Let's solve the given problem step by step.

Given Equation:

$3y - 2x = -6$

Part a: Write the equation in slope-intercept form

The slope-intercept form of a linear equation is: $y = mx + b$ where $m$ is the slope and $b$ is the y-intercept.

1. Start with the given equation: $3y - 2x = -6$

2. Add $2x$ to both sides: $3y = 2x - 6$

3. Divide through by 3: $y = \frac{2}{3}x - 2$

Thus, the slope-intercept form is: $y = \frac{2}{3}x - 2$

Part b: Determine the slope and y-intercept

From the equation $y = \frac{2}{3}x - 2$:

• The slope ($m$) is: $m = \frac{2}{3}$
• The y-intercept ($b$) is: $b = -2$

Part c: Graph the line using the slope and y-intercept

1. Plot the y-intercept: The y-intercept is $-2$. Plot the point $(0, -2)$ on the graph.

2. Use the slope: The slope $\frac{2}{3}$ means:

   • Rise = 2 (move up 2 units)
   • Run = 3 (move right 3 units)

   From the point $(0, -2)$:

   • Move up 2 units and right 3 units to get another point, $(3, 0)$.
   • Move down 2 units and left 3 units to get the point $(-3, -4)$.

3. Draw the line through these points.

Would you like me to provide a detailed drawing of this graph?

Questions to Expand Understanding:

1. What does the slope represent in a real-world context if this equation described a scenario?
2. How do you determine the x-intercept of a line from the slope-intercept form?
3. What happens to the graph if the slope becomes negative?
4. How can you confirm the line's equation if given two points on the graph?
5. Can a line have more than one y-intercept? Why or why not?

Tip:

To quickly graph a line, remember that the slope gives you the "rise over run," and the y-intercept tells you where the line crosses the y-axis.