To find the slopes, y-intercepts, and x-intercepts for each equation, we'll start by rewriting them in slope-intercept form (y = mx + b) where $m$ is the slope and $b$ is the y-intercept. The x-intercept can be found by setting $y = 0$.

1) $3x - 2y = 6$

• Rearranging to slope-intercept form: $-2y = -3x + 6 \implies y = \frac{3}{2}x - 3$
• Slope (m): $\frac{3}{2}$
• Y-intercept (b): $-3$
• X-intercept: Set $y = 0$: $3x = 6 \implies x = 2$

2) $5x + y = 7$

• Rearranging: $y = -5x + 7$
• Slope (m): $-5$
• Y-intercept (b): $7$
• X-intercept: Set $y = 0$: $5x = 7 \implies x = \frac{7}{5}$

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

• Rearranging: $-\frac{1}{3}y = -\frac{3}{2}x + 2 \implies y = \frac{9}{2}x - 6$
• Slope (m): $\frac{9}{2}$
• Y-intercept (b): $-6$
• X-intercept: Set $y = 0$: $\frac{3}{2}x = 2 \implies x = \frac{4}{3}$

4) $y = 7x$

• This is already in slope-intercept form.
• Slope (m): $7$
• Y-intercept (b): $0$
• X-intercept: Set $y = 0$: $0 = 7x \implies x = 0$

5) $2y = 3$

• Rearranging: $y = \frac{3}{2}$
• Slope (m): $0$ (horizontal line)
• Y-intercept (b): $\frac{3}{2}$
• X-intercept: This line does not cross the x-axis.

6) $9x - 3y + 2 = 0$

• Rearranging: $-3y = -9x - 2 \implies y = 3x + \frac{2}{3}$
• Slope (m): $3$
• Y-intercept (b): $\frac{2}{3}$
• X-intercept: Set $y = 0$: $9x + 2 = 0 \implies x = -\frac{2}{9}$

Summary of Results

1. $3x - 2y = 6$: Slope = $\frac{3}{2}$, Y-intercept = $-3$, X-intercept = $2$
2. $5x + y = 7$: Slope = $-5$, Y-intercept = $7$, X-intercept = $\frac{7}{5}$
3. $\frac{3}{2}x - \frac{1}{3}y = 2$: Slope = $\frac{9}{2}$, Y-intercept = $-6$, X-intercept = $\frac{4}{3}$
4. $y = 7x$: Slope = $7$, Y-intercept = $0$, X-intercept = $0$
5. $2y = 3$: Slope = $0$, Y-intercept = $\frac{3}{2}$, X-intercept = None
6. $9x - 3y + 2 = 0$: Slope = $3$, Y-intercept = $\frac{2}{3}$, X-intercept = $-\frac{2}{9}$

Graphing the Lines

Would you like to see the graphs for these equations? If so, I can generate them for you.

Questions to Consider:

1. What methods can you use to solve systems of equations graphically?
2. How does changing the slope of a line affect its steepness?
3. What is the significance of the y-intercept in a real-world context?
4. How do you determine if two lines are parallel or perpendicular?
5. What are the implications of a zero slope in a line?

Tip:

When graphing lines, it can be helpful to plot both the y-intercept and x-intercept, as these points provide clear locations on the graph.