We can use the point-slope formula to find an equation. However, we must first 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 values from the points in the problem and solving gives:
#m = (color(red)(4) - color(blue)(-4))/(color(red)(3) - color(blue)(-1))#
#m = (color(red)(4) + color(blue)(4))/(color(red)(3) + color(blue)(1))#
#m = 8/4 = 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.
We can substitute the slope we calculated and the values from the first point to give:
#(y - color(red)(-4)) = color(blue)(2)(x - color(red)(-1))#
#(y + color(red)(4)) = color(blue)(2)(x + color(red)(1))#
We can also substitute the slope we calculated and the values from the second point to give:
#(y - color(red)(4)) = color(blue)(2)(x - color(red)(3))#
Or we can solve for #y# to put this equation in the more familiar slope-intercept form:
#y - color(red)(4) = (color(blue)(2) xx x) - (color(blue)(2) xx color(red)(3))#
#y - color(red)(4) = 2x - 6#
#y - color(red)(4) + 4 = 2x - 6 + 4#
#y - 0 = 2x - 2#
#y = 2x - 2#
These equations are solutions to this problem:
#(y + color(red)(4)) = color(blue)(2)(x + color(red)(1))#
#(y - color(red)(4)) = color(blue)(2)(x - color(red)(3))#
#y = 2x - 2#
