First, we need to determine the slope of the line. 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)(6) - color(blue)(1))/(color(red)(-3) - color(blue)(4)) = 5/-7 = -5/7#
The standard form of a linear equation is: #color(red)(A)x + color(blue)(B)y = color(green)(C)#
Where, if at all possible, #color(red)(A)#, #color(blue)(B)#, and #color(green)(C)#are integers, and A is non-negative, and, A, B, and C have no common factors other than 1
The slope of an equation in standard form is: #m = -color(red)(A)/color(blue)(B)#
We can substitute the values from the slope for #color(red)(A)# and #color(blue)(B)# giving:
#color(red)(5)x + color(blue)(7)y = color(green)(C)#
We can substitute the values from one of the points in the problem for #x# and #y# to calculate #color(green)(C)#:
#(color(red)(5) xx 4) + (color(blue)(7) xx 1) = color(green)(C)#
#20 + 7 = color(green)(C)#
#27 = color(green)(C)#
Substituting this back into our equation gives:
#color(red)(5)x + color(blue)(7)y = color(green)(27)#
