To find the equation of the line passing through the points $(6, 5)$ and $(1, -5)$, follow these steps:

1. Calculate the Slope ($m$):

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

Substitute the given points $(6, 5)$ and $(1, -5)$: $m = \frac{-5 - 5}{1 - 6} = \frac{-10}{-5} = 2$

2. Use the Point-Slope Form to Find the Equation:

The point-slope form of a line's equation is: $y - y_1 = m(x - x_1)$

Using the slope $m = 2$ and one of the points, say $(6, 5)$: $y - 5 = 2(x - 6)$

3. Simplify to the Slope-Intercept Form:

Expand and simplify this equation to get it in the slope-intercept form $y = mx + b$: $y - 5 = 2x - 12$ $y = 2x - 12 + 5$ $y = 2x - 7$

So, the equation of the line passing through the points $(6, 5)$ and $(1, -5)$ is: $\boxed{y = 2x - 7}$

Would you like more details or have any questions?

Related Questions:

1. How do you find the equation of a line given a slope and one point?
2. How can you determine if two lines are parallel or perpendicular?
3. What is the point-slope form of a line, and how is it derived?
4. How can you find the x-intercept and y-intercept of a line from its equation?
5. How do you convert a linear equation from standard form to slope-intercept form?

Tip: Always verify your calculations by substituting the coordinates of the points back into the final equation to ensure they satisfy it.