Slope of a Line
Calculate slope of a line in Grade 10 algebra using rise over run (m = Δy/Δx), distinguish positive, negative, zero, and undefined slopes from graphs and coordinate pairs.
Key Concepts
The slope $m$ of a line through points $(x 1, y 1)$ and $(x 2, y 2)$ is the ratio of vertical change (rise) to horizontal change (run). $$m = \frac{y 2 y 1}{x 2 x 1} = \frac{\text{rise}}{\text{run}}$$.
For points $( 1, 2)$ and $(1, 6)$: $m = \frac{6 2}{1 ( 1)} = \frac{4}{2} = 2$. The positive slope means the line rises. For points $(2, 5)$ and $(4, 1)$: $m = \frac{1 5}{4 2} = \frac{ 4}{2} = 2$. The negative slope means the line falls. For points $( 3, 4)$ and $(5, 4)$: $m = \frac{4 4}{5 ( 3)} = \frac{0}{8} = 0$. The zero slope means the line is horizontal.
Slope is the secret code for a line's steepness and direction! It’s like a recipe that tells you how many steps to go up or down for every step you take sideways. A positive slope climbs uphill, while a negative one slides downhill.
Common Questions
How do you calculate slope from two points?
Slope m = (y₂-y₁)/(x₂-x₁). For points (1,3) and (4,9): m = (9-3)/(4-1) = 6/3 = 2.
What do positive, negative, zero, and undefined slopes mean?
Positive slope rises left to right. Negative slope falls left to right. Zero slope is a horizontal line. Undefined slope is a vertical line (division by zero).
How is slope related to the rate of change in real-world problems?
Slope represents the constant rate of change — how much y changes for each unit increase in x. In distance-time graphs, slope equals speed.