First, we need to find the slope of the equation. 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)(-8) - color(blue)(0))/(color(red)(6) - color(blue)(3)) = -8/3#
Next, we can 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 calculate and the first point from the problem gives:
#(y - color(red)(0)) = color(blue)(-8/3)(x - color(red)(3))#
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. We will convert the formula above to the standard form as follows:
#y = (color(blue)(-8/3) xx x) - (color(blue)(-8/3) xx color(red)(3))#
#y = -8/3x + 8#
#color(red)(8/3x) + y = color(red)(8/3x) - 8/3x + 8#
#8/3x + y = 0 + 8#
#8/3x + y = 8#
#color(red)(3)(8/3x + y) = color(red)(3) xx 8#
#(color(red)(3) xx 8/3x) + (color(red)(3) xx y) = 24#
#color(red)(8)x + color(blue)(3)y = color(green)(24)#
