The equation provided is:

$y + 3 = \frac{1}{3}(x + 1)$

Step 1: Rearrange the equation into slope-intercept form ($y = mx + b$).

1. Start with: $y + 3 = \frac{1}{3}(x + 1)$

2. Distribute $\frac{1}{3}$ across $(x + 1)$: $y + 3 = \frac{1}{3}x + \frac{1}{3}$

3. Subtract 3 from both sides to isolate $y$: $y = \frac{1}{3}x + \frac{1}{3} - 3$

4. Simplify the constant terms: $y = \frac{1}{3}x - \frac{8}{3}$

Step 2: Identify key features of the line.

• Slope ($m$): $\frac{1}{3}$
• Y-intercept ($b$): $-\frac{8}{3}$, which is approximately $-2.67$.

Step 3: Plot the graph.

1. Start at the y-intercept $(0, -\frac{8}{3})$.

2. Use the slope $\frac{1}{3}$:

   • Rise: 1 unit up.
   • Run: 3 units to the right.
   • From the point $(0, -\frac{8}{3})$, move 1 unit up and 3 units right to find another point on the line.

3. Plot these points and draw a straight line through them.

If you'd like, I can generate an image of the graph for you! Let me know.

Questions for further exploration:

1. What is the significance of the slope in the equation of a line?
2. How would the graph change if the slope were negative?
3. What is the meaning of the y-intercept in this context?
4. How would you graph the line if it were given in standard form (e.g., $Ax + By = C$)?
5. Can you identify another point on the line using substitution?

Tip:

To check if a point lies on the line, substitute the $x$- and $y$-values into the original equation and see if it holds true!