Does the equation 3x = 2 have a solution?
You might think: “What a dumb question! Of course it does! It’s 2/3.” Well, you are not wrong… IF we allow x to be a fraction.
But what if we want x to be an integer? Since 2 is not a multiple of 3, there is no integer x that will make 3x = 2 a true statement, and what does that imply? The equation has no solution.
Now you might think: “All right, fair enough. But why would somebody want to impose a restriction on the solution?”
A good reason is that some restrictions are inherent in the problems we are faced with.
Consider this situation. Suppose you have 2 free movie tickets, and you want to watch the movie with 2 other friends. How can you split the tickets among the 3 of you?
The answer is, you can’t. You can’t simply let each of you take 2/3 of a ticket. To enter a cinema, each person needs a whole number of tickets (actually just one, but you get the idea). Therefore, there is no solution in this case. You either have to get an extra ticket, or get rid of a friend from the movie outing.
All right, so here’s the main point I want to drive home:
The existence of a solution depends not only on the equation itself, but also on what the variables can be.
For the rest of this post, we are going to look at several cases of the number of solutions. To avoid ambiguity, let’s assume that the variables can be any numbers on the usual number line (or “real numbers” in mathematical parlance).
No solution
Consider the equation
If we try to subtract the variable x from both sides, we will arrive at 12 = 0 which is absurd. Hence, there’s no number we can assign to x to make the equation a true statement, meaning the equation has no solution.
There’s also a way to interpret it geometrically. If x were to satisfy the equation, then after we move 12 steps to the right from x, we should be back to x. This is not possible if we move along the number line (see Remark 2.1), and that’s why a solution does not exist.
One unique solution
We’ve already seen one example in this post, so let’s study something different. This time, we have more than one equation and more than one variable.
We call the collection above a system of equations. A solution to the system is an assignment of values to all the variables that makes all the equations true simultaneously (i.e. a common solution). That’s why a system like this is also called simultaneous equations.
Let’s try the pair x = 4 and y = –1 and see what we get:
Since the assignment yields two true statements, it is a solution to the system. Moreover, with some extra work, we can actually show that it is the ONLY solution, but that’s a topic for another day.
More than one solution
Consider the equation
which is an example of a quadratic equation (more on this in future posts). This time, x should be a number whose square is 16, and so both 4 and –4 are valid solutions.
“More than a finite” (or infinite) number of solutions
Recall that in Remark 1.1 we saw the inequality
for which x = 1 is a solution. In fact, after some manipulation, the above inequality can be simplified into 
The discussion of solutions certainly does not end here. There are just so many interesting facets that we can explore. However, I believe this post has had a good number of points in it already, so I’ll just end with two remarks below. But until next time, stay acute, stay positive. See you!
Remark 2.1
When we were discussing the equation x + 12 = x, we claimed that it has no solution because we always get a statement 12 = 0 (i.e. the net effect of moving 12 steps forward is the same as not moving at all) which is simply not true on a number line. But what if we are on a circle with only 12 points?
Wait, what?
No, seriously. This is an example of something called modular arithmetic. And I’m willing to bet that you have seen a real life example at least once in your lifetime.
Let’s go back to the original question, and assume that we are living on this number circle. Say we begin at x = 3, and we move 12 steps clockwise. Where do we land? We’re back to x = 3, aren’t we? Algebraically, doesn’t that translate to x + 12 = x? That means x = 3 is a solution!
Also, there’s really nothing special about x being 3. Begin at any point x you like on the circle, it will always satisfy x + 12 = x. Now, this is interesting because everything on this number circle turns out to be a solution.
Remark 2.2
Hmm, everything is a solution, huh. I wonder if something like this will happen when we are on the usual number line. Well, consider this silly-looking equation
Wait, aren’t both sides of the equation exactly the same? Then of course anything is equal to itself, regardless of what x is. In other words, every x on the number line is a solution to this equation!
What do we call this kind of equations that are always true?
Are there any other nontrivial examples?
What are some interesting aspects of them?
No, you can actually find it out here.
