Is there a mistake with the units here? The explanation sets the cost of the mixture at 70(10). This would mean the mixture is 10 pounds and the problem asks for a 10 oz mixture.

Jaime, if the explanation is confusing, follow this.

Formal algebraic way of solving. Notice that it's useful in problems like this to write out the units of measure.
x = weight of nut 1 in the mix, in ounces
y = weight of nut 2 in the mix, in ounces

Cost of x = 80 cents / 16 ounces * x ounces = 80x/16 cents
Cost of y = 60 cents / 16 ounces * x ounces = 60x/16 cents
Cost of (x+y) = 70 cents / 16 ounces * 10 ounces = 700/16 cents

Cost of x + Cost of y = Cost of (x+y)
80x/16 cents + 60x/16 cents = 700/16 cents; multiply both sides by 16 cents
80x + 60x = 700
140x = 700
x=5

However, if we simply look at the problem, we see that 70 cents / pound is the exact average of 80 cents / pound and 60 cents / pound, so logically if we mix half of each in a 10 ounce mixture, we will get the target cost. So 5 ounces of x and 5 ounces of y! Without even doing any algebraic setup.