To determine which equations are not represented by a straight line, we need to understand the following:

• Linear equations take the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. These equations produce straight lines on a graph.
• Any equation with higher powers of $x$, roots, or other nonlinear components will not produce a straight line.

Now let's analyze the given equations:

1. $2y = 8x^2$
   Divide both sides by 2:
   $y = 4x^2$.
   Since $x$ is squared, this is a parabola (nonlinear).
   ✅ Not a straight line.

2. $y = \frac{1}{\sqrt{x}}$
   The equation involves a square root of $x$, which is a nonlinear function.
   ✅ Not a straight line.

3. $y = 3x \cdot 3x$
   Simplify:
   $y = 9x^2$.
   Since $x$ is squared, this is a parabola (nonlinear).
   ✅ Not a straight line.

4. $y = 3 \cdot 3x$
   Simplify:
   $y = 9x$.
   This equation is in the form $y = mx$, which is linear.
   ❌ Represents a straight line.

5. $y = 5^2 - x$
   Simplify:
   $y = 25 - x$.
   This is in the form $y = -x + 25$, which is linear.
   ❌ Represents a straight line.

6. $y = x^2 + x - 6$
   Since $x$ is squared, this equation represents a parabola (nonlinear).
   ✅ Not a straight line.

Final Answer:

The equations that are not represented by a straight line are:

• $2y = 8x^2$
• $y = \frac{1}{\sqrt{x}}$
• $y = 3x \cdot 3x$
• $y = x^2 + x - 6$

Do you want a breakdown of graph shapes for these equations? Let me know!

Here are 5 related questions for further practice:

1. What are the key differences between linear and quadratic equations?
2. How do you identify a nonlinear equation from its general form?
3. How would you graph $y = x^2 + x - 6$?
4. What kind of graph does $y = \frac{1}{\sqrt{x}}$ produce?
5. How can you determine if an equation is linear by analyzing its degree?

Tip: Equations with $x^2$, $x^3$, roots, or reciprocals of $x$ are typically nonlinear. Always simplify the equation first!