Suppose I have three stones in my hands. If I have two in my left hand, how many are in my right hand?

Denote by $a$ the number of stones in my left hand. Denote by $b$ the number of stones in my right hand. Then the number of stones in both hands is $a + b$. Both $a$ and $b$ are natural numbers. In other words, $a$ is $0$ or $1$ or $2$ or... Likewise for $b$.

We express that I have two stones in my left
hand by the equation $a = 2$.
We express that I have have five stones in
total by the equation that $a + b = 5$.
Because we have identified the names $a$ and
$b$ with tangible objects in my hands, these
equations are statements about tangible objects
in my hands.
But the *numbers* involved are intangible
objects.
In any case, we have two equations in two
unknowns.

We express the question “how many stones do I have in my right hand?” by asking for a solution (see Equations) to the equation $a + b = 5$. We can start by trying natural numbers in order. Is $(a, b) = (0, 0)$ a solution? Well, $0 + 0 = 0$, and $0 \neq 5$, so it is not. Is $(a, b) = (1, 0)$ a solution? Well, $1 + 0 = 1$, and $1 \neq 5$, so it is not. Is $(a, b) = (0, 1)$ a solution? Well, $1 + 0 = 1$, and $1 \neq 5$, so it is not. Likewise for $(0, 2)$, $(1, 2)$ and so on. Some people call this the process of guess and check.

Continuing this way we find that $(3, 2)$ and $(2, 3)$ is a solution. Indeed, $3 + 2 = 2 + 3 = 5$. We are, however, interested in solutions to both equations

\[ \begin{aligned} a &= 3 \\ a + b &= 5 \end{aligned} \]

Both $(3, 2)$ and $(2, 3)$ satisfy the second equation, but only $(3, 2)$ satisfies both. Are there other solutions?- Future editions will expand. ↩︎