Identify linear and nonlinear functions from equations worksheet with multiple-choice options.

Let's solve this step by step.

We are asked to identify whether each equation represents a linear or nonlinear function.

---

🔍 What is a Linear Function?

A linear function has the form:
> $ y = mx + b $
where:
- $ m $ and $ b $ are constants,
- The exponent of $ x $ is 1 (or no exponent, meaning $ x^1 $),
- There are no variables multiplied together, no exponents greater than 1, and no variables in denominators, radicals, etc.

If any variable has an exponent greater than 1, or if there’s a product of variables, it’s nonlinear.

---

Now let’s go through each equation:

---

1. $ y = 3x - 1 $

- $ x $ has exponent 1.
- No higher powers or products.
✔ Linear

---

2. $ 2y + x^3 = 9 $

- $ x^3 $ → exponent is 3 > 1
✘ Nonlinear

---

3. $ y = x^2 - 6 $

- $ x^2 $ → exponent is 2 > 1
✘ Nonlinear

---

4. $ 5x^2 - 3x + 1 = y $

- $ x^2 $ → exponent 2 > 1
✘ Nonlinear

---

5. $ y + 4 = x $

- Simplify: $ y = x - 4 $
- This is in the form $ y = mx + b $
✔ Linear

---

6. $ y = \frac{2}{3}x + 5 $

- Constant slope, exponent of $ x $ is 1
✔ Linear

---

7. $ y - \frac{1}{2} = x^2 $

- Rearranged: $ y = x^2 + \frac{1}{2} $
- $ x^2 $ → exponent 2
✘ Nonlinear

---

8. $ y = x^2 $

- Clearly quadratic
✘ Nonlinear

---

9. $ y = 8 $

- This is a constant function.
- Can be written as $ y = 0x + 8 $
- Still linear (horizontal line)
✔ Linear

---

10. $ 5y = -x $

- Solve for $ y $: $ y = -\frac{1}{5}x $
- Slope-intercept form, exponent 1
✔ Linear

---

11. $ -y + x^3 = 3x $

- Rearranged: $ -y = 3x - x^3 $ → $ y = x^3 - 3x $
- $ x^3 $ → exponent 3 > 1
✘ Nonlinear

---

12. $ x^2 - 4x = 6 + y $

- Rearranged: $ y = x^2 - 4x - 6 $
- $ x^2 $ present → exponent 2
✘ Nonlinear

---

13. $ y = -10x $

- Simple direct proportion, exponent of $ x $ is 1
✔ Linear

---

14. $ x^2 + 5 = y $

- Same as $ y = x^2 + 5 $
- $ x^2 $ → exponent 2
✘ Nonlinear

---

15. $ y = \frac{1}{9}x - 4 $

- Standard linear form
✔ Linear

---

✔ Final Answers:

| # | Equation | Answer |
|---|--------|--------|
| 1 | $ y = 3x - 1 $ | Linear |
| 2 | $ 2y + x^3 = 9 $ | Nonlinear |
| 3 | $ y = x^2 - 6 $ | Nonlinear |
| 4 | $ 5x^2 - 3x + 1 = y $ | Nonlinear |
| 5 | $ y + 4 = x $ | Linear |
| 6 | $ y = \frac{2}{3}x + 5 $ | Linear |
| 7 | $ y - \frac{1}{2} = x^2 $ | Nonlinear |
| 8 | $ y = x^2 $ | Nonlinear |
| 9 | $ y = 8 $ | Linear |
| 10 | $ 5y = -x $ | Linear |
| 11 | $ -y + x^3 = 3x $ | Nonlinear |
| 12 | $ x^2 - 4x = 6 + y $ | Nonlinear |
| 13 | $ y = -10x $ | Linear |
| 14 | $ x^2 + 5 = y $ | Nonlinear |
| 15 | $ y = \frac{1}{9}x - 4 $ | Linear |

---

📌 Summary:

- Linear: 1, 5, 6, 9, 10, 13, 15 → 7 equations
- Nonlinear: 2, 3, 4, 7, 8, 11, 12, 14 → 8 equations

You should circle "Linear" or "Nonlinear" accordingly for each.

Let me know if you'd like this formatted for printing!