standard_description: Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
problem_activity: Task
The King of Syracuse reportedly requested Archimedes' advice for determining
if a crown was made with the appropriate mixture of gold and silver.   Archimedes
devised a method requiring the following measurements:
the volume $V$ of water displaced by the crown when it is submerged
in water,
the volume $V_1$ of water displaced by an equal mass (that is, equal to the
mass of the crown)  of gold when
submerged in the water,
the volume $V_2$ of water displaced by an equal mass of silver when
submerged in water.
How much water would a solid gold crown displace?  What about a solid
silver crown?  What if the crown is half gold (by volume) and half silver (by volume)?
If $p$ denotes the percentage of the crown, by volume, which comes from
gold, how much water will the crown displace?
Show that Archimedes' method can always determine the percentage of gold in the crown from its mass and volume.


aligns: no