We are given the function:

$f(x) = x - \frac{3 + 5x}{3}$

Step 1: Simplifying the function

First, let's simplify the expression for $f(x)$:

$f(x) = x - \frac{3 + 5x}{3}$

We can split the fraction:

$f(x) = x - \left(\frac{3}{3} + \frac{5x}{3}\right)$ $f(x) = x - 1 - \frac{5x}{3}$

Now, combine the terms involving $x$:

$f(x) = \left(1x - \frac{5x}{3}\right) - 1$

We need a common denominator for $x$:

$f(x) = \frac{3x}{3} - \frac{5x}{3} - 1$

Simplify:

$f(x) = \frac{-2x}{3} - 1$

Step 2: Identifying the slope and y-intercept

This is now in the form of a linear equation $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• The slope is $m = \frac{-2}{3}$.
• The y-intercept occurs when $x = 0$. Substituting $x = 0$ into the simplified function:

$f(0) = \frac{-2(0)}{3} - 1 = -1$

So, the y-intercept is $(0, -1)$.

Final Answer:

• Slope: $\frac{-2}{3}$
• y-intercept: $(0, -1)$

Let me know if you want further clarification! Here are some questions that expand on this concept:

1. How do you interpret the slope of a line in a real-world context?
2. What happens to the graph of the function if the slope changes to a positive value?
3. How can we find the x-intercept of this function?
4. What is the significance of the y-intercept in real-world problems?
5. How does the equation change if you add a constant to the function?

Tip: When simplifying linear functions, always try to express them in the form $y = mx + b$ for easier identification of slope and intercepts.