Determine whether each equation represents a linear or nonilinear function
Explain. (exnmpirs 4 and 5)

Ask by Johnson Garrett. in the United Arab Emirates

Jan 18,2025

Question

Determine whether each equation represents a linear or nonilinear function Explain. (exnmpirs 4 and 5) $$ \begin{array}{ll}	ext { 6. } y+7 x=-2 & 	ext { 7. } y=\sqrt{8 x}\end{array} $$

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Let's analyze each equation to determine whether it represents a linear or nonlinear function.

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**6. $$ y + 7x = -2 $$**

**Analysis:**
- **Standard Form:** To identify if this is linear, we can rearrange it into the slope-intercept form $$ y = mx + b $$.
  
  $$
  y = -7x - 2
  $$
  
- **Conclusion:** This equation is **linear** because it fits the form $$ y = mx + b $$, where:
  - $$ m = -7 $$ (slope)
  - $$ b = -2 $$ (y-intercept)

**Why It's Linear:**
- Both $$ y $$ and $$ x $$ are to the first power.
- The graph of this equation is a straight line.

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**7. $$ y = \sqrt{8x} $$**

**Analysis:**
- **Simplification:** The equation can be rewritten using rational exponents:
  
  $$
  y = (8x)^{1/2} = 8^{1/2} \cdot x^{1/2} = 2\sqrt{2} \cdot x^{1/2}
  $$
  
- **Conclusion:** This equation is **nonlinear** because:
  - The variable $$ x $$ is under a square root, which is equivalent to raising $$ x $$ to the power of $$ \frac{1}{2} $$.
  - The exponent of $$ x $$ is not 1, which violates the definition of a linear function.

**Why It's Nonlinear:**
- The presence of the square root introduces a curvature.
- The graph of this equation is a parabola, not a straight line.

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**Summary:**
1. **Equation 6 ($$ y + 7x = -2 $$)** is a **linear** function because it can be expressed in the form $$ y = mx + b $$ with both variables to the first power.
2. **Equation 7 ($$ y = \sqrt{8x} $$)** is a **nonlinear** function due to the square root of $$ x $$, which introduces a fractional exponent and curvature to its graph.

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**Graphical Representation:**

- **Linear Function (Equation 6):**
  ![Linear Graph](https://i.imgur.com/linear.png)  
  *A straight line with slope -7 and y-intercept -2.*

- **Nonlinear Function (Equation 7):**
  ![Nonlinear Graph](https://i.imgur.com/nonlinear.png)  
  *A parabola opening to the right due to the square root function.*

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Feel free to ask if you need further clarification on these concepts!
Equation 6 ($$ y + 7x = -2 $$) is **linear**, while Equation 7 ($$ y = \sqrt{8x} $$) is **nonlinear**.

Determine whether each equation represents a linear or nonilinear function
Explain. (exnmpirs 4 and 5)

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