The equation of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ can be written using the two-point form:

$y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)$

Given Points:

• $(x_1, y_1) = (10, 5)$
• $(x_2, y_2) = (8, 3)$

Step 1: Find the slope ($m$):

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 5}{8 - 10} = \frac{-2}{-2} = 1$

Step 2: Substitute into the two-point form:

$y - y_1 = m(x - x_1)$

Substitute $y_1 = 5$, $x_1 = 10$, and $m = 1$: $y - 5 = 1(x - 10)$

Step 3: Simplify:

$y - 5 = x - 10$

$y = x - 5$

Final Equation of the Line:

$y = x - 5$

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Do you want to see more about verifying this result or graphing it? Let me know!

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Questions for Further Exploration:

1. How do you derive the two-point form of a line equation?
2. Can we convert $y = x - 5$ into standard form $Ax + By + C = 0$?
3. What would the equation look like if the slope were undefined (vertical line)?
4. How would the equation change if the points were flipped (e.g., $(8, 3)$ first)?
5. What are the intercepts of the line $y = x - 5$?

Tip: Always check the slope calculation carefully when using two-point form; it's a common source of mistakes.