We can use the point-slope formula to write an equation. However, we must 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)(-7) - color(blue)(3))/(color(red)(0) - color(blue)(-5))#
#m = (color(red)(-7) - color(blue)(3))/(color(red)(0) + color(blue)(5)) = -10/5 = -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 first point giving:
#(y - color(red)(3)) = color(blue)(-2)(x - color(red)(-5))#
#(y - color(red)(3)) = color(blue)(-2)(x + color(red)(5))#
We can also substitute the slope we calculated and the second point giving:
#(y - color(red)(-7)) = color(blue)(-2)(x - color(red)(0))#
#(y + color(red)(7)) = color(blue)(-2)x#
Or, we can solve this equation for #y# to give an equation in slope-intercept form:
#y + color(red)(7) = color(blue)(-2)x#
#y + color(red)(7) - 7 = color(blue)(-2)x - 7#
#y + 0 = color(blue)(-2)x - 7#
#y = color(blue)(-2)x - 7#
