https://file.aiquickdraw.com/mathbot/file/1727775522385dp26e7xu.jpg

Let's analyze each equation or system of equations and determine whether they are linear.

### Equation 3: 
\[
6x - \sqrt{3}y + \frac{1}{2}z = 0
\]
This equation is **linear** because all variables \(x\), \(y\), and \(z\) appear to the first power, and there are no products or functions (like squares, square roots, or trigonometric functions) involving the variables.

### Equation 4:
\[
x^2 + y^2 + z^2 = 4
\]
This equation is **not linear** because the variables \(x\), \(y\), and \(z\) are squared. A linear equation can only have variables raised to the first power.

### System 5:
\[
\begin{cases}
xy - 3y + z = 5 \\
y^2 + 5z = 0 \\
2x + yz = 3
\end{cases}
\]
This system is **not linear** because:
- In the first equation, there is a product of variables (\(xy\)), which is not allowed in linear equations.
- In the second equation, \(y^2\) is a squared variable, which makes it non-linear.
- The third equation contains \(yz\), another product of variables, making it non-linear.

### System 6:
\[
\begin{cases}
x - 2y + 3z = 10 \\
2x + 5y = 2 \\
y + 2z = 4
\end{cases}
\]
This system is **linear** because all the variables \(x\), \(y\), and \(z\) appear only to the first power, and there are no products of variables or non-linear terms.

### Summary:
- **3.** Linear
- **4.** Not Linear
- **5.** Not Linear
- **6.** Linear

Let me know if you need further details or explanations!

Here are 5 related questions for further understanding:
1. What defines a system of equations as linear?
2. How do products of variables affect the linearity of an equation?
3. Can an equation with square roots of variables be linear?
4. How do we determine if a system of equations is inconsistent?
5. Can a system of linear equations have more than one solution?

**Tip**: In linear systems, variables should only appear in the first degree, and there should be no products of variables or non-linear functions (like squares or square roots).

Analyze and determine the linearity of given equations and systems of equations. This problem covers key concepts in algebra and linear systems, helping students understand the difference between linear and non-linear equations. Suitable for students in Grades 8-10.

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