Let's solve each equation step by step:

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4. $\frac{x}{7} = \frac{3}{21}$

First, simplify the right-hand side:

$\frac{3}{21} = \frac{1}{7}$

Now we have:

$\frac{x}{7} = \frac{1}{7}$

Since both sides have the same denominator, $x = 1$.

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9. $\frac{27}{m} = \frac{9}{2}$

To solve for $m$, cross-multiply:

$27 \cdot 2 = 9 \cdot m$

This gives:

$54 = 9m$

Now, divide both sides by 9:

$m = \frac{54}{9} = 6$

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5. $\frac{3}{5} = \frac{6x}{10}$

Simplify the right-hand side:

$\frac{6x}{10} = \frac{3x}{5}$

Now, since both sides have the same denominator:

$3 = 3x$

Divide both sides by 3:

$x = 1$

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10. $\frac{3k}{5} = \frac{k+5}{10}$

Cross-multiply:

$3k \cdot 10 = 5 \cdot (k + 5)$

Simplify both sides:

$30k = 5(k + 5)$

Distribute on the right-hand side:

$30k = 5k + 25$

Move $5k$ to the left-hand side:

$30k - 5k = 25$

Simplify:

$25k = 25$

Divide both sides by 25:

$k = 1$

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11. $\frac{3x - 4}{5} - \frac{2x + 1}{20} = \frac{-1}{2} - x$

To solve this, first find a common denominator on the left-hand side. The common denominator between 5 and 20 is 20, so rewrite both fractions:

$\frac{3x - 4}{5} = \frac{4(3x - 4)}{20} = \frac{12x - 16}{20}$

Now, the equation becomes:

$\frac{12x - 16}{20} - \frac{2x + 1}{20} = \frac{-1}{2} - x$

Combine the fractions on the left-hand side:

$\frac{(12x - 16) - (2x + 1)}{20} = \frac{-1}{2} - x$

Simplify the numerator:

$\frac{12x - 16 - 2x - 1}{20} = \frac{-1}{2} - x$

This becomes:

$\frac{10x - 17}{20} = \frac{-1}{2} - x$

Now, multiply through by 20 to eliminate the denominator:

$10x - 17 = 10\left(\frac{-1}{2} - x\right)$

Distribute on the right-hand side:

$10x - 17 = -5 - 10x$

Move all terms involving $x$ to one side:

$10x + 10x = -5 + 17$

Simplify:

$20x = 12$

Divide by 20:

$x = \frac{12}{20} = \frac{3}{5}$

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12. $\frac{x}{x-2} + \frac{2}{x+2} - \frac{x^2 - 1}{x^2 - 4} = 0$

First, note that $x^2 - 4 = (x - 2)(x + 2)$, so rewrite the equation:

$\frac{x}{x - 2} + \frac{2}{x + 2} - \frac{x^2 - 1}{(x - 2)(x + 2)} = 0$

Now, express everything with a common denominator of $(x - 2)(x + 2)$:

$\frac{x(x + 2)}{(x - 2)(x + 2)} + \frac{2(x - 2)}{(x - 2)(x + 2)} - \frac{x^2 - 1}{(x - 2)(x + 2)} = 0$

Combine all the terms:

$\frac{x(x + 2) + 2(x - 2) - (x^2 - 1)}{(x - 2)(x + 2)} = 0$

Simplify the numerator:

$x(x + 2) = x^2 + 2x$ $2(x - 2) = 2x - 4$ $x^2 - 1 = x^2 - 1$

Now, the equation becomes:

$\frac{(x^2 + 2x) + (2x - 4) - (x^2 - 1)}{(x - 2)(x + 2)} = 0$

Simplify the numerator:

$x^2 + 2x + 2x - 4 - x^2 + 1 = 4x - 3$

Now, the equation is:

$\frac{4x - 3}{(x - 2)(x + 2)} = 0$

For the fraction to equal 0, the numerator must be 0:

$4x - 3 = 0$

Solve for $x$:

$4x = 3 \quad \Rightarrow \quad x = \frac{3}{4}$

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Let me know if you would like any details about these steps!

Related Questions:

1. How do you determine a common denominator in fraction equations?
2. What are the properties of cross-multiplication?
3. How can you simplify complex fractions?
4. What happens when you cross-multiply terms with variables?
5. How do you work with quadratic expressions in equations?

Tip:

Always simplify fractions early to make the equation easier to solve later on.