But what is a solution?

Welcome to the very first post of Acute Angel! What better way to start a math blog than solving for x!

No wait! I promise you it’s gonna be an easy one.

$\displaystyle 5x-13=3x-7$

Not too bad right? Give it a shot, and continue reading when you are done.

All right, hope you are able to solve the above problem with no sweat. The solution is, surprise, surprise, x = 3. But let’s not stop here and investigate further.

As far as the problem is concerned, what’s so special about the number 3? Why not some other number? What property does 3 possess that no other numbers do?

It all boils down to the next question:

What does a solution actually mean?

Before we try to answer this, here are some terminologies to ease the discussion:

We call the mathematical sentence 5x – 13 = 3x – 7 an equality or equation (some texts draw a distinction between them, but for our purpose we’ll just use them interchangeably).
We call the letter x a variable or unknown.

Now, suppose we’re given the equation 5x – 13 = 3x – 7, and then we are asked, “True or false?”

Well, it’s kinda strange to answer that, isn’t it? I mean, depending on the value we assign to the variable x, the answer varies.

And that’s the point! We have now gotten to the meat and potatoes.

A solution to an equation is a value that we can assign to our variable to make the equation a true statement.

Let’s try to apply this to our example. If we replace all the variables by 3, we get

$\displaystyle 5(3)-13=3(3)-7\quad\Rightarrow\quad 2=2$

which is certainly true. That’s why x = 3 is a solution to the equation. By the way, we often say x = 3 satisfies the equation to mean the same thing as well.

What if we choose some other number, say x = 4? Going through the same process, we obtain

$\displaystyle 5(4)-13=3(4)-7\quad\Rightarrow\quad 7=5$

which is false on the usual number line. And what does that imply? x = 4 is not a solution!

3 Aspects of Solutions

Now that we know what a solution means, let’s dive deeper into this topic.

In general, there are 3 interesting aspects to study:

Existence / Number of solutions
Nature of solutions
Algorithm for solutions

Yes, the concept of a solution is simple, but I’m not trying to make a mountain out of a molehill. As a matter of fact, some famous math theorems and conjectures are closely related to the above aspects: Fermat’s Last Theorem (Existence of solutions), Riemann Hypothesis (Nature of solutions), and Abel-Ruffini Theorem (Algorithm for solutions) to name a few.

I hope to discuss these aspects, directly or indirectly, in future. For now, let me just end this post with a quick remark below. But until next time, stay acute, stay positive. See you!

Remark 1.1

The definition of a solution is not restricted to just equations, but more generally to any types of mathematical sentence. For example, consider the inequality

$\displaystyle 10x-4<5x+2.$