We can use the point-slope formula to find a line passing through these two points.
First, however, we must use the two points to 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 given in the problem produces:
#m = (color(red)(3) - color(blue)(-2))/(color(red)(4) - color(blue)(9))#
#m = (color(red)(3) + color(blue)(2))/(color(red)(4) - color(blue)(9))#
#m = 5/-5 = -1#
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 now substitute the first point from the problem and the slope we calculated to obtain an equation:
#(y - color(red)(-2)) = color(blue)(-1)(x - color(red)(9))#
#(y + color(red)(2)) = color(blue)(-1)(x - color(red)(9))#
We also substitute the second point from the problem and the slope we calculated to obtain another equation:
#(y - color(red)(3)) = color(blue)(-1)(x - color(red)(4))#
We can solve this problem for #y# to obtain the equation in the familiar slope-intercept form:
#y - color(red)(3) = (color(blue)(-1) xx x) - (color(blue)(-1) xx color(red)(4))#
#y - color(red)(3) = -1x + 4#
#y - color(red)(3) + 3 = -1x + 4 + 3#
#y - 0 = -1x + 7#
#y = -1x + 7# or #y = -x + 7#
