Fundamental Concepts of Algebra — Exercise 13(F)

Enclose the given term in bracket as required:

$x - y - z = x$ − (.......................)

Answer

$x - y - z = x - (\bm{y + z})$

Enclose the given term in bracket as required:

$x^2 - xy^2 - 2xy - y^2 = x^2$ − (.......................)

Answer

$x^2 - xy^2 - 2xy - y^2 = x^2 - (\bm{xy^2 + 2xy + y^2})$

Enclose the given term in bracket as required:

$4a - 9 + 2b - 6 = 4a$ − (.......................)

Answer

$4a - 9 + 2b - 6 = 4a - (\bm{9 - 2b + 6})$

Enclose the given term in bracket as required:

$x^2 - y^2 + z^2 + 3x - 2y = x^2$ − (.......................)

Answer

$x^2 - y^2 + z^2 + 3x - 2y = x^2 - (\bm{y^2 - z^2 - 3x + 2y})$

Enclose the given term in bracket as required:

$-2a^2 + 4ab - 6a^2b^2 + 8ab^2 = -2a$ (.......................)

Answer

$-2a^2 + 4ab - 6a^2b^2 + 8ab^2 = -2a(\bm{a - 2b + 3ab^2 - 4b^2})$

Simplify:

$2x - (x + 2y - z)$

Answer

Solving,

$\Rightarrow 2x - (x + 2y - z)\\[1em] \Rightarrow 2x - x - 2y + z\\[1em] \Rightarrow x - 2y + z$

Hence, $\bm{2x - (x + 2y - z) = x - 2y + z}$.

Simplify:

$p + q - (p - q) + (2p - 3q)$

Answer

Solving,

$\Rightarrow p + q - (p - q) + (2p - 3q)\\[1em] \Rightarrow p + q - p + q + 2p - 3q\\[1em] \Rightarrow (1 - 1 + 2)p + (1 + 1 - 3)q\\[1em] \Rightarrow 2p - q$

Hence, $\bm{p + q - (p - q) + (2p - 3q) = 2p - q}$.

Simplify:

$9x - (- 4x + 5)$

Answer

Solving,

$\Rightarrow 9x - (-4x + 5)\\[1em] \Rightarrow 9x + 4x - 5\\[1em] \Rightarrow 13x - 5$

Hence, $\bm{9x - (-4x + 5) = 13x - 5}$.

Simplify:

$6a - (- 5a - 8b) + (3a + b)$

Answer

Solving,

$\Rightarrow 6a - (-5a - 8b) + (3a + b)\\[1em] \Rightarrow 6a + 5a + 8b + 3a + b\\[1em] \Rightarrow (6 + 5 + 3)a + (8 + 1)b\\[1em] \Rightarrow 14a + 9b$

Hence, $\bm{6a - (-5a - 8b) + (3a + b) = 14a + 9b}$.

Simplify:

$(p - 2q) - (3q - r)$

Answer

Solving,

$\Rightarrow (p - 2q) - (3q - r)\\[1em] \Rightarrow p - 2q - 3q + r\\[1em] \Rightarrow p - 5q + r$

Hence, $\bm{(p - 2q) - (3q - r) = p - 5q + r}$.

Simplify:

$9a (2b - 3a + 7c)$

Answer

Solving,

$\Rightarrow 9a(2b - 3a + 7c)\\[1em] \Rightarrow 9a \times 2b - 9a \times 3a + 9a \times 7c\\[1em] \Rightarrow 18ab - 27a^2 + 63ac$

Hence, $\bm{9a(2b - 3a + 7c) = 18ab - 27a^2 + 63ac}$.

Simplify:

$-5m (- 2m + 3n - 7p)$

Answer

Solving,

$\Rightarrow -5m(-2m + 3n - 7p)\\[1em] \Rightarrow -5m \times (-2m) - 5m \times 3n - 5m \times (-7p)\\[1em] \Rightarrow 10m^2 - 15mn + 35mp$

Hence, $\bm{-5m(-2m + 3n - 7p) = 10m^2 - 15mn + 35mp}$.

Simplify:

$-2x (x + y) + x^2$

Answer

Solving,

$\Rightarrow -2x(x + y) + x^2\\[1em] \Rightarrow -2x^2 - 2xy + x^2\\[1em] \Rightarrow -x^2 - 2xy$

Hence, $\bm{-2x(x + y) + x^2 = -x^2 - 2xy}$.

Simplify:

$b\left(2b - \dfrac{1}{b}\right) - 2b\left(b - \dfrac{1}{b}\right)$

Answer

Solving,

$\Rightarrow b\left(2b - \dfrac{1}{b}\right) - 2b\left(b - \dfrac{1}{b}\right)\\[1em] \Rightarrow 2b^2 - 1 - 2b^2 + 2\\[1em] \Rightarrow 1$

Hence, $\bm{b\left(2b - \dfrac{1}{b}\right) - 2b\left(b - \dfrac{1}{b}\right) = 1}$.

Simplify:

$8 (2a + 3b - c) - 10 (a + 2b + 3c)$

Answer

Solving,

$\Rightarrow 8(2a + 3b - c) - 10(a + 2b + 3c)\\[1em] \Rightarrow 16a + 24b - 8c - 10a - 20b - 30c\\[1em] \Rightarrow (16 - 10)a + (24 - 20)b + (-8 - 30)c\\[1em] \Rightarrow 6a + 4b - 38c$

Hence, $\bm{8(2a + 3b - c) - 10(a + 2b + 3c) = 6a + 4b - 38c}$.

Simplify:

$a\left(a + \dfrac{1}{a}\right) - b\left(b - \dfrac{1}{b}\right) - c\left(c + \dfrac{1}{c}\right)$

Answer

Solving,

$\Rightarrow a\left(a + \dfrac{1}{a}\right) - b\left(b - \dfrac{1}{b}\right) - c\left(c + \dfrac{1}{c}\right)\\[1em] \Rightarrow (a^2 + 1) - (b^2 - 1) - (c^2 + 1)\\[1em] \Rightarrow a^2 + 1 - b^2 + 1 - c^2 - 1\\[1em] \Rightarrow a^2 - b^2 - c^2 + 1$

Hence, $\bm{a\left(a + \dfrac{1}{a}\right) - b\left(b - \dfrac{1}{b}\right) - c\left(c + \dfrac{1}{c}\right) = a^2 - b^2 - c^2 + 1}$.

Simplify:

$5x (2x + 3y) - 2x (x - 9y)$

Answer

Solving,

$\Rightarrow 5x(2x + 3y) - 2x(x - 9y)\\[1em] \Rightarrow 10x^2 + 15xy - 2x^2 + 18xy\\[1em] \Rightarrow 8x^2 + 33xy$

Hence, $\bm{5x(2x + 3y) - 2x(x - 9y) = 8x^2 + 33xy}$.

Simplify:

$a + (b + c - d)$

Answer

Solving,

$\Rightarrow a + (b + c - d)\\[1em] \Rightarrow a + b + c - d$

Hence, $\bm{a + (b + c - d) = a + b + c - d}$.

Simplify:

$5 - 8x - (6 - x)$

Answer

Solving,

$\Rightarrow 5 - 8x - (6 - x)\\[1em] \Rightarrow 5 - 8x - 6 + x\\[1em] \Rightarrow -1 - 7x$

Hence, $\bm{5 - 8x - (6 - x) = -1 - 7x}$.

Simplify:

$2a + (b - \overline{a - b})$

Answer

Solving,

$\Rightarrow 2a + (b - \overline{a - b})\\[1em] \Rightarrow 2a + (b - a + b)\\[1em] \Rightarrow 2a + b - a + b\\[1em] \Rightarrow a + 2b$

Hence, $\bm{2a + (b - \overline{a - b}) = a + 2b}$.

Simplify:

$3x + [4x - (6x - 3)]$

Answer

Solving,

$\Rightarrow 3x + [4x - (6x - 3)]\\[1em] \Rightarrow 3x + [4x - 6x + 3] \\[1em] \Rightarrow 3x + 4x - 6x + 3 \\[1em] \Rightarrow x + 3$

Hence, $\bm{3x + [4x - (6x - 3)] = x + 3}$.

Simplify:

$5b - \big\lbrace 6a + (8 - b - a)\big\rbrace$

Answer

Solving,

$\Rightarrow 5b - \big\lbrace 6a + (8 - b - a)\big\rbrace\\[1em] \Rightarrow 5b - \big\lbrace 6a + 8 - b - a\big\rbrace \\[1em] \Rightarrow 5b - \big\lbrace 5a + 8 - b\big\rbrace\\[1em] \Rightarrow 5b - 5a - 8 + b \\[1em] \Rightarrow 6b - 5a - 8$

Hence, $\bm{5b - \big\lbrace6a + (8 - b - a)\big\rbrace = 6b - 5a - 8}$.

Simplify:

$2x - [5y - (3x - y) + x]$

Answer

Solving,

$\Rightarrow 2x - [5y - (3x - y) + x]\\[1em] \Rightarrow 2x - [5y - 3x + y + x] \\[1em] \Rightarrow 2x - [6y - 2x]\\[1em] \Rightarrow 2x - 6y + 2x \\[1em] \Rightarrow 4x - 6y$

Hence, $\bm{2x - [5y - (3x - y) + x] = 4x - 6y}$.

Simplify:

$6a - 3 (a + b - 2)$

Answer

Solving,

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

Hence, $\bm{6a - 3(a + b - 2) = 3a - 3b + 6}$.

Simplify:

$8[m + 2n - p - 7(2m - n + 3p)]$

Answer

Solving,

$\Rightarrow 8[m + 2n - p - 7(2m - n + 3p)]\\[1em] \Rightarrow 8[m + 2n - p - 14m + 7n - 21p] \\[1em] \Rightarrow 8[(1 - 14)m + (2 + 7)n + (-1 - 21)p]\\[1em] \Rightarrow 8[-13m + 9n - 22p]\\[1em] \Rightarrow -104m + 72n - 176p$

Hence, $\bm{8[m + 2n - p - 7(2m - n + 3p)] = -104m + 72n - 176p}$.

Simplify:

$\big\lbrace 9 - (4p - 6q)\big\rbrace - \big\lbrace 3q - (5p - 10)\big\rbrace$

Answer

Solving,

$\Rightarrow \big\lbrace 9 - (4p - 6q)\big\rbrace - \big\lbrace 3q - (5p - 10)\big\rbrace \\[1em] \Rightarrow \big\lbrace 9 - 4p + 6q\big\rbrace - \big\lbrace 3q - 5p + 10\big\rbrace \\[1em] \Rightarrow 9 - 4p + 6q - 3q + 5p - 10 \\[1em] \Rightarrow (-4 + 5)p + (6 - 3)q + (9 - 10) \\[1em] \Rightarrow p + 3q - 1$

Hence, $\bm{ \big\lbrace 9 - (4p - 6q)\big\rbrace - \big\lbrace 3q - (5p - 10)\big\rbrace = p + 3q - 1}$.

Simplify:

$2[a - 3 \big\lbrace a + 5(a - 2) + 7\big\rbrace]$

Answer

Solving,

$\Rightarrow 2[a - 3 \big\lbrace a + 5(a - 2) + 7\big\rbrace]\\[1em] \Rightarrow 2[a - 3 \big\lbrace a + 5a - 10 + 7\big\rbrace] \\[1em] \Rightarrow 2[a - 3 \big\lbrace 6a - 3\big\rbrace]\\[1em] \Rightarrow 2[a - 18a + 9] \\[1em] \Rightarrow 2[-17a + 9]\\[1em] \Rightarrow -34a + 18$

Hence, $\bm{2[a - 3 \big\lbrace a + 5(a - 2) + 7\big\rbrace] = -34a + 18}$.

Simplify:

$5a - [6a - \big\lbrace 9a - (10a - \overline{4a - 3a})\big\rbrace]$

Answer

Solving,

$\Rightarrow 5a - [6a - \big\lbrace 9a - (10a - \overline{4a - 3a})\big\rbrace]\\[1em] \Rightarrow 5a - [6a - \big\lbrace 9a - (10a - a)\big\rbrace] \\[1em] \Rightarrow 5a - [6a - \big\lbrace 9a - 9a\big\rbrace] \\[1em] \Rightarrow 5a - [6a - 0] \\[1em] \Rightarrow 5a - 6a\\[1em] \Rightarrow -a$

Hence, $\bm{5a - [6a - \big\lbrace 9a - (10a - \overline{4a - 3a})\big\rbrace] = -a}$.

Simplify:

$9x + 5 - [4x - \big\lbrace 3x - 2(4x - 3)\big\rbrace]$

Answer

Solving,

$\Rightarrow 9x + 5 - [4x - \big\lbrace 3x - 2(4x - 3)\big\rbrace]\\[1em] \Rightarrow 9x + 5 - [4x - \big\lbrace 3x - 8x + 6\big\rbrace] \\[1em] \Rightarrow 9x + 5 - [4x - \big\lbrace -5x + 6\big\rbrace]\\[1em] \Rightarrow 9x + 5 - [4x + 5x - 6] \\[1em] \Rightarrow 9x + 5 - [9x - 6]\\[1em] \Rightarrow 9x + 5 - 9x + 6 \\[1em] \Rightarrow 11$

Hence, $\bm{9x + 5 - [4x - \big\lbrace 3x - 2(4x - 3)\big\rbrace] = 11}$.

Simplify:

$(x + y - z)x + (z + x - y)y - (x + y - z)z$

Answer

Solving,

$\Rightarrow (x + y - z)x + (z + x - y)y - (x + y - z)z\\[1em] \Rightarrow (x^2 + xy - xz) + (yz + xy - y^2) - (xz + yz - z^2)\\[1em] \Rightarrow x^2 + xy - xz + yz + xy - y^2 - xz - yz + z^2\\[1em] \Rightarrow x^2 - y^2 + z^2 + (1 + 1)xy + (-1 - 1)xz + (1 - 1)yz\\[1em] \Rightarrow x^2 - y^2 + z^2 + 2xy - 2xz$

Hence, $\bm{(x + y - z)x + (z + x - y)y - (x + y - z)z = x^2 - y^2 + z^2 + 2xy - 2xz}$.

Simplify:

$-1[a - 3 \big\lbrace b - 4(\overline{a - b} - 8) + 4a\big\rbrace + 10]$

Answer

Solving,

$\Rightarrow -1[a - 3 \big\lbrace b - 4(\overline{a - b} - 8) + 4a\big\rbrace + 10]\\[1em] \Rightarrow -1[a - 3 \big\lbrace b - 4(a - b - 8) + 4a\big\rbrace + 10] \\[1em] \Rightarrow -1[a - 3 \big\lbrace b - 4a + 4b + 32 + 4a\big\rbrace + 10] \\[1em] \Rightarrow -1[a - 3 \big\lbrace 5b + 32\big\rbrace + 10]\\[1em] \Rightarrow -1[a - 15b - 96 + 10] \\[1em] \Rightarrow -1[a - 15b - 86]\\[1em] \Rightarrow -a + 15b + 86$

Hence, $\bm{-1[a - 3 \big\lbrace b - 4(\overline{a - b} - 8) + 4a\big\rbrace + 10] = -a + 15b + 86}$.

Simplify:

$p^2 - [x^2 - \big\lbrace x^2 - (q^2 - \overline{x^2 - q^2}) - 2y^2\big\rbrace]$

Answer

Solving,

$\Rightarrow p^2 - [x^2 - \big\lbrace x^2 - (q^2 - \overline{x^2 - q^2}) - 2y^2\big\rbrace]\\[1em] \Rightarrow p^2 - [x^2 - \big\lbrace x^2 - (q^2 - x^2 + q^2) - 2y^2\big\rbrace] \\[1em] \Rightarrow p^2 - [x^2 - \big\lbrace x^2 - (2q^2 - x^2) - 2y^2\big\rbrace]\\[1em] \Rightarrow p^2 - [x^2 - \big\lbrace x^2 - 2q^2 + x^2 - 2y^2\big\rbrace] \\[1em] \Rightarrow p^2 - [x^2 - \big\lbrace 2x^2 - 2q^2 - 2y^2\big\rbrace]\\[1em] \Rightarrow p^2 - [x^2 - 2x^2 + 2q^2 + 2y^2] \\[1em] \Rightarrow p^2 - [-x^2 + 2q^2 + 2y^2] \\[1em] \Rightarrow p^2 + x^2 - 2q^2 - 2y^2$

Hence, $\bm{p^2 - [x^2 - \big\lbrace x^2 - (q^2 - \overline{x^2 - q^2}) - 2y^2\big\rbrace] = p^2 + x^2 - 2q^2 - 2y^2}$.

Simplify:

$10 - \big\lbrace 4a - (7 - \overline{a - 5}) - (5a - \overline{1 + a})\big\rbrace$

Answer

Solving,

$\Rightarrow 10 - \big\lbrace 4a - (7 - \overline{a - 5}) - (5a - \overline{1 + a})\big\rbrace\\[1em] \Rightarrow 10 - \big\lbrace 4a - (7 - a + 5) - (5a - 1 - a)\big\rbrace \\[1em] \Rightarrow 10 - \big\lbrace 4a - (12 - a) - (4a - 1)\big\rbrace\\[1em] \Rightarrow 10 - \big\lbrace 4a - 12 + a - 4a + 1\big\rbrace \\[1em] \Rightarrow 10 - \big\lbrace a - 11\big\rbrace\\[1em] \Rightarrow 10 - a + 11 \\[1em] \Rightarrow 21 - a$

Hence, $\bm{10 - \big\lbrace 4a - (7 - \overline{a - 5}) - (5a - \overline{1 + a})\big\rbrace = 21 - a}$.

Simplify:

$7a - [8a - \big\lbrace 11a - (12a - \overline{6a - 5a})\big\rbrace]$

Answer

Solving,

$\Rightarrow 7a - [8a - \big\lbrace 11a - (12a - \overline{6a - 5a})\big\rbrace]\\[1em] \Rightarrow 7a - [8a - \big\lbrace 11a - (12a - a)\big\rbrace] \\[1em] \Rightarrow 7a - [8a - \big\lbrace 11a - 11a\big\rbrace] \\[1em] \Rightarrow 7a - [8a - 0] \\[1em] \Rightarrow 7a - 8a\\[1em] \Rightarrow -a$

Hence, $\bm{7a - [8a - \big\lbrace 11a - (12a - \overline{6a - 5a})\big\rbrace] = -a}$.

Simplify:

$8x - [4y - \big\lbrace 4x + (2x - \overline{2y - 2x})\big\rbrace]$

Answer

Solving,

$\Rightarrow 8x - [4y - \big\lbrace 4x + (2x - \overline{2y - 2x})\big\rbrace]\\[1em] \Rightarrow 8x - [4y - \big\lbrace 4x + (2x - 2y + 2x)\big\rbrace] \\[1em] \Rightarrow 8x - [4y - \big\lbrace 4x + (4x - 2y)\big\rbrace]\\[1em] \Rightarrow 8x - [4y - \big\lbrace 4x + 4x - 2y\big\rbrace] \\[1em] \Rightarrow 8x - [4y - \big\lbrace 8x - 2y\big\rbrace]\\[1em] \Rightarrow 8x - [4y - 8x + 2y] \\[1em] \Rightarrow 8x - [6y - 8x]\\[1em] \Rightarrow 8x - 6y + 8x \\[1em] \Rightarrow 16x - 6y$

Hence, $\bm{8x - [4y - \big\lbrace 4x + (2x - \overline{2y - 2x})\big\rbrace] = 16x - 6y}$.

Simplify:

$x - (3y - \overline{4z - 3x} + 2z - \overline{5y - 7x})$

Answer

Solving,

$\Rightarrow x - (3y - \overline{4z - 3x} + 2z - \overline{5y - 7x})\\[1em] \Rightarrow x - (3y - 4z + 3x + 2z - 5y + 7x) \\[1em] \Rightarrow x - [(3 + 7)x + (3 - 5)y + (-4 + 2)z]\\[1em] \Rightarrow x - (10x - 2y - 2z)\\[1em] \Rightarrow x - 10x + 2y + 2z \\[1em] \Rightarrow -9x + 2y + 2z$

Hence, $\bm{x - (3y - \overline{4z - 3x} + 2z - \overline{5y - 7x}) = -9x + 2y + 2z}$.