Difficulty: Easy
Correct Answer: -1
Explanation:
Introduction / Context:
This problem uses the concept of slope in coordinate geometry and the fact that all lines parallel to a given non-vertical line share the same slope. The question asks you first to find the slope of a given line and then use this to determine the slope of any line parallel to it.
Given Data / Assumptions:
Concept / Approach:
The slope m of a line through points (x1, y1) and (x2, y2) is:
m = (y2 − y1) / (x2 − x1)
If another line is parallel to this one and is not vertical, then its slope is the same number m.
Step-by-Step Solution:
Take A(4, -2) and B(-3, 5).
Compute slope of AB: m = (5 − (-2)) / (-3 − 4).
Numerator: 5 − (-2) = 5 + 2 = 7.
Denominator: -3 − 4 = -7.
So m = 7 / (-7) = -1.
Therefore, any line parallel to AB also has slope -1.
Verification / Alternative check:
We can see that a line with slope -1 rises by 1 unit when moving 1 unit to the left and falls by 1 unit when moving 1 unit to the right. Starting from (4, -2), moving left 7 units goes to (-3, 5) (since y increases by 7). This behaviour is consistent with slope -1 and confirms our calculation.
Why Other Options Are Wrong:
3/7 and -3/7 would correspond to lines with much gentler slopes, not matching the steepness of the line from (4, -2) to (-3, 5). A slope of 1 would give a line rising one unit for each move to the right, opposite in sign to the actual line, which slopes downward from left to right.
Common Pitfalls:
Students sometimes reverse the order of points or miscalculate numerator or denominator signs. Remember that as long as you keep the order consistent (y2 − y1 over x2 − x1), you will get the correct slope. Changing the order of both numerator and denominator simultaneously still yields the same result.
Final Answer:
The slope of any line parallel to the given line is -1.
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