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.