Slope: Rise Over Run

The slope of a line measures how steeply it rises (or falls). Numerically, slope is the ratio of the vertical change ("rise") to the horizontal change ("run") between any two points on the line:

$m = \dfrac{\text{rise}}{\text{run}} = \dfrac{y_2 - y_1}{x_2 - x_1}$

The slope is a property of the line, not of the two points you pick — any two points give the same answer.

Slope Sign

• $m > 0$ → line goes up as $x$ increases
• $m < 0$ → line goes down as $x$ increases
• $m = 0$ → horizontal line
• $m$ undefined → vertical line (run = 0, division by zero)

Step-by-Step Solution

Given: A line passes through $(1, 2)$ and $(5, 8)$.

Find: The slope.

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Step 1 — Identify the two points.

$(x_1, y_1) = (1, 2)$

$(x_2, y_2) = (5, 8)$

Step 2 — Compute the rise.

$\text{rise} = y_2 - y_1 = 8 - 2 = 6$

Step 3 — Compute the run.

$\text{run} = x_2 - x_1 = 5 - 1 = 4$

Step 4 — Divide.

$m = \dfrac{\text{rise}}{\text{run}} = \dfrac{6}{4} = 1.5$

The slope is 1.5, meaning the line rises 1.5 units for every 1 unit of horizontal travel. Or equivalently, 3 units up for every 2 units right — that's a "3:2" slope.

Step 5 — Find the equation of the line.

Using point-slope form with the slope $m = 1.5$ and point $(1, 2)$:

$y - 2 = 1.5(x - 1)$

$y = 1.5x - 1.5 + 2$

$y = 1.5x + 0.5$

That's the slope-intercept form: $y = 1.5x + 0.5$, with $y$-intercept at $(0, 0.5)$.

Step 6 — Verify with both points.

At $x = 1$: $y = 1.5(1) + 0.5 = 2.0$ ✓

At $x = 5$: $y = 1.5(5) + 0.5 = 8.0$ ✓

Step 7 — Try a different pair of points on the same line.

If you use $(3, 5)$ and $(7, 11)$ (both on this line — verify with the equation):

$m = \dfrac{11 - 5}{7 - 3} = \dfrac{6}{4} = 1.5 \;\;\checkmark$

Same slope. Choosing different points doesn't change the answer.

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Answer:

$\boxed{\;m = \dfrac{\text{rise}}{\text{run}} = \dfrac{8 - 2}{5 - 1} = \dfrac{6}{4} = 1.5\;}$

The line through $(1, 2)$ and $(5, 8)$ has slope $1.5$. Its equation is $y = 1.5x + 0.5$.

Try It

• Adjust the two points with the sliders.
• Watch the line and the slope triangle update.
• The HUD shows the rise, run, and resulting slope.
• Try aligning the two points horizontally — slope becomes 0.
• Try aligning them vertically — slope becomes undefined (the formula divides by zero).