Given two points we can use the point-slope formula to find an equation for a line.
First, determine 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 gives:
#m = (color(red)(3) - color(blue)(2))/(color(red)(-7) - color(blue)(5))#
#m = 1/-12 = -1/12#
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 use the first point and the slope we calculate to give:
#(y - color(red)(2)) = color(blue)(-1/12)(x - color(red)(5))#
We can also use the second point and the slope we calculate to give:
#(y - color(red)(3)) = color(blue)(-1/12)(x - color(red)(-7))#
#(y - color(red)(3)) = color(blue)(-1/12)(x + color(red)(7))#
We can also solve this for #y# to give an equation in slope-intercept form:
#y - color(red)(3) = (color(blue)(-1/12) xx x) + (color(blue)(-1/12) xx color(red)(7))#
#y - color(red)(3) = color(blue)(-1/12)x - 7/12#
#y - color(red)(3) + 3 = color(blue)(-1/12)x - 7/12 + 3#
#y - 0 = color(blue)(-1/12)x - 7/12 + 36/12#
#y = -1/12x + 29/12#
