If the line has the origin, (0, 0), as a point on the line and a slope of #-6# 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 given in the problem and the origin gives:

#(y - color(red)(0)) = color(blue)(-6)(x - color(red)(0))#

#y = -6x#

We can convert this to the standard form for a linear equation. 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 add #color(red)(6x)# to each side of the equation to start the conversion and keep the equation balanced:

#color(red)(6x) + y = color(red)(6x) - 6x#

#6x + y = 0#

Or

#color(red)(6)x + color(blue)(1)y = color(green)(0)#