First, we need to find the slope of the equation. 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)(-3) - color(blue)(2))/(color(red)(3) - color(blue)(-7)) = (color(red)(-3) - color(blue)(2))/(color(red)(3) + color(blue)(7)) = -5/10 = -1/2#

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.

We know #m = -1/2# and can substitute this into the equation giving:

#y = color(red)(-1/2)x + color(blue)(b)#

We can also substitute the values from the first point in the problem and solve for #b#:

#2 = (color(red)(-1/2) xx -7) + color(blue)(b)#

#2 = 7/2 + color(blue)(b)#

#color(red)(-7/2) + 2 = color(red)(-7/2) + 7/2 + color(blue)(b)#

#color(red)(-7/2) + (2/2 xx 2) = 0 + color(blue)(b)#

#color(red)(-7/2) + 4/2 = color(blue)(b)#

#-3/2 = color(blue)(b)#

We can now substitute this along with the slope into the slope-intercept form to give:

#y = color(red)(-1/2)x + color(blue)(-3/2)#

#y = color(red)(-1/2)x - color(blue)(3/2)#