In all the equations that we've solved so far, it has always been the case that the last line of work has given us a specific number or expression for the variable of interest. Let's look at solving the equation \(3x + 7 = -2\text{.}\)
\begin{equation*}
\begin{aligned}
3x + 7 \amp = -2 \\
3x \amp = -9 \amp \eqnspacer \amp \text{Subtract $7$ from both sides} \\
x \amp = -3 \amp \amp \text{Divide both sides by $3$} \\
\end{aligned}
\end{equation*}
We have been interpreting the last line as telling us that when \(x = -3\text{,}\) the original equation is true. We can even check this by explicit substitution.
How can we interpret the this? The first thing to observe is that the equation is false. The numbers 11 and 7 are definitely not the same.
What this is telling us is that there are no values of the variable that make the equation true. The equation will always be false, no matter what value we choose the variable to be. A concise way of saying this is to say that there are no solutions to the equation. We can pick some values of and check this.
Once again, the variable disappeared. But this time, we have a true equation. What this means is that regardless of the value of the variable, the equation will always be true. A shorter way of saying that is that can be any real number. We will check a few examples to demonstrate this.
As we continue with various types of algebraic calculations and manipulations, there will be other times when a variable will be missing from an equation or an expression. For example, consider the following: Substitute \(x = 4\) and \(y = 2\) into the expression \(2x - 7\text{.}\) The substitution into the \(x\) term is clear, but what does it mean to substitute for when there is no \(y\) variable?
The insight comes from looking at our calculations above. What caused the variable terms to disappear? It's both obvious and subtle at the same time. In the last step, we subtracted off a certain quantity that caused the variable terms to cancel out. Let's break that down more carefully:
The last two arithmetic steps are what cause the variable term to go away. We're using the fact that zero multiplied by any number is zero and that zero added to any number does not change the value. And by these observations, we are explaining why it makes sense for us to simplify the expression so that the variable is not shown. But this does not say that the variable doesn't exist anymore. In some sense, the term is still there. We're just not bothering to write it down because it doesn't have any impact on the quantity being expressed.
And that brings us back to the two-variable substitution. Even though we're only writing \(2x - 7\text{,}\) you should really be thinking about it as something like \(2x + 0y - 7\text{.}\) We say that the expression \(2x - 7\) is \emph{independent of the variable and that simply means that changing the value of has no impact on the value of the expression.
The first place that students run into this is when we define functions. For example, let's say we define the function \(f(x) = 4\text{.}\) What is \(f(2)\text{?}\) We know that this means we're supposed to plug in 2 for \(x\) but there are no \(x\) variables anywhere. The expression is independent of and so \(f(x) = 4\) regardless of what value of \(x\) is chosen.
This may seem like a small point, but the concept of functions or other expressions being independent of certain variables comes up regularly in both pure and applied mathematical settings. So it's helpful to understand this idea sooner rather than later.
