First, we need to determine the slope of the line for the point-slope form and the slope intercept form. 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 points from the problem and calculating gives:
#m = (color(red)(2) - color(blue)(0))/(color(red)(0) - color(blue)(1)) = 2/-1 = -2#
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. Substituting the slope we calculate and the first point from the problem gives:
#(y - color(red)(0)) = color(blue)(-2)(x - color(red)(1))# or #y = -2(x - 1)#
We can also substitute the slope we calculate and the second point from the problem gives:
#(y - color(red)(2)) = color(blue)(-2)(x - color(red)(0))# or #(y - 2) = -2x#
Because we have the slope and by definition the y-intercept is the point #(0, 2)# we can substitute this into the slope-intercept formula. 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.
#y = color(red)(-2)x + color(blue)(2)#
