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To determine which of the following equations are linear, we need to check if each equation fits the definition of a linear equation. 

### Definition of a Linear Equation:
A linear equation is an equation where:
1. The variables are only raised to the first power.
2. There are no products of variables (e.g., no \(xy\)).
3. There are no square roots or any other non-linear functions of the variables.

Let's go through the list one by one.

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1. **\(-3y = 0\)**  
   Linear? **Yes** (Only a first-degree term for \(y\)).

2. **\(\sqrt{2}x + 6y = -25\)**  
   Linear? **Yes** (There are no powers greater than 1, and \(\sqrt{2}\) is just a constant).

3. **\(-5x + 24y = 200\)**  
   Linear? **Yes** (Both \(x\) and \(y\) are of degree 1).

4. **\(-x^2 + 3y = 40\)**  
   Linear? **No** (\(x^2\) makes it non-linear).

5. **\(2x - 3xy + 5y = 10\)**  
   Linear? **No** (The term \(xy\) is a product of two variables).

6. **\(\sqrt{4x} - 3y = -45\)**  
   Linear? **No** (The square root of \(x\) makes it non-linear).

7. **\(u - 3v = 20\)**  
   Linear? **Yes** (Both variables \(u\) and \(v\) are first degree).

8. **\(r/2 + s/5 = \frac{3}{7}\)**  
   Linear? **Yes** (Both \(r\) and \(s\) are first-degree terms).

9. **\(m/2 + (2m - 3n)/5 = 0\)**  
   Linear? **Yes** (Both \(m\) and \(n\) are first-degree terms).

10. **\((x + 2y)/3 - 3x/4 = 2x - 5y\)**  
   Linear? **Yes** (All terms are first-degree and can be simplified).

11. **\(40 - 3y = \sqrt{24}\)**  
   Linear? **Yes** (Both terms involve only constants and \(y\)).

12. **\(0.0003x - 2.3245y = x + y - 3.2543\)**  
   Linear? **Yes** (All terms are first-degree).

13. **\(2x_1 - 3x_2 + x_3 = 0\)**  
   Linear? **Yes** (All terms are first-degree).

14. **\((x_1 - 3x_2 + 5x_3 - 2x_4 + x_5)/25 = 300\)**  
   Linear? **Yes** (All terms are first-degree).

15. **\((x_1 + x_2 - x_3 x_1) = 5\)**  
   Linear? **No** (The product \(x_3 x_1\) makes it non-linear).

16. **\(3x_2 - 4x_1 = 5x_3 + 2x_2 - x_4 + 36\)**  
   Linear? **Yes** (All terms are first-degree).

17. **\(\sqrt{x^2 + 2xy + y^2} = 25\)**  
   Linear? **No** (Square root and powers higher than 1 make it non-linear).

18. **\((2x_1 - 3x_2 + x_3)/4 = (x_2 - 2x_4)/5 + 90\)**  
   Linear? **Yes** (All terms are first-degree).

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### Summary of Linear Equations:
- Linear: 1, 2, 3, 7, 8, 9, 10, 11, 12, 13, 14, 16, 18.
- Non-linear: 4, 5, 6, 15, 17.

Would you like further details on any of these equations or how I determined their linearity?

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**5 Relative Questions:**
1. What are the characteristics of a linear equation in three variables?
2. Can linear equations have irrational numbers like \(\sqrt{2}\)?
3. How do products of variables affect the classification of linearity?
4. Why are square roots of variables considered non-linear?
5. Can an equation with fractional coefficients still be linear?

**Tip:** Always look for variables being multiplied or raised to powers to quickly identify non-linear equations.

Determine which of the following equations are linear.

This problem involves identifying linear equations based on algebraic properties, including first-degree terms, products of variables, and non-linear functions like square roots. Suitable for students in Grades 9-12.

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