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

Let's analyze each equation to determine whether it represents a linear or nonlinear function. --- **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. --- **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. --- **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. --- **Graphical Representation:** - **Linear Function (Equation 6):**  *A straight line with slope -7 and y-intercept -2.* - **Nonlinear Function (Equation 7):**  *A parabola opening to the right due to the square root function.* --- Feel free to ask if you need further clarification on these concepts!