Simplify:

$(i) (7 x - 8) (3 x + 2) (i i) (p x - q) (p x + q) (i i i) (5 a + 5 b - c) (2 b - 3 c) (i v) (4 x - 5 y) (5 x - 4 y) (v) (3 y + 4 z) (3 y - 4 z) + (2 y + 7 z) (y + z)$

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Solution

$(i) (7 x - 8) (3 x + 2) = 7 x (3 x + 2) - 8 (3 x + 2) = 21 x^{2} + 14 x - 24 x - 16 = 21 x^{2} - 10 x - 16 (i i) (p x - q) (p x + q) = p x (p x + q) - q (p x + q) = p^{2} x^{2} + p x q - p q x - q^{2} = p^{2} x^{2} - q^{2} (i i i) (5 a + 5 b - c) (2 b - 3 c) = 5 a (2 b - 3 c) + 5 b (2 b - 3 c) - c (2 b - 3 c) = 10 a b - 15 a c + 10 b^{2} - 15 b c - 2 b c + 3 c^{2} = 10 a b + 10 b^{2} - 17 b c - 15 a c + 3 c^{2} (i v) (4 x - 5 y) (5 x - 4 y) = 4 x (5 x - 4 y) - 5 y (5 x - 4 y) = 20 x^{2} - 16 x y - 25 x y + 20 y^{2} = 20 x^{2} - 41 x y + 20 y^{2} (v) (3 y + 4 z) (3 y - 4 z) + (2 y + 7 z) (y + z) = 3 y (3 y - 4 z) + 4 z (3 y - 4 z) + 2 y (y + z) + 7 z (y + z) = 9 y^{2} - 12 y z + 12 y z - 16 z^{2} + 2 y^{2} + 2 y z + 7 y z + 7 z^{2} = (9 + 2) y^{2} + (- 12 + 12 + 2 + 7) y z + (- 16 + 7) z^{2} = 11 y^{2} + 9 y z - 9 z^{2}$

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(3 y + 4 z) (3 y - 4 z) + (2 y + 7 z) (y + z)

11 y^{2} + 9 y z - 9 z^{2}

\begin{matrix} x - 3 y - 2 z 5 x + 7 y - z - 7 x - 2 y + 4 z \end{matrix}

x - 3 y - 2 z, 5 x + 7 y - z, - 7 x - 2 y + 4 z

7 x + 6 y + 4 z = 122

4 x + 5 y + 3 z = 88

9 x + 2 y + z = 78

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