Understanding the Solutions of a System of Equations

When it comes to systems of equations, the question of how many solutions a given system has can often be a sticking point. Let’s break it down using a simple example: the equations **5x + 4y = 9** and **6x - 3y = 18**. You might be asking yourself, “So, how many solutions are we looking at here?” The answer might surprise you—it’s only **one unique solution**. Let’s uncover why this is the case!  

Now, try to picture the equations as lines plotted on a graph. The first equation, **5x + 4y = 9**, can easily be rearranged to find its slope and intercept. Similarly, the equation **6x - 3y = 18** can also be manipulated to uncover its own line. Here’s the catch: if these two lines cross—like shaking hands—they represent the same solution. That’s the intersection point you’re looking for.  

You might wonder, “What if there was no intersection?” Well, if the lines were parallel (same slope but different intercepts), they would never meet, meaning there’d be **no solution**—that’s option A, which we can discard for our scenario. On the flip side, if the lines did happen to lie on top of each other perfectly, they wouldn’t just intersect at one point, they would have **infinitely many solutions**; they’re essentially the same line. So, option D isn’t relevant to our example either.  

And what about two solutions? Well, that’s option C—and that’s simply impossible in a linear system unless we’re toeing that line of a quadratic equation! In our case, we are dealing strictly with linear equations. If two distinct lines have two distinct intersection points, I’d consider that a one-way ticket to a plot twist!  

This analytical journey might sound a bit rigorous, but it’s the nature of algebra! The beauty lies in the depths of the intersection, where the lines meet; it’s like finding common ground with a friend. Wouldn’t you agree that finding common ground—whether in math or conversations—can be magical?  

Now, let’s circle back to our solution: it’s a unique one! And, just for a clarity boost, the lines intersect at precisely one point, meaning they are neither parallel nor overlapping. Once you get comfortable with this concept, you’ll find that solving systems of equations can become second nature. That’s the goal, right?  

When you’re preparing for your College Algebra exams, keep repeating this mantra: lines can dance together, each with its own personality, but they can only collide at one point unless otherwise coordinated. Remembering how these systems behave will set you up for success.  

So as you gear up for your exam prep, think of each equation as a story waiting to unfold on the graph. Be confident {in} your sounds understanding of intersections, and remember—every equation has a tale to tell if you just take the time to listen!