We will use the point-slope formula to first define the equation. First however we need to first use the two points from the problem to find the slope:
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 two points from the problem gives the slope as:
#m = (color(red)(-3) - color(blue)(4))/(color(red)(1) - color(blue)(2))#
#m = -7/-1 = 7#
Now that we have the slope we can use it and one of the points in the point-slope formula to get an equation for the line.
The point-slope formula states: #(y - color(red)(y_1)) = color(blue)(m)(x - color(red)(x_1))#
Where #color(blue)(m)# is the slope and #color(red)(((x_1, y_1)))# is a point the line passes through.
#(y - color(red)(-3)) = color(blue)(7)(x - color(red)(1))#
#(y + color(red)(3)) = color(blue)(7)(x - color(red)(1))#
We can now transform this into the standard form for a linear equation by doing the necessary mathematics.
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
#y + color(red)(3) = (color(blue)(7) xx x) - (color(blue)(7) xx color(red)(1))#
#y + 3 = 7x - 7#
#y + 3 - color(red)(7x) - color(blue)(3) = 7x - 7 - color(red)(7x) - color(blue)(3)#
# - color(red)(7x) + y + 3 - color(blue)(3) = 7x - color(red)(7x) - 7 - color(blue)(3)#
# -7x + y + 0 = 0 - 7 - 3#
# -7x + y = -10#
#-1(-7x + y) = -1 xx -10#
#7x - y = 10#
#color(red)(7)x - color(blue)(1)y = color(green)(10)#
