Each of our next two sample problems entails a set of purchases that can be analyzed as a system of linear relationships.

The first is stated in terms of the *sous*, a coin mentioned earlier, and the *setier*, a measure of volume for liquids and grains that is roughly two gallons (it was one of the units swept away by the French Revolution). See whether you can figure it out.

**Problem 3. **A merchant paid 10 *liras*, which is 200 *sous*, for two manners of grain, namely wheat and oats, and he purchased each *setier* of wheat for 10 *sous* and each *setier* of oats for 5 *sous*. This merchant turned around and sold his grain, selling each *setier* of oats for 4 *sous* and each *setier* of wheat for 12 *sous*, and realized a profit of 10 *sous*. I ask how many *setiers* of wheat and how many of oats he purchased, and how much money he paid for each grain. (Sesiano 1984, p. 56)

Solutions to Problem 3 by double false position, by algebra (simple elimination), and by matrix algebra

The other sample problem mentions a gold coin of France called a “lamb of gold” (*moto d’aur* in Languedocian or *mouton d’or* in modern French, cognate with the English word “mutton”). It was worth 12½ *sous*. The coin contained an image of a lamb accompanied by a Latin inscription to Christ, who is often referred to by Christians as Agnus Dei, the Lamb of God. See the photograph of a *moto* (a *moto photo*!) below.

**Figure 9.** A “lamb of gold” coin from France, this one struck in the year 1311. Abbreviated in Latin is an invocation to Christ, *Agnus dei qui tollit peccata mundi miserere nobis* (“Lamb of God who bears the sins of the world, take pity on us”). (Image: Classical Numismatic Group, Inc. / Wikimedia Commons, licensed under the GNU Free Documentation License.)

**Problem 4.** A merchant purchased three pieces of cloth that cost him [a total of] 30 *motos*, and doesn’t know with certainty what each of the pieces cost, but does know that the second cost double the first and 4 more; the third cost three times as much as the second, less 7. I ask what each one cost. (Sesiano 1984, p. 55)

Solutions to Problem 4 by double false position and by algebra