Blog

It’s been a while since the last update, so let’s warm up a bit! Observe the following three equations and see if you can identify something in common among them.

“There are alphabets and numbers.”

Well, I can’t argue with that, but we can definitely go deeper.

Here’s the deal. Notice how in each equation the variable is raised to just the first power. If so, we say that the equation is linear (the first four letters kind of suggest something geometrical, which we’ll talk about sooner or later).

Moreover, through just the four basic operations, we can reformulate each of them into the form of

where A and B are constants, called the standard form (probably because, you know, it looks standard, duh).

In general, one would like to come up with some systematic approach (i.e. algorithm) to solve the equation, not just for some specific values of A and B, but for any values of A and B.

To do this, we first subtract B from both sides to get

Next, dividing both sides by A yields

Voilà! Given a linear equation Ax + B = 0, we always get the solution as x = −B/A…

.

.

.

.

except that it is NOT ALWAYS true!

A more careful look at the algorithm

Time to troubleshoot! Fortunately it’s not too difficult as the whole algorithm merely involves two steps:

Step 1: Subtract B from both sides.
This step is always valid. There’s nothing wrong with it.

Step 2: Divide both sides by A.
This step is also always valid This step is valid ONLY when A is not zero (see here)!

Thus, if A ≠ 0, it’s true that x = −B/A is the solution to Ax + B = 0. However, if A turns out to be 0, then we have two more possibilities:

If A = 0 and B = 0, then the equation degenerates into the identity 0 = 0, meaning every number x is a solution.

If A = 0 but B ≠ 0, then the equation reduces to B = 0 which is inconsistent (e.g. 3 = 0), meaning no number x is a solution.

More on the uniqueness of the solution

In this section, we will focus on the case of Ax + B = 0 where A ≠ 0. The short algorithm we saw above kinda reveals that the solution is unique, i.e. x = −B/A. Just in case you are not convinced yet, I will quickly prove it now before advancing the discussion.

Let’s suppose we have two candidates that solve Ax + B = 0, say x₁ and x₂. Our goal is to show that x₁ = x₂ necessarily.

Recalling what solution means, we obtain two true statements as follows:

In particular, Ax₁ + B = Ax₂ + B. Subtracting B from both sides yields

Finally, remembering the assumption that A ≠ 0, we’re allowed to divide both sides by A to get

Q.E.D.

At any rate, the solution to any non-degenerate linear equation (i.e. A ≠ 0 in standard form) must be unique. For instance, 7x − 2 = 0 has only one solution x = 2/7.

What about 7x − 2 = 1? Same thing happens, except this time the solution is x = 3/7.

And 7x − 2 = −9124.86? No difference. It still has only one solution, though nastier.

In general, given a number y, there is only one number x that satisfies the equation 7x − 2 = y. This phenomenon is called a one-to-one or injective relationship, which is pivotal in the topic of functions (more on this in future article).

There we go! Hope you find our very first investigation of linear equation fruitful. I feel like it shouldn’t be treated perfunctorily (as it often is) and deserves a lot more scrutiny in spite of … no, BECAUSE OF its simple structure. Next time, we’ll change gear and solve some practical problem using what we’ve learnt. Until then, stay acute, stay hygienic. See you!