Question

A unicorn bought ten pieces of candy and paid $16. Large candies cost $4 each, medium-sized candies cost $2 a piece, and small candies come at $1 each. How many large candies did the unicorn buy?

Answer 1

There are three possible solutions:

1. 6 medium-sized candies and 4 small candies.
2. 1 large candy, 3 medium-sized candies and 6 small candies.
3. 2 large candies and 8 small candies.

How to find possible solution from a system of equations whose number of variables is greater than the number of equations

Let be $x$, $y$ and $z$ the quantities of large, medium-sized and small candies bought by the unicorn., which represent non-negative integers. After a quick reading, we build the following system of linear equations:

$x + y + z = 10$  (1)

$4\cdot x +2\cdot y +z = 16$   (2)

We notice that the number of variables is greater than the number of linear equations. Thus, the system have several solutions. By (1):

$z = 10 - x - y$

(1) in (2):

$4\cdot x + 2\cdot y + 10 - x - y = 16$

$3\cdot x + y = 6$ (3)

After some inspections, we find three possible solutions:

1. $x = 0$, $y = 6$, $z = 4$
2. $x = 1$, $y = 3$, $z = 6$
3. $x = 2$, $y = 0$, $z = 8$

There are three possible solutions:

1. 6 medium-sized candies and 4 small candies.
2. 1 large candy, 3 medium-sized candies and 6 small candies.
3. 2 large candies and 8 small candies.

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