Let’s determine whether each given function is linear or nonlinear:

Criteria for Linear and Nonlinear Functions:

• A linear function has the general form $y = mx + b$ (or can be rearranged into this form), where $m$ and $b$ are constants. Variables $x$ and $y$ appear only to the first power, and there are no products or divisions between them.
• A nonlinear function involves powers, roots, or products of variables (e.g., $x^2, x^3, \sqrt{x}, xy$).

Analysis:

1. $5y = 7x + 6$:

   • Rearranging: $y = \frac{7}{5}x + \frac{6}{5}$.
   • This is a linear function because it is in the form $y = mx + b$.

2. $y = 4 + \frac{x^5}{3}$:

   • The term $\frac{x^5}{3}$ involves $x$ raised to the 5th power.
   • This is nonlinear due to the $x^5$ term.

3. $y - \frac{x}{9} = 12$:

   • Rearranging: $y = \frac{x}{9} + 12$.
   • This is a linear function because it is in the form $y = mx + b$.

4. $2x^3 = y - 11$:

   • Rearranging: $y = 2x^3 + 11$.
   • This is nonlinear due to the $x^3$ term.

5. $9x^4 - y = 13$:

   • Rearranging: $y = 9x^4 - 13$.
   • This is nonlinear due to the $x^4$ term.

6. $y = -5(x + 7)$:

   • Expanding: $y = -5x - 35$.
   • This is a linear function because it is in the form $y = mx + b$.

7. $y + 8 = \frac{x}{2}$:

   • Rearranging: $y = \frac{x}{2} - 8$.
   • This is a linear function because it is in the form $y = mx + b$.

8. $2y = 6x^2 - x + 3$:

   • Dividing by 2: $y = 3x^2 - \frac{x}{2} + \frac{3}{2}$.
   • This is nonlinear due to the $x^2$ term.

Summary:

1. Linear
2. Nonlinear
3. Linear
4. Nonlinear
5. Nonlinear
6. Linear
7. Linear
8. Nonlinear

Would you like further clarification or detailed steps?

5 Related Questions:

1. What is the difference between linear and affine functions?
2. How can you identify a quadratic function from its equation?
3. Can a function with fractional powers of $x$ be linear?
4. How do you graph linear vs. nonlinear functions?
5. What real-world problems can be modeled using linear equations?

1 Tip:

Always check the highest degree of $x$ or the presence of products of variables to determine whether a function is linear or nonlinear.