Final answer:
The equation of the line that passes through points A (3, 2) and B (-2, 4) with the x-axis bisecting the line segment is y = (-2/5)x + 16/5.
Step-by-step explanation:
To find the equation of the line that passes through points A (3, 2) and B (-2, 4), we need to determine the slope (m) of the line. The slope of a line that goes through two points (x1, y1) and (x2, y2) is calculated using the formula m = (y2 - y1) / (x2 - x1). Plugging in our points, we get m = (4 - 2) / (-2 - 3) = 2 / -5 = -2/5. With the slope and one point, we can use the point-slope form of the equation of a line, which is y - y1 = m(x - x1). Using point A and our slope, we get: y - 2 = (-2/5)(x - 3). Expanding and simplifying this equation, we get:
y - 2 = (-2/5)x + 6/5
y = (-2/5)x + 6/5 + 10/5
y = (-2/5)x + 16/5
This is the equation of the line that passes through points A and B. The line's equation in slope-intercept form is y = (-2/5)x + 16/5, where -2/5 is the slope and 16/5 is the y-intercept.