To find the value of $x$ or $y$ so that the line through the points $(12, 9)$ and $(x, 8)$ has the given slope $m = -\frac{1}{7}$, we use the formula for the slope of a line passing through two points:

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

Substituting the given values:

$-\frac{1}{7} = \frac{8 - 9}{x - 12}$

$-\frac{1}{7} = \frac{-1}{x - 12}$

Solve for $x$:

1. Cross-multiply to eliminate the fractions: $-1 \cdot (x - 12) = -7 \cdot 1$

2. Simplify: $-x + 12 = -7$

3. Rearrange to isolate $x$: $-x = -7 - 12$

$-x = -19$

$x = 19$

Final Answer:

The value of $x$ is 19.

Would you like further explanation or additional examples?

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Related Questions:

1. How is the slope formula applied in different coordinate systems?
2. What are alternate ways to derive the slope-intercept form from a slope calculation?
3. How can we determine if three points lie on the same straight line using slopes?
4. How can this problem change if $x$ or $y$ was a parameter instead of a variable?
5. How do vertical and horizontal lines differ when calculating slope?

Tip: Always cross-check your slope calculation to avoid sign errors, especially with negative values.