Ok, I get that, but the value would not be greater than the sum of the 3, because the price of the metals are price per ounce, and the item will have less than an ounce of each metal. Maybe a gram or 2. So will that still work? If I am looking at this right, x would be the constant, ie the amount of metals in the item?
Of course the weight of the unit exceeds the weight of some of its materials. But that is not what the equation is about.
You seem to be assuming that the price at which an item is exchanged equals the cost of the materials. This is generally not even close to correct. Do you think that the price of a car is simply the sum of the prices of the parts. In fact, the greatest component of the price of any physical good is almost always labor cost, not cost of materials.
Cost of materials and of labor place a floor under the sustainable price of any item. They do not impose a ceiling.
The variables [MATH]m_1, \ m_2, \ m_3[/MATH] represent weights of metals used in each item expressed in a consistent unit of weight, say ounces of metal used per item.
The functions [MATH]p_1(t), \ p_2(t), \ p_3(t)[/MATH] represent units of money per unit of weight of metal, say dollars per ounce of metal.
And v(t) represents the price of one of the items. Everything then is expressed in monetary terms, and x does not represent a weight. It represents how much greater the price of the item is than the cost of the metals used in manufacturing the item.
It is about finding how much greater is the price of the item than the cost of those three materials. If that differential in value is constant in time, and only then, you can determine the weights of metal used by solving a system of four simultaneous equations.
P.S. After reading Dr. Peterson's response, if your observations are close in time and the percentage differences in the price of the item are small relative to the percentage changes in the costs of the metal, then the assumption that x is constant will be a very plausible assumption. Did you understand his comment about extra data to check the validity of the assumption?