Understand and calculate slope between two points. Math 10-C Alberta mathematics.
Lesson 6.1 of Linear Relations and Functions in Math 10C — Alberta curriculum lessons.
Definition — Slope. The slope of a line measures its rate of change — how much the y-value changes for each unit of change in x.
The letter m is used to represent slope.
Rise and Run. Rise is the vertical change (up or down).
Run is the horizontal change (left or right).
Rise is positive when you move up, negative when you move down.
Run is positive when you move right, negative when you move left.
How Slope Sign Determines Direction. The sign of the slope tells you which way the line tilts:
Positive slope: line goes up from left to right (rises to the right).
Negative slope: line goes down from left to right (falls to the right).
Zero slope: line is perfectly horizontal — no rise, so m = (0)/(run) = 0.
Undefined slope: line is perfectly vertical — no run, so m = (rise)/(0), which is undefined.
Why Is Vertical Slope Undefined — Not Infinity?. Division by zero does not produce a number at all — it is not defined in mathematics.
Saying the slope is "undefined" is more precise than saying it is infinite, because infinity is not a real number either.
Strategy — Reading Slope from a Graph. Choose two points where the line crosses grid intersections so you can read the coordinates exactly.
From one point to the other, count the rise (vertical change) and the run (horizontal change).
Then apply: m = (rise)/(run)
Example 1 — Positive Slope. A line passes through (-5, -5) and (-3, -2).
Step 1 — Find the rise:
From y = -5 up to y = -2: rise = +3.
Step 2 — Find the run:
From x = -5 right to x = -3: run = +2.
Step 3 — Calculate:
m = (rise)/(run) = (3)/(2)
Answer: m = (3)/(2) — for every 2 units right, the line rises 3 units.
Example 2 — Negative Slope. A line passes through (5, -1) and (2, 0).
Step 1 — Find the rise:
From y = -1 up to y = 0: rise = +1.
Step 2 — Find the run:
From x = 5 left to x = 2: run = -3.
Step 3 — Calculate:
m = (1)/(-3) = -(1)/(3)
Answer: m = -(1)/(3) — the slope is negative, so the line falls to the right.
Example 3 — Negative Slope: (2, -3) and (-4, 1). Step 1 — Label the points:
(x_1, y_1) = (2, -3) and (x_2, y_2) = (-4, 1).
Step 2 — Substitute into the formula:
m = (1 - (-3))/(-4 - 2)
Step 3 — Simplify the numerator:
m = (1 + 3)/(-6) = (4)/(-6)
Step 4 — Reduce:
Answer: m = -(2)/(3)
Example 4 — Whole-Number Slope: (3, -3) and (4, 1). Step 1 — Substitute:
m = (1 - (-3))/(4 - 3)
Step 2 — Simplify:
m = (4)/(1)
Answer: m = 4
Example 5 — Undefined Slope: (2, 3) and (2, -6). Step 1 — Substitute:
m = (-6 - 3)/(2 - 2)
Step 2 — Simplify:
m = (-9)/(0)
Answer: m is undefined. Both x-values are 2, so this is a vertical line.
Example 6 — Zero Slope: (3, 4) and (2, 4). Step 1 — Substitute:
m = (4 - 4)/(2 - 3)
Step 2 — Simplify:
m = (0)/(-1)
Answer: m = 0. Both y-values are 4, so this is a horizontal line.
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