To graph the line with slope $-3$ passing through the point $(1, 1)$, we need to follow these steps:

1. Point-Slope Form: The equation of a line in point-slope form is:

   $y - y_1 = m(x - x_1)$

   where $m$ is the slope and $(x_1, y_1)$ is a point on the line. In this case, the point is $(1, 1)$ and the slope $m = -3$. Substituting into the formula:

   $y - 1 = -3(x - 1)$

2. Simplifying:

   $y - 1 = -3x + 3$ $y = -3x + 4$

   This is the equation of the line in slope-intercept form, $y = -3x + 4$, which tells us the slope is $-3$ and the y-intercept is $4$.

3. Plotting the Graph:

   • Start by plotting the point $(1, 1)$ on the graph.
   • From this point, use the slope $-3$, which means for every $1$ unit you move to the right (increase in $x$), you move $3$ units down (decrease in $y$).

4. Finding another point:

   • From $(1, 1)$, move right $1$ unit to $x = 2$, then move down $3$ units to $y = -2$. So, another point on the line is $(2, -2)$.

5. Drawing the Line:

   • Now, draw a straight line passing through the points $(1, 1)$ and $(2, -2)$.

Would you like further details or clarification on how to graph the line?

Here are 5 related questions:

1. How do you determine the slope of a line from its graph?
2. What is the general form of a linear equation, and how does it relate to slope-intercept form?
3. How would the graph change if the slope was positive instead of negative?
4. How do you find the x-intercept of a line given its equation?
5. What is the difference between point-slope form and slope-intercept form of a line?

Tip: When graphing lines, always remember that the slope tells you how much the line rises or falls as you move horizontally across the graph.