Question

A fruit stand has to decide what to charge for their produce. They need
\[\$10\] for
\[4\] apples and
\[4\] oranges. They also need
\[\$15\] for
\[6\] apples and
\[6\] oranges. We put this information into a system of linear equations.
Can we find a unique price for an apple and an orange?

Answer 1

Let a be the price of an apple and o the price of an orange (in dollars). The information gives the system
4a + 4o = 10
6a + 6o = 15.

Divide the first equation by 4 and the second by 6 to get
a + o = 2.5
a + o = 2.5.

The two equations are the same (the second is 1.5 times the first), so the system is dependent and has infinitely many solutions. Any pair (a,o) with a + o = 2.5 satisfies the data (e.g. a = 1.00, o = 1.50 or a = 1.25, o = 1.25). Thus you cannot determine unique prices for the apple and the orange from this information alone.