To find the slope $m$ of a line passing through two points, we use the formula for slope:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Given points are $(0, 1)$ and $\left( -\frac{1}{6}, \frac{7}{8} \right)$.

Substitute these values into the formula:

$m = \frac{\frac{7}{8} - 1}{-\frac{1}{6} - 0}$

First, simplify the numerator:

$\frac{7}{8} - 1 = \frac{7}{8} - \frac{8}{8} = -\frac{1}{8}$

Now simplify the denominator:

$-\frac{1}{6} - 0 = -\frac{1}{6}$

Now divide the two fractions:

$m = \frac{-\frac{1}{8}}{-\frac{1}{6}} = \frac{1}{8} \times \frac{6}{1} = \frac{6}{8} = \frac{3}{4}$

Thus, the slope $m$ is:

$m = \frac{3}{4}$

Let me know if you want more details or have any questions.

Here are five related questions:

1. How do you find the slope if one of the points is the origin?
2. What happens to the slope if the x-coordinates of both points are the same?
3. Can two points on a vertical line have a defined slope?
4. How does changing the order of points affect the slope calculation?
5. What is the significance of a slope of 0?

Tip: The slope formula is a ratio, so understanding fractions simplifies slope calculations.