RS Aggarwal 2019,2020 solution class 7 chapter 6 Algebraic Expressions Exercise 6A

Exercise 6A

Page-99

Question 1:

Add the following expressions:

(i) 5x, 7x, −6x
(ii) $\frac{3}{5}x,\frac{2}{3}x,\frac{-4}{5}x$
(iii) 5a²b, −8a²b, 7a²b
(iv) $\frac{3}{4}{x}^2,5x^2,-3x^2,-\frac{1}{4}{x}^2$
(v) x − 3y + 4z, y − 2x − 8z, 5x − 2y − 3z
(vi) 2x² − 3y², 5x² + 6y², − 3x² − 4y²
(vii) 5x − 2x² − 8, 8x² − 7x − 9, 3 + 7x² − 2x
(viii) $\frac{2}{3}a-\frac{4}{5}b+\frac{3}{5}c,-\frac{3}{4}a-\frac{5}{2}b+\frac{2}{3}c,\frac{5}{2}a+\frac{7}{4}b-\frac{5}{6}c$
(ix) $\frac{8}{5}x+\frac{11}{7}y+\frac{9}{4}xy,-\frac{3}{2}x-\frac{5}{3}y-\frac{9}{5}xy$
(x) $\frac{3}{2}{x}^3-\frac{1}{4}{x}^2+\frac{5}{3},-\frac{5}{4}{x}^3+\frac{3}{5}{x}^2-x+\frac{1}{5},-x^2+\frac{3}{8}x-\frac{8}{15}$

Answer 1:

(i)
  5x + 7x + (-6x)
= 5x + 7x -6x
= 6x

(ii)

 $$\begin{gathered} \frac{3}{5}\mathrm{x} +\frac{2}{3}\mathrm{x} +\frac{-4}{5}\mathrm{x} \\ =\frac{9\mathrm{x} +10\mathrm{x} -12\mathrm{x}}{15}=\frac{7\mathrm{x}}{15} \end{gathered}$$

(iii) 
5a²b +( −8a²b)  + 7a²b
= 5a²b − 8a²b + 7a²b
4a²b

(iv)
  $$\begin{gathered} \frac{3}{4}{\mathrm{x}}^2 + 5{\mathrm{x}}^2+ (-3{\mathrm{x}}^{2 }) + (-\frac{1}{4}{\mathrm{x}}^2) \\ =\frac{3}{4}{\mathrm{x}}^2 - \frac{1}{4}{\mathrm{x}}^2 + 5{\mathrm{x}}^2-3{\mathrm{x}}^2 \\ =\frac{1}{2}{\mathrm{x}}^2 + 2{\mathrm{x}}^2 = \frac{5}{2}{\mathrm{x}}^2 \end{gathered}$$


(v) Collecting like terms and adding them:

x − 3y + 4z + y − 2x − 8z + 5x − 2y − 3z
= x- 2x + 5x - 3y + y - 2y + 4z - 8z - 3z        
= 4x -4y -7z

(vi) Collecting like terms and adding them:
2x² − 3y² + 5x² + 6y² + (− 3x² − 4y²)
 $$\begin{gathered} = 2{\mathrm{x}}^2 + 5{\mathrm{x}}^2 - 3{\mathrm{x}}^2 - 3{\mathrm{y}}^2 +6{\mathrm{y}}^2 -4{\mathrm{y}}^2 \\ = 4{\mathrm{x}}^2 -{\mathrm{y}}^2 \end{gathered}$$                   

(vii) Collecting like terms and adding them:
5x − 2x² − 8 +  8x² − 7x − 9 +  3 + 7x² − 2x
$$\begin{gathered} = -2{\mathrm{x}}^{2 }+8{\mathrm{x}}^2 + 7{\mathrm{x}}^2 + 5\mathrm{x} - 7\mathrm{x} - 2\mathrm{x} -8 -9 + 3 \\ = 13{\mathrm{x}}^2 -4\mathrm{x} -14 \end{gathered}$$      


(viii) Collecting like terms and adding them:

$$\begin{gathered} \frac{2}{3}\mathrm{a}-\frac{4}{5}\mathrm{b}+\frac{3}{5}\mathrm{c} +\left(-\frac{3}{4}\mathrm{a} -\frac{5}{2}\mathrm{b}+\frac{2}{3}\mathrm{c} \right)+ \frac{5}{2}\mathrm{a}+\frac{7}{4}\mathrm{b}- \frac{5}{6}\mathrm{c} \\ \mathrm{b} -\frac{5}{2}\mathrm{b} +\frac{7}{4}\mathrm{b} +\frac{3}{5}\mathrm{c} +\frac{2}{3}\mathrm{c} - \frac{5}{6}\mathrm{c} \\ =\frac{(8 -9 +30)\mathrm{a}}{12} + \frac{(-16 - 50 +35)\mathrm{b}}{20} + \frac{(18 +20 -25)\mathrm{c}}{30} \\ =\frac{29}{12}\mathrm{a} - \frac{31}{20}\mathrm{b} + \frac{13}{30}\mathrm{c} \end{gathered}$$


(ix) Collecting like terms and adding them:
$$\begin{gathered} \frac{8}{5}\mathrm{x}+ \frac{11}{7}\mathrm{y}+\frac{9}{4}\mathrm{xy} + \left(- \frac{3}{2}\mathrm{x}- \frac{5}{3}\mathrm{y}-\frac{9}{5}\mathrm{xy}\right) \\ =\frac{8}{5}\mathrm{x} - \frac{3}{2}\mathrm{x}+ \frac{11}{7}\mathrm{y} - \frac{5}{3}\mathrm{y} +\frac{9}{4}\mathrm{xy} -\frac{9}{5}\mathrm{xy} \\ = \frac{1}{10}\mathrm{x} - \frac{2}{21}\mathrm{y} + \frac{9}{20}\mathrm{xy} \end{gathered}$$

(x) Collecting like terms and adding them:
$$\begin{gathered} \frac{3}{2}{\mathrm{x}}^3-\frac{1}{4}{\mathrm{x}}^2 +\frac{5}{3}+\left(- \frac{5}{4}{\mathrm{x}}^3+\frac{3}{5}{\mathrm{x}}^2- \mathrm{x}+\frac{1}{5}\right)+\left(- {\mathrm{x}}^{2 }+\frac{3}{8}\mathrm{x}-\frac{8}{15}\right) \\ = \frac{3}{2}{\mathrm{x}}^3 - \frac{5}{4}{\mathrm{x}}^3 -\frac{1}{4}{\mathrm{x}}^2 +\frac{3}{5}{\mathrm{x}}^2- {\mathrm{x}}^{2 }- \mathrm{x} +\frac{3}{8}\mathrm{x} +\frac{5}{3}+\frac{1}{5}-\frac{8}{15} \\ =\frac{1}{4}{\mathrm{x}}^3 - \frac{13}{20}{\mathrm{x}}^2 -\frac{5}{8}\mathrm{x} +\frac{4}{3} \end{gathered}$$x+117y+94xy,−32x−53y−9

Question 2:

Subtract:

(i) −8xy from 7xy
(ii) x² from − 3x²
(iii) (x − y) from (4y − 5x)
(iv) (a² + b² − 2ab) from (a² + b² + 2ab)
(v) (x² − y²) from (2x² − 3y² + 6xy)
(vi) (x − y + 3z) from (2z − x − 3y)

Answer 2:

(i) 7xy- (-8xy)
= 7xy+ 8xy
= 15xy

(ii)  - 3x² - x²
= -4x<sup>2

(iii)  (4y - 5x) - (x- y)
= 4y - 5x - x + y
= 5y - 6x

(iv)</sup> (a² + b² + 2ab) - (a² + b² − 2ab)
= $a^2 -a^2 +b^2 -b^2 + 2ab + 2ab$    (Collecting like terms and adding them)
= 4ab

(v) (2x² − 3y² + 6xy) -  (x² − y²)
$$\begin{gathered} 2x^2 - x^2 - 3y^2 + y^2 +6xy \\ =x^2 -2y^2 +6xy \end{gathered}$$     (Collecting like terms and adding them)

(vi)  (2z -x -3y) - (x - y +3z)
= 2z -3z -x -x -3y +y       (Collecting like terms and adding them)
= -z -2x - 2y 

Question 3:

Subtract (2a − 3b + 4c) from the sum of (a + 3b − 4c), (4a − b + 9c) and (−2b + 3c − a).

Answer 3:

(a + 3b − 4c) + (4a − b + 9c) + (−2b + 3c − a)
= a + 4a - a + 3b -b -2b -4c +9c + 3c
= 4a + 8c

Now,  (4a + 8c ) - (2a − 3b + 4c)
= 4a - 2a + 3b + 8c - 4c
= 2a + 3b + 4c

Question 4:

Subtract the sum of (8m − 7n + 6p²) and (−3m − 4n − p²) from the sum of (2m + 4n − 3p²) and (− m − n − p²).

Answer 4:

(8m − 7n + 6p²) + (−3m − 4n − p²)
 $$\begin{gathered} =8\mathrm{m} - 3\mathrm{m} - 7\mathrm{n} -4\mathrm{n} + 6{\mathrm{p}}^2 - \mathrm{p}{ }^2 \\ =5\mathrm{m} -11\mathrm{n} +5{\mathrm{p}}^2 \end{gathered}$$

(2m + 4n − 3p²) + (− m − n − p²).
 $$\begin{gathered} = 2\mathrm{m} -\mathrm{m} + 4\mathrm{n} -\mathrm{n} -3{\mathrm{p}}^2 -{\mathrm{p}}^2 \\ =\mathrm{m} + 3\mathrm{n} -4{\mathrm{p}}^2 \end{gathered}$$


$$\begin{gathered} \mathrm{Now}, (\mathrm{m}+ 3\mathrm{n} - 4{\mathrm{p}}^2) - (5\mathrm{m} -11\mathrm{n} +5{\mathrm{p}}^2) \\ =-4\mathrm{m} +14\mathrm{n} -9{\mathrm{p}}^2 \end{gathered}$$

Question 5:

Subtract the sum of (8a − 6a² + 9) and (−10a − 8 + 8a²) from −3.

Answer 5:

(8a − 6a² + 9)+  (−10a − 8 + 8a²)

Collecting like terms and adding them:

 $$\begin{gathered} 8\mathrm{a} -10\mathrm{a} -6{\mathrm{a}}^2 + 8{\mathrm{a}}^2 +9 -8 \\ = -2\mathrm{a} + 2{\mathrm{a}}^2 + 1 \\ \mathrm{Now}, -3 -(-2\mathrm{a} + 2{\mathrm{a}}^2 + 1) \\ = 2\mathrm{a} - 2{\mathrm{a}}^2 -4 \end{gathered}$$

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Question 6:

Simplify:

(i) (5x −9y) − (−7x + y)
(ii) $\left(x^2-x\right)-\frac{1}{2}\left(x-3+3x^2\right)$
(iii) [7 − 2x + 5y − (x − y)] − (5x + 3y − 7)
(iv) $\left(\frac{1}{3}{y}^2-\frac{4}{7}y+5\right)-\left(\frac{2}{7}y-\frac{2}{3}{y}^2+2\right)-\left(\frac{1}{7}y-3+2y^2\right)$

Answer 6:

Collecting like terms and adding them:

(i)  5x + 7x - 9y -y
= 12x -10y

(ii)
$$\begin{gathered} x^2 -\frac{3}{2}{x}^2 -x -\frac{1}{2}x +\frac{3}{2} \\ =-\frac{1}{2}{x}^2 - \frac{3}{2}x + \frac{3}{2} \end{gathered}$$

(iii)  7 + 7 - 2x -x - 5x + 5y + y - 3y
= 14 - 8x -3y

(iv) 
$$\begin{gathered} \frac{1}{3}{\mathrm{y}}^2 +\frac{2}{3}{\mathrm{y}}^2 -2{\mathrm{y}}^2 - \frac{4}{7}\mathrm{y} - \frac{2}{7}\mathrm{y} -\frac{1}{7}\mathrm{y} +5 -2 +3 \\ = - {\mathrm{y}}^2 - \mathrm{y} +6 \end{gathered}$$