Finding the slope from two points: 3 worked examples

The slope $m$ of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is calculated using the formula:
$m = \frac{y_2 - y_1}{x_2 - x_1}$

Given the points $(2, 5)$ and $(6, 13)$, we assign:
$x_1 = 2$, $y_1 = 5$
$x_2 = 6$, $y_2 = 13$

Substituting these values into the formula:
$m = \frac{13 - 5}{6 - 2}$
$m = \frac{8}{4}$
$m = 2$

To find the slope of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$, we use the slope formula:
$m = \frac{y_2 - y_1}{x_2 - x_1}$

Given the points $(-3, 4)$ and $(5, -2)$, we identify the coordinates as:
$x_1 = -3$
$y_1 = 4$
$x_2 = 5$
$y_2 = -2$

Substitute these values into the slope formula:
$m = \frac{-2 - 4}{5 - (-3)}$

Simplify the numerator and the denominator:
$m = \frac{-6}{5 + 3}$
$m = \frac{-6}{8}$

Finally, simplify the fraction by dividing the numerator and the denominator by their greatest common divisor, which is $2$:
$m = -\frac{3}{4}$

To find the slope of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$, we use the slope formula:
$m = \frac{y_2 - y_1}{x_2 - x_1}$

Given the points $(1, -7)$ and $(4, -7)$, we identify the coordinates:
$x_1 = 1$
$y_1 = -7$
$x_2 = 4$
$y_2 = -7$

Substitute these values into the formula:
$m = \frac{-7 - (-7)}{4 - 1}$
$m = \frac{-7 + 7}{3}$
$m = \frac{0}{3}$
$m = 0$

Since the $y$-coordinates are the same, the line is a horizontal line, which always has a slope of $0$.