Fundamental Concepts of Algebra — Exercise 12(B)

Fill in the blanks:

(i) The sum of −2 and −5 = ............... and the sum of $−2x$ and $−5x$ = ...............

(ii) The sum of 8 and −3 = ............... and the sum of $8ab$ and $−3ab$ = ...............

(iii) The sum of −15 and −4 = ............... and the sum of $−15x$ and $−4y$ = ...............

(iv) 15 + 8 + 3 = ............... and $15x + 8y + 3x$ = ...............

(v) 12 − 9 + 15 = ............... and $12ab − 9ab + 15ba$ = ...............

(vi) 25 − 7 − 9 = ............... and $25xy − 7xy − 9yx$ = ...............

Answer

For the addition or subtraction of like terms, the rules are the same as those for the addition or subtraction of integers. Unlike terms cannot be added or subtracted to get a single term.

(i) The sum of -2 and -5 = -2 - 5 = −7 and the sum of $-2x$ and $-5x = -2x - 5x = (-2 - 5)x = \bm{-7x}$.

(ii) The sum of 8 and -3 = 8 - 3 = 5 and the sum of $8ab$ and $-3ab = 8ab - 3ab = (8 - 3)ab = \bm{5ab}$.

(iii) The sum of -15 and -4 = -15 - 4 = −19 and the sum of $-15x$ and $-4y = \bm{-15x - 4y}$, as $-15x$ and $-4y$ are unlike terms.

(iv) 15 + 8 + 3 = 26 and $15x + 8y + 3x = (15 + 3)x + 8y = \bm{18x + 8y}$.

(v) 12 - 9 + 15 = 18 and $12ab - 9ab + 15ba = (12 - 9 + 15)ab = \bm{18ab}$.

(vi) 25 - 7 - 9 = 9 and $25xy - 7xy - 9yx = (25 - 7 - 9)xy = \bm{9xy}$.

Add:

$8xy$ and $3xy$

Answer

Solving,

$\Rightarrow 8xy + 3xy \\[1em] \Rightarrow (8 + 3)xy \\[1em] \Rightarrow 11xy$

Hence, the required sum = $\bm{11xy}$.

Add:

$2xyz, xyz$ and $6xyz$

Answer

Solving,

$\Rightarrow 2xyz + xyz + 6xyz \\[1em] \Rightarrow (2 + 1 + 6)xyz \\[1em] \Rightarrow 9xyz$

Hence, the required sum = $\bm{9xyz}$.

Add:

$2a, 3a$ and $4b$

Answer

Solving,

$\Rightarrow 2a + 3a + 4b \\[1em] \Rightarrow (2 + 3)a + 4b \\[1em] \Rightarrow 5a + 4b$

Hence, the required sum = $\bm{5a + 4b}$.

Add:

$3x$ and $2y$

Answer

Solving,

$\Rightarrow 3x + 2y$

As $3x$ and $2y$ are unlike terms, they cannot be added to get a single term.

Hence, the required sum = $\bm{3x + 2y}$.

Add:

$5m, 3n$ and $4p$

Answer

Solving,

$\Rightarrow 5m + 3n + 4p$

As $5m, 3n$ and $4p$ are unlike terms, they cannot be added to get a single term.

Hence, the required sum = $\bm{5m + 3n + 4p}$.

Add:

$6a, 3a$ and $9ab$

Answer

Solving,

$\Rightarrow 6a + 3a + 9ab \\[1em] \Rightarrow (6 + 3)a + 9ab \\[1em] \Rightarrow 9a + 9ab$

Hence, the required sum = $\bm{9a + 9ab}$.

Add:

$3p, 4q$ and $9q$

Answer

Solving,

$\Rightarrow 3p + 4q + 9q \\[1em] \Rightarrow 3p + (4 + 9)q \\[1em] \Rightarrow 3p + 13q$

Hence, the required sum = $\bm{3p + 13q}$.

Add:

$5ab, 4ba$ and $6b$

Answer

Solving,

$\Rightarrow 5ab + 4ba + 6b \\[1em] \Rightarrow (5 + 4)ab + 6b \\[1em] \Rightarrow 9ab + 6b$

Hence, the required sum = $\bm{9ab + 6b}$.

Add:

$50pq, 30pq$ and $10pr$

Answer

Solving,

$\Rightarrow 50pq + 30pq + 10pr \\[1em] \Rightarrow (50 + 30)pq + 10pr \\[1em] \Rightarrow 80pq + 10pr$

Hence, the required sum = $\bm{80pq + 10pr}$.

Add:

$-2y, -y$ and $-3y$

Answer

Solving,

$\Rightarrow -2y + (-y) + (-3y) \\[1em] \Rightarrow (-2 - 1 - 3)y \\[1em] \Rightarrow -6y$

Hence, the required sum = $\bm{-6y}$.

Add:

$-3b$ and $-b$

Answer

Solving,

$\Rightarrow -3b + (-b) \\[1em] \Rightarrow (-3 - 1)b \\[1em] \Rightarrow -4b$

Hence, the required sum = $\bm{-4b}$.

Add:

$-2c, -c$ and $-5c$

Answer

Solving,

$\Rightarrow -2c + (-c) + (-5c) \\[1em] \Rightarrow (-2 - 1 - 5)c \\[1em] \Rightarrow -8c$

Hence, the required sum = $\bm{-8c}$.

Evaluate:

$6a - a - 5a - 2a$

Answer

Solving,

$\Rightarrow 6a - a - 5a - 2a \\[1em] \Rightarrow (6 - 1 - 5 - 2)a \\[1em] \Rightarrow -2a$

Hence, the value of the given expression = $\bm{-2a}$.

Evaluate:

$2b - 3b - b + 4b$

Answer

Solving,

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

Hence, the value of the given expression = $\bm{2b}$.

Evaluate:

$3x - 2x - 4x + 7x$

Answer

Solving,

$\Rightarrow 3x - 2x - 4x + 7x \\[1em] \Rightarrow (3 - 2 - 4 + 7)x \\[1em] \Rightarrow 4x$

Hence, the value of the given expression = $\bm{4x}$.

Evaluate:

$5ab + 2ab - 6ab + ab$

Answer

Solving,

$\Rightarrow 5ab + 2ab - 6ab + ab \\[1em] \Rightarrow (5 + 2 - 6 + 1)ab \\[1em] \Rightarrow 2ab$

Hence, the value of the given expression = $\bm{2ab}$.

Evaluate:

$8x - 5y - 3x + 10y$

Answer

Solving,

$\Rightarrow 8x - 5y - 3x + 10y \\[1em] \Rightarrow (8 - 3)x + (-5 + 10)y \\[1em] \Rightarrow 5x + 5y$

Hence, the value of the given expression = $\bm{5x + 5y}$.

Evaluate:

$-7x + 9x + 2x - 2x$

Answer

Solving,

$\Rightarrow -7x + 9x + 2x - 2x \\[1em] \Rightarrow (-7 + 9 + 2 - 2)x \\[1em] \Rightarrow 2x$

Hence, the value of the given expression = $\bm{2x}$.

Evaluate:

$5ab - 2ab - 8ab + 6ab$

Answer

Solving,

$\Rightarrow 5ab - 2ab - 8ab + 6ab \\[1em] \Rightarrow (5 - 2 - 8 + 6)ab \\[1em] \Rightarrow ab$

Hence, the value of the given expression = $\bm{ab}$.

Evaluate:

$-8a - 3a + 12a + 13a - 6a$

Answer

Solving,

$\Rightarrow -8a - 3a + 12a + 13a - 6a \\[1em] \Rightarrow (-8 - 3 + 12 + 13 - 6)a \\[1em] \Rightarrow 8a$

Hence, the value of the given expression = $\bm{8a}$.

Subtract the first term from the second:

$4ab, 6ba$

Answer

Solving,

$\Rightarrow 6ba - 4ab \\[1em] \Rightarrow (6 - 4)ab \\[1em] \Rightarrow 2ab$

Hence, the required difference = $\bm{2ab}$.

Subtract the first term from the second:

$4.8b, 6.8b$

Answer

Solving,

$\Rightarrow 6.8b - 4.8b \\[1em] \Rightarrow (6.8 - 4.8)b \\[1em] \Rightarrow 2b$

Hence, the required difference = $\bm{2b}$.

Subtract the first term from the second:

$3.5abc, 10.5abc$

Answer

Solving,

$\Rightarrow 10.5abc - 3.5abc \\[1em] \Rightarrow (10.5 - 3.5)abc \\[1em] \Rightarrow 7abc$

Hence, the required difference = $\bm{7abc}$.

Subtract the first term from the second:

$3\dfrac{1}{2}mn, 8\dfrac{1}{2}nm$

Answer

Solving,

$\Rightarrow 8\dfrac{1}{2}nm - 3\dfrac{1}{2}mn \\[1em] \Rightarrow \dfrac{17}{2}mn - \dfrac{7}{2}mn \\[1em] \Rightarrow \left(\dfrac{17 - 7}{2}\right)mn \\[1em] \Rightarrow \dfrac{10}{2}mn \\[1em] \Rightarrow 5mn$

Hence, the required difference = $\bm{5mn}$.

Simplify:

$2a^2b^2 + 5ab^2 + 8a^2b^2 - 3ab^2$

Answer

Solving,

$\Rightarrow 2a^2b^2 + 5ab^2 + 8a^2b^2 - 3ab^2 \\[1em] \Rightarrow (2 + 8)a^2b^2 + (5 - 3)ab^2 \\[1em] \Rightarrow 10a^2b^2 + 2ab^2$

Hence, the simplified expression = $\bm{10a^2b^2 + 2ab^2}$.

Simplify:

$4a + 3b - 2a - b$

Answer

Solving,

$\Rightarrow 4a + 3b - 2a - b \\[1em] \Rightarrow (4 - 2)a + (3 - 1)b \\[1em] \Rightarrow 2a + 2b$

Hence, the simplified expression = $\bm{2a + 2b}$.

Simplify:

$2xy + 4yz + 5xy + 3yz - 6xy$

Answer

Solving,

$\Rightarrow 2xy + 4yz + 5xy + 3yz - 6xy \\[1em] \Rightarrow (2 + 5 - 6)xy + (4 + 3)yz \\[1em] \Rightarrow xy + 7yz$

Hence, the simplified expression = $\bm{xy + 7yz}$.

Simplify:

$ab + 15ab - 11ab - 2ab$

Answer

Solving,

$\Rightarrow ab + 15ab - 11ab - 2ab \\[1em] \Rightarrow (1 + 15 - 11 - 2)ab \\[1em] \Rightarrow 3ab$

Hence, the simplified expression = $\bm{3ab}$.

Simplify:

$6a^2 - 3b^2 + 2a^2 + 5b^2 - 4a^2$

Answer

Solving,

$\Rightarrow 6a^2 - 3b^2 + 2a^2 + 5b^2 - 4a^2 \\[1em] \Rightarrow (6 + 2 - 4)a^2 + (-3 + 5)b^2 \\[1em] \Rightarrow 4a^2 + 2b^2$

Hence, the simplified expression = $\bm{4a^2 + 2b^2}$.

Simplify:

$8abc + 2ab - 4abc + ab$

Answer

Solving,

$\Rightarrow 8abc + 2ab - 4abc + ab \\[1em] \Rightarrow (8 - 4)abc + (2 + 1)ab \\[1em] \Rightarrow 4abc + 3ab$

Hence, the simplified expression = $\bm{4abc + 3ab}$.

Simplify:

$9xyz + 15yxz - 10zyx - 2zxy$

Answer

Solving,

$\Rightarrow 9xyz + 15yxz - 10zyx - 2zxy \\[1em] \Rightarrow (9 + 15 - 10 - 2)xyz \\[1em] \Rightarrow 12xyz$

Hence, the simplified expression = $\bm{12xyz}$.

Simplify:

$13pqr + 2p + 4q - 6pqr + 5pqr$

Answer

Solving,

$\Rightarrow 13pqr + 2p + 4q - 6pqr + 5pqr \\[1em] \Rightarrow (13 - 6 + 5)pqr + 2p + 4q \\[1em] \Rightarrow 12pqr + 2p + 4q$

Hence, the simplified expression = $\bm{12pqr + 2p + 4q}$.

Simplify:

$4ab + 0 - 2ba$

Answer

Solving,

$\Rightarrow 4ab + 0 - 2ba \\[1em] \Rightarrow (4 - 2)ab \\[1em] \Rightarrow 2ab$

Hence, the simplified expression = $\bm{2ab}$.