To graph the line described by the equation $y + 3 = \frac{1}{6}(x + 3)$, we can first rewrite the equation in a more familiar form, such as the slope-intercept form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Step 1: Simplify the equation

Starting with the equation: $y + 3 = \frac{1}{6}(x + 3)$

Distribute the $\frac{1}{6}$ on the right-hand side: $y + 3 = \frac{1}{6}x + \frac{1}{2}$

Now, subtract 3 from both sides to solve for $y$: $y = \frac{1}{6}x + \frac{1}{2} - 3$

Simplify: $y = \frac{1}{6}x - \frac{5}{2}$

Step 2: Identify the slope and y-intercept

• The slope $m = \frac{1}{6}$, meaning the line rises by 1 unit for every 6 units it moves horizontally.
• The y-intercept $b = -\frac{5}{2}$, which means the line crosses the y-axis at $y = -2.5$.

Step 3: Plotting key points

1. Start at the y-intercept: $(0, -2.5)$.
2. Use the slope $\frac{1}{6}$ to find another point. From $(0, -2.5)$, move 6 units to the right (to $x = 6$) and 1 unit up (to $y = -1.5$). This gives the point $(6, -1.5)$.
3. You can also move left and down according to the slope. From $(0, -2.5)$, move 6 units left (to $x = -6$) and 1 unit down (to $y = -3.5$), giving the point $(-6, -3.5)$.

Step 4: Draw the line

Now, connect the points you've plotted with a straight line. This will represent the graph of the equation.

Would you like to see more details about any of these steps, or have specific questions?

Here are some related questions:

1. How does changing the slope affect the steepness of the line?
2. What would happen to the graph if the y-intercept were a positive number?
3. How do you calculate the slope between two points?
4. What are the conditions for two lines to be parallel or perpendicular?
5. How can you determine if a point lies on a line given its equation?

Tip: Always plot multiple points to ensure accuracy when graphing a line, especially if the slope is a fraction.