A straight line passes through the points (5, -1) and (4, -4). What is the slope of any line that is parallel to this line?

Introduction / Context:
This coordinate geometry problem asks for the slope of a line parallel to a given line. The given line is defined by two points, and the slope of any parallel line is the same as the slope of the original line. Therefore, we must first find the slope of the line passing through the two points using the slope formula, and then state that value as the slope of any parallel line.

Given Data / Assumptions:
- The given line passes through points (5, -1) and (4, -4).- We are working in the standard Cartesian coordinate plane.- Slope of a line through points (x1, y1) and (x2, y2) is given by (y2 - y1) / (x2 - x1).- Parallel lines have equal slopes.

Concept / Approach:
To find the slope m of the given line, we use m = (change in y) / (change in x). We will label the points, compute the numerator and denominator carefully, and simplify the fraction. Once the slope of the original line is found, we know that any line parallel to it must have the same slope. Thus the answer is simply this computed slope.

Step-by-Step Solution:
Step 1: Let the two points be (x1, y1) = (5, -1) and (x2, y2) = (4, -4).Step 2: Use the slope formula m = (y2 - y1) / (x2 - x1).Step 3: Compute y2 - y1 = -4 - (-1) = -4 + 1 = -3.Step 4: Compute x2 - x1 = 4 - 5 = -1.Step 5: So m = (-3) / (-1) = 3.Step 6: Therefore, the slope of the line passing through the two points is 3.Step 7: Any line parallel to this line must have the same slope.Step 8: Hence, the slope of any parallel line is also 3.

Verification / Alternative check:
We can interchange the roles of the points and compute the slope again to ensure consistency. Taking (x1, y1) = (4, -4) and (x2, y2) = (5, -1), we get m = (y2 - y1) / (x2 - x1) = (-1 - (-4)) / (5 - 4) = (3) / (1) = 3. The slope is the same regardless of the order in which we choose the points, confirming the correctness of m = 3. Therefore, any line parallel to this one also has slope 3.

Why Other Options Are Wrong:
- -3: This would be the slope if only the change in y were used without correctly considering the change in x sign.- -1/3: This might result from inverting the fraction and also changing the sign incorrectly.- 1/3: This results from incorrectly swapping the changes in x and y or miscomputing the differences.- 0: This would correspond to a horizontal line, which is not the case since the y values clearly change as x changes.

Common Pitfalls:
Students often mix up the order of subtraction in the slope formula, which can lead to sign mistakes. However, using consistent ordering for both numerator and denominator avoids such errors. Another pitfall is to mistakenly invert the slope or think that parallel lines require a different slope, when in fact they require the same slope. Careful use of the formula and attention to signs will yield the correct result.

Final Answer:
The slope of any line parallel to the given line is 3.