To find the equation of a line passing through two points, we can use the point-slope form:

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

where $m$ is the slope of the line, and $(x_1, y_1)$ is one of the points on the line. We are given the points $(4, -2)$ and $(6, 1)$.

Step 1: Find the slope, $m$

The slope formula for a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

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

Substituting the coordinates:

$m = \frac{1 - (-2)}{6 - 4} = \frac{1 + 2}{2} = \frac{3}{2}$

Step 2: Use the point-slope form

Now that we have the slope $m = \frac{3}{2}$, we can substitute it along with one of the points (say, $(4, -2)$) into the point-slope form:

$y - (-2) = \frac{3}{2}(x - 4)$

Simplify:

$y + 2 = \frac{3}{2}x - 6$

Step 3: Write in slope-intercept form

To convert this to slope-intercept form $y = mx + b$, subtract 2 from both sides:

$y = \frac{3}{2}x - 8$

Final Answer

The equation of the line passing through the points $(4, -2)$ and $(6, 1)$ is:

$y = \frac{3}{2}x - 8$

Would you like further details, or do you have any questions?

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Further Practice Questions:

1. How would you find the equation of a line with a given slope that passes through a specific point?
2. Can you find the equation of a line that passes through the origin and a given point?
3. How do you determine if two lines are parallel or perpendicular using their equations?
4. What would the slope be for a line that is vertical or horizontal?
5. Can you derive the distance formula using points on a line?

Tip: Remember, the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ helps determine whether lines are parallel (same slope) or perpendicular (negative reciprocal slopes).