First, we need to determine the slope of the line passing through the two points in the problem. 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)(1) - color(blue)(4))/(color(red)(-1) - color(blue)(1)) = (-3)/-2 = 3/2#
Now, use the point-slope formula to write and 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.
Substituting the slope we calculated and the first point from the problem gives:
#(y - color(red)(4)) = color(blue)(3/2)(x - color(red)(1))#
We can also substitute the slope we calculated and the second point from the problem giving:
#(y - color(red)(1)) = color(blue)(3/2)(x - color(red)(-1))#
#(y - color(red)(1)) = color(blue)(3/2)(x + color(red)(1))#
Or, we can solve either of these equations for #y# to put the equation into slope-intercept form. 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)(1) = (color(blue)(3/2) xx x) + (color(blue)(3/2) xx color(red)(1))#
#y - color(red)(1) = 3/2x + 3/2#
#y - color(red)(1) + 1 = 3/2x + 3/2 + 1#
#y - 0 = 3/2x + 3/2 + (2/2 xx 1)#
#y = 3/2x + 3/2 + 2/2#
#y = color(red)(3/2)x + color(blue)(5/2)#
