Natural Equations
Why
Suppose I have three stones in my hands. If I have two in my left hand, how many are in my right hand?
Discussion
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
\[
\begin{aligned}
a &= 3 \\
a + b &= 5
\end{aligned}
\]
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