3a−b\begin{array}{rrr} &-2a &+ b \\ &+a &+ 2b \\ &- &- \\ \hline &-3a &- b \end{array}

Hence, (−3a−b)\bm{(-3a - b)} must be added to a+2b\bm{a + 2b} to get b−2a\bm{b - 2a}.

Question 14

What must be added to 5x2+6x−35x^2 + 6x - 3 to get 3x+23x + 2?

Answer

Required expression =(3x+2)−(5x2+6x−3)= (3x + 2) - (5x^2 + 6x - 3)

First we arrange the like terms columnwise. Change the sign of each term to be subtracted and then add.

+3x+2+5x2+6x−3−−+−5x2−3x+5\begin{array}{rrrr} & &+ 3x &+ 2 \\ &+5x^2 &+ 6x &- 3 \\ &- &- &+ \\ \hline &-5x^2 &- 3x &+ 5 \end{array}

Hence, (−5x2−3x+5)\bm{(-5x^2 - 3x + 5)} must be added to 5x2+6x−3\bm{5x^2 + 6x - 3} to get 3x+2\bm{3x + 2}.

Question 15

Subtract 5x2−8x+65x^2 - 8x + 6 from the sum of 3x2−4x+23x^2 - 4x + 2 and 7−5x2−6x7 - 5x^2 - 6x.

Answer

Adding the two expressions and then subtracting 5x2−8x+65x^2 - 8x + 6 from their sum columnwise, we get:

+3x2−4x+2+−5x2−6x+7−2x2−10x+9+5x2−8x+6−+−−7x2−2x+3\begin{array}{rrrr} &+3x^2 &- 4x &+ 2 \\ {+} &-5x^2 &- 6x &+ 7 \\ \hline &-2x^2 &- 10x &+ 9 \\ &+5x^2 &- 8x &+ 6 \\ &- &+ &- \\ \hline &-7x^2 &- 2x &+ 3 \end{array}

Hence, the required difference is −7x2−2x+3\bm{-7x^2 - 2x + 3}.

Question 16

Subtract 3y2−5y−43y^2 - 5y - 4 from the sum of y−5y2+1y - 5y^2 + 1 and 3y+4y2−53y + 4y^2 - 5.

Answer

Adding the two expressions and then subtracting 3y2−5y−43y^2 - 5y - 4 from their sum columnwise, we get:

−5y2+y+1++4y2+3y−5−y2+4y−4+3y2−5y−4−++−4y2+9y\begin{array}{rrrr} &-5y^2 &+ y &+ 1 \\ {+} &+4y^2 &+ 3y &- 5 \\ \hline &-y^2 &+ 4y &- 4 \\ &+3y^2 &- 5y &- 4 \\ &- &+ &+ \\ \hline &-4y^2 &+ 9y & \end{array}

Hence, the required difference is −4y2+9y\bm{-4y^2 + 9y}.

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