Find the sum of the weights of a green and a red ball. As before, we have $ 5g + 2r =10 $ and $ g+4r=7 $. Before solving this system of equations, we note that we seek $ 8g+8r, $ which equals $ 8(g+r) $. So, if we can find $ g+r, $ we can find the total weight of the balls $ \textit{without finding the weight of each ball} $. Looking at our equations, we see a total of $ 6g $ and $ 6r $ on the left, so adding the two equations can get us to $ g+r $. Adding the equations gives $ 6g+6r = 17, $ and dividing both sides by $ 6 $ gives $$g+r = \frac{17}{6}.$$ Therefore, we have $$8g+8r= 8(g+r) = 8\cdot\frac{17}{6} = \boxed{\frac{68}{3}\text{ pounds}}.$$