We are asked to find the slope ($m$) and the $y$-intercept ($b$) of the line given by the equation:

$4x - 3y = 9$

Step 1: Rewrite the equation in slope-intercept form

The slope-intercept form of a line is:

$y = mx + b$

To do this, solve for $y$ in terms of $x$:

1. Start with the original equation:
   $4x - 3y = 9$

2. Subtract $4x$ from both sides:
   $-3y = -4x + 9$

3. Divide everything by $-3$:
   $y = \frac{-4}{-3}x + \frac{9}{-3}$ Simplifying: $y = \frac{4}{3}x - 3$

Step 2: Identify the slope and y-intercept

Now that the equation is in the form $y = mx + b$, we can see:

• The slope $m = \frac{4}{3}$
• The y-intercept $b = -3$

Step 3: Graph the line

• y-intercept: Start at the point $(0, -3)$ on the y-axis.
• Slope: The slope $\frac{4}{3}$ means that for every 3 units you move horizontally (to the right), you move 4 units vertically (up). So, from $(0, -3)$, move right 3 units and up 4 units to get the next point $(3, 1)$.

I can help visualize the graph as well if you'd like!

Would you like to see the graph? Also, do you want further details or have any questions?

Here are 5 related questions to expand on this concept:

1. How do you find the slope between two given points?
2. What is the relationship between parallel and perpendicular slopes?
3. How do you find the equation of a line given a point and a slope?
4. How do you convert from slope-intercept form to standard form?
5. How do you calculate the x-intercept of a line from its equation?

Tip: Always check the units of the slope — it’s a ratio of vertical change to horizontal change.