The slope measures a line’s steepness; formula: $m=\frac{y_2-y_1}{x_2-x_1}$.
Explanation
- Meaning: Slope $m$ = rise (change in $y$) divided by run (change in $x$). Positive slope rises left→right, negative falls, zero is horizontal, undefined is vertical.
- From two points: For points $(x_1,y_1)$ and $(x_2,y_2)$ use $m=\dfrac{y_2-y_1}{x_2-x_1}$.
- From slope-intercept form: If the equation is $y=mx+b$, the slope is the coefficient $m$.
- From standard form: For $Ax+By=C$, rewrite or use $m=-\dfrac{A}{B}$ (provided $B\neq0$).
- Parallel/perpendicular: Lines parallel to slope $m$ have slope $m$. A line perpendicular to slope $m$ (with $m\neq0$) has slope $-\dfrac{1}{m}$.
Examples (step-by-step)
Example 1 — slope between two points
Points: $(2,3)$ and $(5,11)$.
Steps:
- Compute changes: $\Delta y=11-3=8$, $\Delta x=5-2=3$.
- Apply formula: $$m=\frac{y_2-y_1}{x_2-x_1}=\frac{11-3}{5-2}=\frac{8}{3}.$$
Therefore the slope is $\frac{8}{3}$.
Example 2 — slope from an equation in standard form
Equation: $3x-2y=6$.
Steps:
- Use $m=-\dfrac{A}{B}$ where $A=3$, $B=-2$: $$m=-\frac{A}{B}=-\frac{3}{-2}=\frac{3}{2}.$$
- (Optional) Solve for $y$: $-2y=-3x+6\Rightarrow y=\frac{3}{2}x-3$ confirms $m=\frac{3}{2}$.
Therefore the slope is $\frac{3}{2}$.
Example 3 — horizontal and vertical lines
- Horizontal line: $y=4$ — slope $m=0$ (no change in $y$).
- Vertical line: $x=-1$ — slope is undefined (change in $x$ is $0$, division by zero).
Example 4 — perpendicular slope
If a line has slope $m_1=2$, a line perpendicular to it has slope $$m_2=-\frac{1}{2}.$$
Quick how-to checklist
- If you have two points: use $m=\dfrac{y_2-y_1}{x_2-x_1}$ (watch order — use same order for both coordinates).
- If equation is $y=mx+b$: slope is $m$.
- If equation is $Ax+By=C$: slope is $-\dfrac{A}{B}$.
- Horizontal → $m=0$. Vertical → slope undefined.
- Parallel → same $m$. Perpendicular → negative reciprocal $-\dfrac{1}{m}$.
If you want, give me a specific pair of points or an equation and I’ll compute the slope step-by-step.