First, we need to determine the slope. The slope can be found by using the formula: #m = (color(red)(y_2) - color(blue)(y_1))/(color(red)(x_2) - color(blue)(x_1))#

Where #m# is the slope and (#color(blue)(x_1, y_1)#) and (#color(red)(x_2, y_2)#) are the two points on the line.

Substituting the values from the points in the problem gives:

#m = (color(red)(4) - color(blue)(-2))/(color(red)(6) - color(blue)(3)) = (color(red)(4) + color(blue)(2))/(color(red)(6) - color(blue)(3)) = 6/3 = 2#

Next, we can use the point-slope formula to write and equation for the line. The point-slope formula states: #(y - color(red)(y_1)) = color(blue)(m)(x - color(red)(x_1))#

Where #color(blue)(m)# is the slope and #color(red)(((x_1, y_1)))# is a point the line passes through.

Substituting the slope we calculated and the first point from the problem gives:

#(y - color(red)(-2)) = color(blue)(2)(x - color(red)(3))#

#(y + color(red)(2)) = color(blue)(2)(x - color(red)(3))#

We can also substitute the slope we calculated and the second point from the problem giving:

#(y - color(red)(4)) = color(blue)(2)(x - color(red)(6))#

We can also solve this equation for #y# to put it in the slope-intercept form. The slope-intercept form of a linear equation is: #y = color(red)(m)x + color(blue)(b)#

Where #color(red)(m)# is the slope and #color(blue)(b)# is the y-intercept value.

#y - color(red)(4) = (color(blue)(2) xx x) - (color(blue)(2) xx color(red)(6))#

#y - color(red)(4) = 2x - 12#

#y - color(red)(4) + 4 = 2x - 12 + 4#

#y - 0 = 2x - 8#

#y = color(red)(2)x - color(blue)(8)#