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)(-10) - color(blue)(-7))/(color(red)(0) - color(blue)(2)) = (color(red)(-10) + color(blue)(7))/(color(red)(0) - color(blue)(2)) = (-3)/-2 = 3/2#
We know from the problem the #y# intercept is #(0, -10)# or #-10#
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.
Substituting the slope we calculated and the #y#-intercept gives:
#y = color(red)(3/2)x - color(blue)(10)#
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.
We can substitute the slope we calculated and the values from the first point in the problem giving:
#(y - color(blue)(-7)) = color(red)(3/2)(x - color(blue)(2))#
#(y + color(blue)(7)) = color(red)(3/2)(x - color(blue)(2))#
We can also substitute the slope we calculated and the values from the second point in the problem giving:
#(y - color(blue)(-10)) = color(red)(3/2)(x - color(blue)(0))#
#(y + color(blue)(10)) = color(red)(3/2)x#
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
We can take this last point slope equation and convert it to standard form as follows:
#2(y + color(blue)(10)) = 2 xx color(red)(3/2)x#
#2y + 20 = 3x#
#2y - 2y + 20 = 3x - 2y#
#0 + 20 = 3x - 2y#
#20 = 3x - 2y#
#3x - 2y = 20#
The are no constraints on the value #x# can take in this equation therefore the Domain is the set of All Real Numbers or #{RR}#
Because this is a pure linear transformation the Range of this equation is also the set of All Real Numbers or #{RR}#
