Algebraic equations and their solutions presented in a structured format.

It looks like you've uploaded an image containing a set of rational equations and their corresponding solutions. Let's go through each equation step-by-step to verify the correctness of the given solutions and explain how to solve them properly.

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🔹 General Approach for Solving Rational Equations

To solve equations of the form:
$$
\frac{A}{B} = \frac{C}{D}
$$
We can use cross-multiplication:
$$
A \cdot D = B \cdot C
$$
Then solve the resulting linear or simple algebraic equation for $ x $. Always check that the solution doesn't make any denominator zero (undefined).

Let’s analyze each row:

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✔ 1. $\frac{x}{7} = \frac{3}{2}$

Cross-multiply:
$$
x \cdot 2 = 7 \cdot 3 \Rightarrow 2x = 21 \Rightarrow x = \frac{21}{2} = 10.5
$$

✔ Correct: $ x = 10.5 $ or $ \frac{21}{2} $

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✔ 2. $\frac{2x}{7} = \frac{3}{2}$

Cross-multiply:
$$
2x \cdot 2 = 7 \cdot 3 \Rightarrow 4x = 21 \Rightarrow x = \frac{21}{4} = 5.25
$$

✔ Correct: $ x = 5.25 $ or $ \frac{21}{4} $

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✔ 3. $\frac{2x}{7} = \frac{3}{14}$

Cross-multiply:
$$
2x \cdot 14 = 7 \cdot 3 \Rightarrow 28x = 21 \Rightarrow x = \frac{21}{28} = \frac{3}{4} = 0.75
$$

✔ Correct: $ x = 0.75 $ or $ \frac{3}{4} $

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✔ 4. $\frac{2x}{14} = \frac{3}{14}$

Simplify both sides:
$$
\frac{x}{7} = \frac{3}{14}
$$
Multiply both sides by 14:
$$
2x = 3 \Rightarrow x = \frac{3}{2} = 1.5
$$

✔ Correct: $ x = 1.5 $ or $ \frac{3}{2} $

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✔ 5. $\frac{2x}{14} = \frac{3}{7}$

Simplify left side: $ \frac{x}{7} = \frac{3}{7} $

Multiply both sides by 7:
$$
x = 3
$$

✔ Correct: $ x = 3 $

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✔ 6. $\frac{14}{2x} = \frac{7}{3}$

Cross-multiply:
$$
14 \cdot 3 = 7 \cdot 2x \Rightarrow 42 = 14x \Rightarrow x = 3
$$

✔ Correct: $ x = 3 $

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Now let's look at the right column:

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✔ 7. $\frac{14}{2x + 1} = \frac{7}{3}$

Cross-multiply:
$$
14 \cdot 3 = 7 \cdot (2x + 1) \Rightarrow 42 = 14x + 7
\Rightarrow 35 = 14x \Rightarrow x = \frac{35}{14} = \frac{5}{2} = 2.5
$$

✔ Correct: $ x = 2.5 $ or $ \frac{5}{2} $

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✔ 8. $\frac{14}{2x - 1} = \frac{7}{3}$

Cross-multiply:
$$
14 \cdot 3 = 7 \cdot (2x - 1) \Rightarrow 42 = 14x - 7
\Rightarrow 49 = 14x \Rightarrow x = \frac{49}{14} = \frac{7}{2} = 3.5
$$

✔ Correct: $ x = 3.5 $ or $ \frac{7}{2} $

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✔ 9. $\frac{7}{2x - 1} = \frac{7}{3}$

Since numerators are equal, set denominators equal:
$$
2x - 1 = 3 \Rightarrow 2x = 4 \Rightarrow x = 2
$$

✔ Correct: $ x = 2 $

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✔ 10. $\frac{2x - 1}{7} = \frac{7}{3}$

Cross-multiply:
$$
(2x - 1) \cdot 3 = 7 \cdot 7 \Rightarrow 6x - 3 = 49
\Rightarrow 6x = 52 \Rightarrow x = \frac{52}{6} = \frac{26}{3} \approx 8.\overline{6}
$$

✔ Correct: $ x = 8.\dot{6} $ or $ \frac{26}{3} $

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✔ 11. $\frac{2x - 1}{6} = \frac{7}{3}$

Cross-multiply:
$$
(2x - 1) \cdot 3 = 6 \cdot 7 \Rightarrow 6x - 3 = 42
\Rightarrow 6x = 45 \Rightarrow x = \frac{45}{6} = \frac{15}{2} = 7.5
$$

✔ Correct: $ x = 7.5 $ or $ \frac{15}{2} $

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✔ 12. $\frac{1 - 2x}{6} = \frac{7}{3}$

Cross-multiply:
$$
(1 - 2x) \cdot 3 = 6 \cdot 7 \Rightarrow 3 - 6x = 42
\Rightarrow -6x = 39 \Rightarrow x = -\frac{39}{6} = -\frac{13}{2} = -6.5
$$

✔ Correct: $ x = -6.5 $ or $ -\frac{13}{2} $

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✔ Final Summary:

All the given solutions are correct! Each equation was solved using proper algebraic techniques — primarily cross-multiplication and simplification.

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📌 Key Takeaways:

- Cross-multiplication is valid when both sides are fractions.
- Always simplify expressions before solving.
- Check for extraneous solutions (e.g., values that make a denominator zero), though none occurred here.
- Fractions can be expressed as decimals or mixed numbers, but exact fractions are preferred in math.

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✔ Conclusion: All solutions provided in the image are correct. Well done!

Let me know if you'd like this turned into a worksheet or need help solving similar problems.