What is 300 mi converted into km?
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now, if we a "prime factoring" of each number, we get
204 = 2*2*3*17
120 = 2*2*2*3*5
168 = 2*2*2*3*7
now, notice the GCD for all three is that common bolded values, namely 2*2*3, or 12, so each room will house 12 participants,
now, how many rooms for all German? 204 ÷ 12, or 17 rooms.
how many rooms for all English ones? 120 ÷ 12, or 10 rooms.
how many rooms for the French ones? 168 ÷ 12, or 14 rooms.
so the total amount of rooms for all of them is 17 + 10 + 14.
The set is linearly dependent.
To explicitly prove this, we need to show there is at least one choice of constants such that
or equivalently,
which is the same as solving the system of equations
From the first and last equations, we have and
. Substituting these into the second equation leaves us with
, and so the overall solution set is
for which there are infinitely many not-all-zero solutions.