Let's analyze the problem in your image. The task is to:

Write an equation in slope-intercept form of the line that passes through the given point (-9, -2) and is parallel to the graph of the given equation $y = -4x + 1$.

Solution:

1. Find the slope of the given equation: The given equation is $y = -4x + 1$, which is already in slope-intercept form ($y = mx + b$), where $m$ is the slope.

   • Here, $m = -4$.

   Since the new line is parallel to this line, it will have the same slope: $\text{slope of new line } = m = -4.$

2. Use the point-slope form: The point-slope form of a line is: $y - y_1 = m(x - x_1),$ where $m$ is the slope, and $(x_1, y_1)$ is the given point.

   Substituting $m = -4$, $x_1 = -9$, and $y_1 = -2$: $y - (-2) = -4(x - (-9)).$ Simplify: $y + 2 = -4(x + 9).$

3. Simplify to slope-intercept form: Distribute the slope ($-4$) on the right-hand side: $y + 2 = -4x - 36.$ Subtract 2 from both sides: $y = -4x - 38.$

Final Answer:

The equation of the line in slope-intercept form is: $y = -4x - 38.$

Would you like a deeper explanation or have any questions?

Here are some related questions for further exploration:

1. How do you determine if two lines are parallel or perpendicular based on their slopes?
2. Can you write the equation of a line that passes through a point and is perpendicular to a given line?
3. What is the difference between point-slope form and slope-intercept form?
4. How do you find the slope of a line given two points?
5. Can you derive the equation of a line given its slope and the y-intercept directly?

Tip: Remember, parallel lines always have the same slope, but their y-intercepts will differ.