The Guess and Check Method for Solving Equations

What It Means to Solve Equations

(If you already read the main lesson before clicking this topic, skip this section)

Perhaps before you’ve seen a problem like “Solve for \(x\) in the equation \(3x+1=7\).” You’ve probably also learned a few ways you could go about “solving for \(x\)” in order to get the problem done. But what does it actually mean to solve for \(x\)? What does that \(x\) mean once we have it?

Suppose you are given a function like \(f(x)=3x+1\). Recall a function is nothing more than a “machine” that takes an input (\(x\)-value) and gives you some output (\(y\)-value). So when you see an equation like \(3x+1=7\), and are asked to “find \(x\),” you are being asked “when does the function \(f(x)=3x+1\) produce an output \(y=7\)?” or rather “for what \(x\)-value does the function \(f(x)=3x+1\) produce an output of \(7\)?” Put in yet another alternative way: “what \(x\)-value(s) make the equation true?”

Similarly, if you are given a pair of functions, say \(f(x)=3x+1\) and \(g(x)=x^2+1\), and are asked to solve for \(x\) in the equation \(f(x)=g(x)\) (or rather \(3x+1=x^2+1\). What this is asking is “when are these two functions equal?” or “for what \(x\)-value are the \(y\)-values the same?” or, more simply: “When are the functions’ outputs the same?”

Regardless of the situation in which you are solving for \(x\), any \(x\)-value for which the equation is true is called a solution of the equation.

Solving Equations by Guess-and-Check

Guess-and-check is perhaps the most inefficient and (forgive me) outright dumb method for solving equations… at least by hand. It takes a while, is often tedious unless one recognizes a pattern quickly, and sometimes leads you away from the correct answer. Many naiive computer algorithms rely on clever variants of the guess-and-check method (or other table search procedure) so we cover it here.

Often, one uses this method in their head, by just plugging different numbers into the function to see what works. Sometimes it’s obvious what you need to plug in. Sometimes it’s not. Let’s explore two examples illustrating this.

Finding when two functions are equal

In the last example, it was fairly easy to guess which \(x\)-values worked and solved the equation. When you are trying to determine when two functions are equal to one another, it is useful to organize things in a table.

Ugly solutions

The last few examples had convenient whole-number solutions. More often than not, in practice, the solution one finds to an equation is an approximate solution. Approximate solutions can often be found using guess and check as well. However, it often requires that you plug in decimal values for inputs as opposed to merely whole numbers, and paying very close attention to what the outputs of the function are doing compared to the one you want.

The implied existence of solutions

When we ask a question like “Solve \(4x+1=x^2-5\) for \(x\),” we are implicitly stating that there actually is an \(x\)-value that makes this equation true. However, just because we can set two functions equal to one another, it does NOT mean that there is a solution! For instance, there is no real number \(x\) that solves the equation \(x^2+1=0\); No \(x\)-value will make \(x^2+1\) equal to \(0\). You will be guessing and checking forever. Unfortunately, the guess-and-check method does not really give us a means that consistently can tell us whether or not a solution exists. So when you use this method, you are seeking a solution to an equation in faith that a solution indeed exists (when it very well might not).