A direct variation equation is one that requires y varies directly as x and looks like this in equation form:
[tex] \frac{y}{x} =k [/tex]
where k is the constant of variation. If we solve this for y, we have y = kx, which happens to be a linear function... a line. k here, then, serves as the slope. So what we are given as points on a direct variation function are actually points on a line. The equation for this requires that we find the slope and then rewrite the formula accordingly. First the slope:
[tex] m(k)=\frac{-4-(-3)}{-12-(-9)}=\frac{-4+3}{-12+9}=\frac{1}{3} [/tex]
Now we need to write the equation by using one of the points' coordinates. I picked the first point that has an x coordinate of -9 and a y coordinate of -3. Fitting those into the slope-intercept form of a line,
[tex] -3=\frac{1}{3}(-9)+b [/tex]
which simplifies to
-3 = -3 + b and b = 0. That means that the equation of direct variation is
[tex] y=\frac{1}{3}x+0 [/tex] or just
[tex] y=\frac{1}{3}x [/tex]
