Let's analyze each equation to determine if it is linear or not.

Definition:

A linear equation is one where each term is either a constant or the product of a constant and a single variable. No variables should be raised to powers greater than 1, multiplied together, or appear in the denominator.

Now let's evaluate the given equations:

(a) $9x + 7 + 8y = x - 5$

• Rearrange: $9x - x + 8y + 7 = -5$, simplifying to $8x + 8y + 7 = -5$, which is a linear equation.
• Answer: Linear (Yes)

(b) $y = 9xy + 6$

• There is a product term $9xy$, which makes it nonlinear because variables $x$ and $y$ are multiplied together.
• Answer: Nonlinear (No)

(c) $x + \frac{7}{y} = 0$

• The variable $y$ is in the denominator, making this equation nonlinear.
• Answer: Nonlinear (No)

(d) $y^2 + 2x = 9$

• The term $y^2$ (a variable raised to the power of 2) makes this equation nonlinear.
• Answer: Nonlinear (No)

Summary of answers:

• (a) Yes
• (b) No
• (c) No
• (d) No

Would you like further explanation on any of the points?

Here are 5 related questions you could explore:

1. How can you graph a linear equation?
2. What makes an equation nonlinear?
3. Can two variables in a product ever form a linear equation?
4. How does rearranging an equation help determine its linearity?
5. What are examples of systems of linear equations?

Tip: For an equation to be linear, no variables should be squared, cubed, or appear in denominators or multiplied together.