First, we find the slope of the line.
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)(-5) - color(blue)(4))/(color(red)(2) - color(blue)(-5))#
#m = (color(red)(-5) - color(blue)(4))/(color(red)(2) + color(blue)(5))#
#m = -9/7#
We can now use the point-slope formula to find an 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)(-9/7)(x - color(red)(-5))#
#(y - color(red)(4)) = color(blue)(-9/7)(x + color(red)(5))#
Now, we must transform this equation to standard form.
The standard form of a linear equation is:
#color(red)(A)x + color(blue)(B)y = color(green)(C)#
where, if at all possible, #color(red)(A)#, #color(blue)(B)#, and #color(green)(C)#are integers, and A is non-negative, and, A, B, and C have no common factors other than 1
First, multiply both sides of the equation by #color(red)(7)# to eliminate the fraction:
#color(red)(7)(y - 4) = color(red)(7) xx -9/7(x + 5)#
#7y - 28 = cancel(color(red)(7)) xx -9/color(red)(cancel(color(black)(7)))(x + 5)#
#7y - 28= -9((x + 5)#
#7y - 28 = -9x - 45#
#color(red)(9x) + 7y - 28 + color(blue)(28) = color(red)(9x) + - 9x - 45 + color(blue)(28)#
#9x + 7y - 0 = 0 - 45 + 28#
#9x + 7y = -17#
