Let's analyze each problem to determine if $y$ can be defined as a linear function of $x$.

Problem 4:

The equation is: $-9x - 8y = 9$

To determine if this defines $y$ as a linear function of $x$, we need to solve for $y$ in terms of $x$.

1. Rearrange the equation to isolate $y$: $-8y = 9 + 9x$

2. Divide by $-8$ to solve for $y$: $y = -\frac{9}{8}x - \frac{9}{8}$

Since the equation is now in the form $y = mx + b$ (where $m = -\frac{9}{8}$ and $b = -\frac{9}{8}$), this does define $y$ as a linear function of $x$. Therefore, the answer is Yes, and $y = -\frac{9}{8}x - \frac{9}{8}$.

Problem 5:

The equation is: $10x = 3y + 4$

Again, let’s rearrange to see if we can express $y$ in terms of $x$.

1. Move terms involving $y$ to one side: $3y = 10x - 4$

2. Divide by 3 to solve for $y$: $y = \frac{10}{3}x - \frac{4}{3}$

This equation is also in the form $y = mx + b$ (where $m = \frac{10}{3}$ and $b = -\frac{4}{3}$), meaning it also defines $y$ as a linear function of $x$. The answer here is Yes, and $y = \frac{10}{3}x - \frac{4}{3}$.

Would you like further details or have any specific questions about these steps?

Further Practice Questions:

1. How can we identify linear equations quickly without rearranging terms?
2. What are the characteristics of a non-linear function?
3. How would the process change if the equation had a $y^2$ term?
4. What happens if both $x$ and $y$ have squared terms in an equation?
5. Can the constant term in a linear equation ever be negative?

Tip: When rearranging equations to find if they are linear, always aim for the form $y = mx + b$. If you can achieve this form, it confirms the function is linear.