https://file.aiquickdraw.com/mathbot/file/1729095753709hsrsdqrg.jpg

Let's analyze each function to determine if it's linear or nonlinear:

### 1. \( 3x + y = 12 \)
- This is a linear equation because it can be written in the form \( y = mx + b \), where \( m \) and \( b \) are constants.

### 2. \( y = \frac{x}{2} - 3 \)
- This is also a linear equation because it has the form \( y = mx + b \), where \( m = \frac{1}{2} \) and \( b = -3 \).

### 3. \( y = x \)
- This is a linear equation with a slope of 1 and no constant term, making it linear.

### 4. \( y = 3x^3 + 5 \)
- This is a nonlinear equation because it contains an \( x^3 \) term, which is a higher power of \( x \), making the function nonlinear.

### 5. \( y = \frac{6}{x} - 2 \)
- This is a nonlinear equation because it contains a division by \( x \), which is not characteristic of linear functions.

### Classification:
- **Linear**: 
  1. \( 3x + y = 12 \)
  2. \( y = \frac{x}{2} - 3 \)
  3. \( y = x \)

- **Nonlinear**: 
  1. \( y = 3x^3 + 5 \)
  2. \( y = \frac{6}{x} - 2 \)

Let me know if you'd like further explanations! Here are 5 related questions to extend your learning:

1. How do you convert a standard form equation into slope-intercept form?
2. What makes a function nonlinear mathematically?
3. Can a polynomial equation of degree 2 be linear?
4. How can you graphically identify whether a function is linear or nonlinear?
5. What is the general form of a linear equation in multiple variables?

**Tip**: A function is linear if it forms a straight line when graphed and all the variables appear to the first power without multiplication of variables.

Determine if each function is linear or nonlinear. Drag each function into a box to correctly classify it.

Learn how to classify functions as linear or nonlinear by analyzing key properties such as powers of the variable or division by x. Understand the difference between linear equations in the form y = mx + b and nonlinear equations that include higher powers of x or division. Suitable for students in Grades 8-10.

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