First, we need to determine the slope of the line running through the two points. 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)(385) - color(blue)(169))/(color(red)(10) - color(blue)(4)) = 216/6 = 36#
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 can substitute the slope we calculated for #m# and the values from one of the points can be substituted for #x# and #y# and we can solve for #b#:
#385 = (color(red)(36) * 10) + color(blue)(b)#
#385 = 360 + color(blue)(b)#
#-color(red)(360) + 385 = -color(red)(360) + 360 + color(blue)(b)#
#25 = 0 + color(blue)(b)#
#25 = color(blue)(b)#
#color(blue)(b) = 25#
We can now substitute the slope and value for #b# we calculated into the formula to obtain the formula for the line:
#y = color(red)(36)x + color(blue)(25)#
