First, we need to determine the slope of the line. The formula for find the slope of a line is:

#m = (color(red)(y_2) - color(blue)(y_1))/(color(red)(x_2) - color(blue)(x_1))#

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

Substituting the values from the points in the problem gives:

#m = (color(red)(4) - color(blue)(0))/(color(red)(4) - color(blue)(-12)) = (color(red)(4) - color(blue)(0))/(color(red)(4) + color(blue)(12)) = 4/16 = 1/4#

Now, we can use the point-slope formula to write and equation for the line. The point-slope form of a linear equation is: #(y - color(blue)(y_1)) = color(red)(m)(x - color(blue)(x_1))#

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

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

#(y - color(blue)(0)) = color(red)(1/4)(x - color(blue)(-12))#

#y = color(red)(1/4)(x + color(blue)(12))#

We can modify this result to put the equation in 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)(1/4)(x + color(blue)(12))#

#y = (color(red)(1/4) xx x) + (color(red)(1/4) xx color(blue)(12))#

#y = color(red)(1/4)x + color(blue)(12)/(color(red)(4)#

#y = color(red)(1/4)x + color(blue)(3)#