Fundamental Concepts of Algebra — Exercise 12(E)

Add:

$9x, 7x$

Answer

Adding,

$\Rightarrow 9x + 7x\\[1em] \Rightarrow (9 + 7)x\\[1em] \Rightarrow \bm{16x}$

Hence, $\bm{9x + 7x = 16x}$

Add:

$13p, 6p, 2p$

Answer

Adding,

$\Rightarrow 13p + 6p + 2p\\[1em] \Rightarrow (13 + 6 + 2)p\\[1em] \Rightarrow \bm{21p}$

Hence, $\bm{13p + 6p + 2p = 21p}$

Add:

$8xy, 7xy, xy$

Answer

Adding,

$\Rightarrow 8xy + 7xy + xy\\[1em] \Rightarrow (8 + 7 + 1)xy\\[1em] \Rightarrow \bm{16xy}$

Hence, $\bm{8xy + 7xy + xy = 16xy}$

Add:

$5y, -13y$

Answer

Adding,

$\Rightarrow 5y + (-13y)\\[1em] \Rightarrow (5 - 13)y\\[1em] \Rightarrow \bm{-8y}$

Hence, $\bm{5y + (-13y) = -8y}$

Add:

$4x^2, -9x^2$

Answer

Adding,

$\Rightarrow 4x^2 + (-9x^2)\\[1em] \Rightarrow (4 - 9)x^2\\[1em] \Rightarrow \bm{-5x^2}$

Hence, $\bm{4x^2 + (-9x^2) = -5x^2}$

Add:

$3a^2b, -5a^2b$

Answer

Adding,

$\Rightarrow 3a^2b + (-5a^2b)\\[1em] \Rightarrow (3 - 5)a^2b\\[1em] \Rightarrow \bm{-2a^2b}$

Hence, $\bm{3a^2b + (-5a^2b) = -2a^2b}$

Add:

$\dfrac{1}{2}mn, 2mn$

Answer

Adding,

$\Rightarrow \dfrac{1}{2}mn + 2mn\\[1em] \Rightarrow \left(\dfrac{1}{2} + 2\right)mn\\[1em] \Rightarrow \dfrac{1 + 4}{2}mn\\[1em] \Rightarrow \bm{\dfrac{5}{2}mn}$

Hence, $\bm{\dfrac{1}{2}mn + 2mn = \dfrac{5}{2}mn}$

Add:

$\dfrac{1}{3}x^2y, 3x^2y$

Answer

Adding,

$\Rightarrow \dfrac{1}{3}x^2y + 3x^2y\\[1em] \Rightarrow \left(\dfrac{1}{3} + 3\right)x^2y\\[1em] \Rightarrow \dfrac{1 + 9}{3}x^2y\\[1em] \Rightarrow \bm{\dfrac{10}{3}x^2y}$

Hence, $\bm{\dfrac{1}{3}x^2y + 3x^2y = \dfrac{10}{3}x^2y}$

Add:

$5ab^2, -ab^2$

Answer

Adding,

$\Rightarrow 5ab^2 + (-ab^2)\\[1em] \Rightarrow (5 - 1)ab^2\\[1em] \Rightarrow \bm{4ab^2}$

Hence, $\bm{5ab^2 + (-ab^2) = 4ab^2}$

Add:

$5a, -3a, 4a$

Answer

Adding,

$\Rightarrow 5a + (-3a) + 4a\\[1em] \Rightarrow (5 - 3 + 4)a\\[1em] \Rightarrow \bm{6a}$

Hence, $\bm{5a + (-3a) + 4a = 6a}$

Add:

$-2b, b, -3b$

Answer

Adding,

$\Rightarrow (-2b) + b + (-3b)\\[1em] \Rightarrow (-2 + 1 - 3)b\\[1em] \Rightarrow \bm{-4b}$

Hence, $\bm{(-2b) + b + (-3b) = -4b}$

Add:

$4ab, -5ab, -3ab$

Answer

Adding,

$\Rightarrow 4ab + (-5ab) + (-3ab)\\[1em] \Rightarrow (4 - 5 - 3)ab\\[1em] \Rightarrow \bm{-4ab}$

Hence, $\bm{4ab + (-5ab) + (-3ab) = -4ab}$

Add:

$6xy^2, -8xy^2, -3xy^2$

Answer

Adding,

$\Rightarrow 6xy^2 + (-8xy^2) + (-3xy^2)\\[1em] \Rightarrow (6 - 8 - 3)xy^2\\[1em] \Rightarrow \bm{-5xy^2}$

Hence, $\bm{6xy^2 + (-8xy^2) + (-3xy^2) = -5xy^2}$

Add:

$4m^2n, -m^2n, -6m^2n$

Answer

Adding,

$\Rightarrow 4m^2n + (-m^2n) + (-6m^2n)\\[1em] \Rightarrow (4 - 1 - 6)m^2n\\[1em] \Rightarrow \bm{-3m^2n}$

Hence, $\bm{4m^2n + (-m^2n) + (-6m^2n) = -3m^2n}$

Add:

$2xyz, -5xyz, -xyz$

Answer

Adding,

$\Rightarrow 2xyz + (-5xyz) + (-xyz)\\[1em] \Rightarrow (2 - 5 - 1)xyz\\[1em] \Rightarrow \bm{-4xyz}$

Hence, $\bm{2xyz + (-5xyz) + (-xyz) = -4xyz}$

Add:

$-6x, -9x$

Answer

Adding,

$\Rightarrow (-6x) + (-9x)\\[1em] \Rightarrow (-6 - 9)x\\[1em] \Rightarrow \bm{-15x}$

Hence, $\bm{(-6x) + (-9x) = -15x}$

Add:

$-3y, -y, -7y$

Answer

Adding,

$\Rightarrow (-3y) + (-y) + (-7y)\\[1em] \Rightarrow (-3 - 1 - 7)y\\[1em] \Rightarrow \bm{-11y}$

Hence, $\bm{(-3y) + (-y) + (-7y) = -11y}$

Add:

$-8xy, -4xy, 6xy$

Answer

Adding,

$\Rightarrow (-8xy) + (-4xy) + 6xy\\[1em] \Rightarrow (-8 - 4 + 6)xy\\[1em] \Rightarrow \bm{-6xy}$

Hence, $\bm{(-8xy) + (-4xy) + 6xy = -6xy}$

Add:

$7m^2n, -2m^2n, -3m^2n$

Answer

Adding,

$\Rightarrow 7m^2n + (-2m^2n) + (-3m^2n)\\[1em] \Rightarrow (7 - 2 - 3)m^2n\\[1em] \Rightarrow \bm{2m^2n}$

Hence, $\bm{7m^2n + (-2m^2n) + (-3m^2n) = 2m^2n}$

Add:

$-2ab^2, -8ab^2, 5ab^2$

Answer

Adding,

$\Rightarrow (-2ab^2) + (-8ab^2) + 5ab^2\\[1em] \Rightarrow (-2 - 8 + 5)ab^2\\[1em] \Rightarrow \bm{-5ab^2}$

Hence, $\bm{(-2ab^2) + (-8ab^2) + 5ab^2 = -5ab^2}$

Add:

$\dfrac{1}{2}yz, 2yz, -yz$

Answer

Adding,

$\Rightarrow \dfrac{1}{2}yz + 2yz + (-yz)\\[1em] \Rightarrow \left(\dfrac{1}{2} + 2 - 1\right)yz\\[1em] \Rightarrow \dfrac{1 + 4 - 2}{2}yz\\[1em] \Rightarrow \bm{\dfrac{3}{2}yz}$

Hence, $\bm{\dfrac{1}{2}yz + 2yz + (-yz) = \dfrac{3}{2}yz}$

Add:

$5a^3, -3a^3, -\dfrac{1}{2}a^3$

Answer

Adding,

$\Rightarrow 5a^3 + (-3a^3) + \left(-\dfrac{1}{2}a^3\right)\\[1em] \Rightarrow \left(5 - 3 - \dfrac{1}{2}\right)a^3\\[1em] \Rightarrow \dfrac{10 - 6 - 1}{2}a^3\\[1em] \Rightarrow \bm{\dfrac{3}{2}a^3}$

Hence, $\bm{5a^3 + (-3a^3) + \left(-\dfrac{1}{2}a^3\right) = \dfrac{3}{2}a^3}$

Add:

$-\dfrac{1}{2}a^2, -\dfrac{1}{3}a^2, 2a^2$

Answer

Adding,

$\Rightarrow \left(-\dfrac{1}{2}a^2\right) + \left(-\dfrac{1}{3}a^2\right) + 2a^2\\[1em] \Rightarrow \left(-\dfrac{1}{2} - \dfrac{1}{3} + 2\right)a^2\\[1em] \Rightarrow \dfrac{-3 - 2 + 12}{6}a^2\\[1em] \Rightarrow \bm{\dfrac{7}{6}a^2}$

Hence, $\bm{\left(-\dfrac{1}{2}a^2\right) + \left(-\dfrac{1}{3}a^2\right) + 2a^2 = \dfrac{7}{6}a^2}$

Find the sum of the following algebraic expressions:

$8a - 3b + 11c, 5b - 6a + 7c$ and $9c + 2a - 8b$.

Answer

Adding columnwise, we get:

$\begin{array}{rrrr} & 8a & - 3b & + 11c \\ & -6a & + 5b & + 7c \\ {+} & 2a & - 8b & + 9c \\ \hline & 4a & - 6b & + 27c \end{array}$

Hence, the required sum is $\bm{4a - 6b + 27c}$.

Find the sum of the following algebraic expressions:

$3y + 4z - 6x, 2x - 4y + z$ and $5z - 8y - x$.

Answer

Adding columnwise, we get:

$\begin{array}{rrrr} & -6x & + 3y & + 4z \\ & 2x & - 4y & + z \\ {+} & -x & - 8y & + 5z \\ \hline & -5x & - 9y & + 10z \end{array}$

Hence, the required sum is $\bm{-5x - 9y + 10z}$.

Find the sum of the following algebraic expressions:

$8x - 5y + 3z, 6y - 11x - z, 7y - 2z$ and $x - 2y$.

Answer

Adding columnwise, we get:

$\begin{array}{rrrr} & 8x &- 5y &+ 3z \\ & -11x &+ 6y &- z \\ & & 7y &- 2z \\ {+} &x &- 2y& \\ \hline &-2x &+ 6y& \end{array}$

Hence, the required sum is $\bm{-2x + 6y}$.

Find the sum of the following algebraic expressions:

$10x^2 + 8x - 7, 3x - 13x^2 + 5$ and $9 - 6x - 5x^2$.

Answer

Adding columnwise, we get:

$\begin{array}{rrrr} & 10x^2 & + 8x & - 7 \\ & -13x^2 & + 3x & + 5 \\ {+} & -5x^2 & - 6x & + 9 \\ \hline & -8x^2 & + 5x & + 7 \end{array}$

Hence, the required sum is $\bm{-8x^2 + 5x + 7}$.

Find the sum of the following algebraic expressions:

$5x^3 - 8x^2 + 6x - 3, x^2 - 2x + 15$ and $16 - 2x^3$.

Answer

Adding columnwise, we get:

$\begin{array}{rrrrr} & 5x^3 & - 8x^2 & + 6x & - 3 \\ & & x^2 & - 2x & + 15 \\ {+} & -2x^3 & & & + 16 \\ \hline & 3x^3 & - 7x^2 & + 4x & + 28 \end{array}$

Hence, the required sum is $\bm{3x^3 - 7x^2 + 4x + 28}$.

Find the sum of the following algebraic expressions:

$3x^2 - \dfrac{1}{5}x + \dfrac{7}{3}, -\dfrac{1}{4}x^2 + \dfrac{1}{3}x - \dfrac{1}{6}, -2x^2 - \dfrac{1}{2}x + \dfrac{1}{15}$

Answer

Collecting the like terms, we get:

Coefficient of $x^2 = 3 - \dfrac{1}{4} - 2 = 1 - \dfrac{1}{4} = \dfrac{4 - 1}{4} = \dfrac{3}{4}$

Coefficient of $x = -\dfrac{1}{5} + \dfrac{1}{3} - \dfrac{1}{2} = \dfrac{-6 + 10 - 15}{30} = -\dfrac{11}{30}$

Constant term $= \dfrac{7}{3} - \dfrac{1}{6} + \dfrac{1}{15} = \dfrac{70 - 5 + 2}{30} = \dfrac{67}{30}$

Hence, the required sum is $\bm{\dfrac{3}{4}x^2 - \dfrac{11}{30}x + \dfrac{67}{30}}$.

Find the sum of the following algebraic expressions:

$2a^2 - 3ab + 4bc, 6ab - a^2 - 5bc$ and $13bc - 8a^2 + ab$.

Answer

Adding columnwise, we get:

$\begin{array}{rrrr} & 2a^2 & - 3ab & + 4bc \\ & -a^2 & + 6ab & - 5bc \\ {+} & -8a^2 & + ab & + 13bc \\ \hline & -7a^2 & + 4ab & + 12bc \end{array}$

Hence, the required sum is $\bm{-7a^2 + 4ab + 12bc}$.

Find the sum of the following algebraic expressions:

$6x^2 + 2xy - y^2, xy - x^2 - 2y^2$ and $5y^2 - 8x^2 + 3xy$.

Answer

Adding columnwise, we get:

$\begin{array}{rrrr} & 6x^2 & + 2xy & - y^2 \\ & -x^2 & + xy & - 2y^2 \\ {+} & -8x^2 & + 3xy & + 5y^2 \\ \hline & -3x^2 & + 6xy & + 2y^2 \end{array}$

Hence, the required sum is $\bm{-3x^2 + 6xy + 2y^2}$.

Find the sum of the following algebraic expressions:

$2x^2 - 3y^2 + 6z^2, y^2 - 2x^2 + z^2, z^2 - 6x^2 - 8y^2$ and $x^2 - y^2$.

Answer

Adding columnwise, we get:

$\begin{array}{rrrr} & 2x^2 & - 3y^2 & + 6z^2 \\ & -2x^2 & + y^2 & + z^2 \\ & -6x^2 & - 8y^2 & + z^2 \\ {+} & x^2 & - y^2 & \\ \hline & -5x^2 & - 11y^2 & + 8z^2 \end{array}$

Hence, the required sum is $\bm{-5x^2 - 11y^2 + 8z^2}$.

Find the sum of the following algebraic expressions:

$6m^2n + 3mn - 2n^2, n^2 - mn - m^2n$ and $mn - 3n^2 - 4m^2n$.

Answer

Adding columnwise, we get:

$\begin{array}{rrrr} & 6m^2n & + 3mn & - 2n^2 \\ & -m^2n & - mn & + n^2 \\ {+} & -4m^2n & + mn & - 3n^2 \\ \hline & m^2n & + 3mn & - 4n^2 \end{array}$

Hence, the required sum is $\bm{m^2n + 3mn - 4n^2}$.

Simplify:

$2m^3 - 5m^2 + 3m - 6 - 5m^3 + 2m^2 - 8m + 2 + 6m + 5$.

Answer

Given,

$2m^3 - 5m^2 + 3m - 6 - 5m^3 + 2m^2 - 8m + 2 + 6m + 5$

Re-arranging and collecting the like terms, we get:

$\Rightarrow (2m^3 - 5m^3) + (-5m^2 + 2m^2) + (3m - 8m + 6m) + (-6 + 2 + 5)\\[1em] \Rightarrow (2 - 5)m^3 + (-5 + 2)m^2 + (3 - 8 + 6)m + 1\\[1em] \Rightarrow \bm{-3m^3 - 3m^2 + m + 1}$

Hence, the simplified expression is $\bm{-3m^3 - 3m^2 + m + 1}$.

Simplify:

$2x^4 - 5x^3 + 2x - 5 + 6x^3 - 3x + 7 - x^4 - 2x^3 + 2$.

Answer

Given,

$2x^4 - 5x^3 + 2x - 5 + 6x^3 - 3x + 7 - x^4 - 2x^3 + 2$

Re-arranging and collecting the like terms, we get:

$\Rightarrow (2x^4 - x^4) + (-5x^3 + 6x^3 - 2x^3) + (2x - 3x) + (-5 + 7 + 2)\\[1em] \Rightarrow (2 - 1)x^4 + (-5 + 6 - 2)x^3 + (2 - 3)x + 4\\[1em] \Rightarrow \bm{x^4 - x^3 - x + 4}$

Hence, the simplified expression is $\bm{x^4 - x^3 - x + 4}$.

Simplify:

$3p^2 - 2pq + 5p - 4q + 5pq - 3p + 4p^2 + 3q$.

Answer

Given,

$3p^2 - 2pq + 5p - 4q + 5pq - 3p + 4p^2 + 3q$

Re-arranging and collecting the like terms, we get:

$\Rightarrow (3p^2 + 4p^2) + (-2pq + 5pq) + (5p - 3p) + (-4q + 3q)\\[1em] \Rightarrow (3 + 4)p^2 + (-2 + 5)pq + (5 - 3)p + (-4 + 3)q\\[1em] \Rightarrow \bm{7p^2 + 3pq + 2p - q}$

Hence, the simplified expression is $\bm{7p^2 + 3pq + 2p - q}$.