What is the slope of a line that is parallel to the line passing through the points (-2, -1) and (4, -3)?

Introduction / Context:
In coordinate geometry, the slope of a line is a key measure that tells us how steep the line is and in what direction it inclines or declines. When two lines are parallel, they never intersect and they always have the same slope. This question checks your understanding of how to compute the slope from two given points and how to use that result to find the slope of any parallel line.

Given Data / Assumptions:

• We have a line passing through the points A(-2, -1) and B(4, -3).
• We need the slope of a line that is parallel to this line.
• The coordinate plane is standard with x as horizontal axis and y as vertical axis.

Concept / Approach:
The slope m of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:
m = (y2 - y1) / (x2 - x1) For two lines to be parallel in a plane, their slopes must be equal. Therefore, once we compute the slope of the line through the two given points, that same value will be the slope of every line parallel to it.

Step-by-Step Solution:
Step 1: Identify the coordinates of the two points: A(-2, -1) and B(4, -3). Step 2: Use the slope formula m = (y2 - y1) / (x2 - x1). Step 3: Substitute the values: m = (-3 - (-1)) / (4 - (-2)). Step 4: Simplify the numerator: -3 - (-1) = -3 + 1 = -2. Step 5: Simplify the denominator: 4 - (-2) = 4 + 2 = 6. Step 6: So m = -2 / 6 = -1 / 3. Step 7: Any line parallel to this one must have slope -1/3.

Verification / Alternative check:
We can check the sign of the slope by examining the movement from A to B. As x increases from -2 to 4 (a positive change), y decreases from -1 to -3 (a negative change). A positive change in x and a negative change in y indicates a negative slope. This matches our computed value of -1/3. The fraction is already in simplest form, so no further reduction is required.

Why Other Options Are Wrong:

• 1/3: This is the same magnitude but wrong sign; the line clearly slopes downward as x increases, so the slope cannot be positive.
• -3: This represents a much steeper negative slope than the actual line and does not match the ratio of change in y to change in x.
• 3: This is both positive and much larger in magnitude, so it does not describe this line.
• 0: A slope of zero would correspond to a horizontal line, which is not the case here.

Common Pitfalls:
Students often reverse the order of subtraction in the slope formula or mix x and y differences incorrectly. Another common mistake is ignoring signs while subtracting negative numbers. It is also easy to forget that parallel lines must have exactly the same slope, not just the same magnitude. Carefully applying the formula and simplifying step by step prevents these errors.

Final Answer:
Hence, the slope of a line parallel to the line passing through the points (-2, -1) and (4, -3) is -1/3.