What is the equation of the line between #(-1,12)# and #(31,16)#?

Fist, 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)(16) - color(blue)(12))/(color(red)(31) - color(blue)(-1)) = (color(red)(16) - color(blue)(12))/(color(red)(31) + color(blue)(1)) = 4/32 = 1/8#

Now, we can use this point-slope formula to write an 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)(12)) = color(red)(1/8)(x - color(blue)(-1))#

#(y - color(blue)(12)) = color(red)(1/8)(x + color(blue)(1))#

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

#(y - color(blue)(16)) = color(red)(1/8)(x - color(blue)(31))#