1.

[tex] \frac{ {x}^{2} }{9x} = \frac{x}{9} [/tex]

[tex] \frac{4ax}{4xy} = \frac{a}{y} [/tex]

2.

[tex] \frac{ x - 2}{y} + \frac{2}{y} = \frac{x - 2 + 2}{y} = \frac{x}{y} [/tex]

[tex] \frac{x}{7} - \frac{2}{y} = \frac{xy - 14}{7y} [/tex]

3.

1)

x + 2 ≠ 0

x ≠ -2

2)

[tex] {x}^{2} - 3x[/tex]

x(x - 3) ≠ 0

x ≠ 0

x - 3 ≠ 0

x ≠ 3

4.

[tex] \frac{42 {a}^{5} {b}^{2} }{7 {a}^{9} {b}^{3} } = \frac{6}{ {a}^{4}b } [/tex]

[tex] \frac{5x + 3 {x}^{2} }{7 {x}^{2} - 8x } = \frac{x(5 + 3x)}{x(7x - 8)} = \frac{5 + 3x}{7x - 8} [/tex]

[tex] \frac{4x - 5}{15 - 12x} = \frac{4x - 5}{ - 3(4x - 5)} = - \frac{1}{3} [/tex]

[tex] \frac{ {x}^{2} - 36 }{3x - 18} = \frac{(x - 6)(x + 6)}{3(x - 6)} = \frac{x + 6}{3} [/tex]

5.

[tex] \frac{10}{8 + x} - \frac{2 - x}{8 + x} = \frac{10 - 2 + x}{8 + x} = \frac{8 + x}{8 + x} = 1[/tex]

[tex] \frac{x + 2}{x - 5} + \frac{x - 12}{x - 5} = \frac{x + 2 + x - 12}{x - 5} = \frac{2x - 10}{x - 5} = \frac{2(x - 5)}{x - 5} = 2[/tex]

6.

[tex] \frac{18 {x}^{2} - y }{3x} - 6x = \frac{18 {x}^{2} - y - 18 {x}^{2} }{3x} = - \frac{y}{3x} [/tex]

[tex] \frac{3x}{ {y}^{2} - 9 {x}^{2} } + \frac{1}{3x - y} = \frac{3x}{(y - 3x)(y + 3x)} + \frac{1}{ - (y - 3x)} = \frac{3x - 1}{(y - 3x)(y + 3x)} = \frac{3x - 1}{ {y}^{2} - 9 {x}^{2} } [/tex]

[tex] \frac{4}{ {x}^{2} - 2x } - \frac{x}{x - 2} + \frac{x + 2}{x} = \frac{4}{x(x - 2)} - \frac{x}{x - 2} + \frac{x + 2}{x} = \frac{4 - {x}^{2} + (x - 2)(x + 2) }{x(x - 2)} = \frac{4 - {x}^{2} + {x}^{2} - 4 }{ {x}^{2} - 2x} = \frac{0}{ {x}^{2} - 2x } = 0[/tex]