Consider the line in the figure which passes through A (−4, 5) and B (−1, 2). Find the slope of the line.
Step-by-step solution
Idea: Use slope = y2 − y1x2 − x1. Moving from A to B the line goes 3 right and 3 down, so the slope is negative. Swapping the points changes the sign of both top and bottom, so the answer does not change.
- Take A(−4, 5) as (x1, y1) and B(−1, 2) as (x2, y2). Let P(−4, 2) be the corner below A, level with B.½ mark
- Slope = y2 − y1x2 − x1 = 2 − 5−1 − (−4) = −33 = −1.1 mark
- Reverse the order: B as (x1, y1), A as (x2, y2): 5 − 2−4 − (−1) = 3−3 = −1. Same answer.1 mark
- So the slope is −1 whatever order we choose. It is negative because the line goes down from left to right.½ mark
Check: The line is x + y = 1, i.e. y = −x + 1; its slope m = −1 ✓. Both points satisfy it: −4 + 5 = 1, −1 + 2 = 1 ✓.
Answer to write in the exam
Slope = y2 − y1x2 − x1
= 2 − 5−1 − (−4) = −33 = −1
Taking points in reverse: 5 − 2−4 − (−1) = 3−3 = −1
∴ Slope = −1
Common mistakes that cost marks
- Writing −1 − (−4) as −5 instead of 3.
- Giving the slope as +1 by measuring lengths only. Lengths are positive, but the rise here is a fall, so the slope is negative.
- Mixing orders, e.g. 2 − 5−4 − (−1) = 1. Top and bottom must start with the same point.
How this can come in the exam
The slope of the line through (−3, 4) and (1, −4) is
- 2
- −2
- 12
- −12
Show answer
(B) −2
−4 − 41 − (−3) = −84 = −2.
Assertion (A): A line falling from left to right has a negative slope.
Reason (R): Slope does not depend on which of two points is taken first.
- Both Assertion (A) and Reason (R) are true, and R is the correct explanation of A.
- Both Assertion (A) and Reason (R) are true, but R is not the correct explanation of A.
- Assertion (A) is true, but Reason (R) is false.
- Assertion (A) is false, but Reason (R) is true.
Show answer
(B) Both Assertion (A) and Reason (R) are true, but R is not the correct explanation of A.
Both are true, but R (order does not matter) does not explain why the sign is negative; the fall (negative rise for a positive run) does.
Try one yourself
Find the slope of the line through (−5, 1) and (−2, −5).
Show answer
−5 − 1−2 − (−5) = −63 = −2.
More questions like this
- But why does ‘m’ represent the slope?
- Consider the line 5x + y = 3. Rewrite it as y = −5x + 3. Can you now find its slope? What does it mean?
- Can you use the points A, B and C to verify that the slope of this line is indeed −5?
- The following diagrams represent ski hills. Rank the hills in order of their steepness, from least to greatest.
- The ramp at a loading dock rises 2.5 metres over a run of 4 metres. Find the slope of the ramp.
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