Mastering Systems of Equations: A Guide to Solving College Algebra Questions

Have you ever found yourself staring at a system of equations, wondering where to start? Equations like 4x + 6y = 24 and 3x - 4y = 2 can feel daunting, but with the right approach, they become much more manageable. Let's break it down together. 

Solving a system of equations is essentially finding the values of x and y that make both equations true. It’s like a puzzle, where every piece matters. And just like with any puzzle, you have methods to help you piece together the solution—two common methods are substitution and elimination.

**So, How Do You Solve This?**

First, let’s take a look at the first equation: 4x + 6y = 24. To use the substitution method, we can rearrange it to solve for y. You might be thinking, “That’s easier said than done!” But hang in there; we can simplify this to: 

y = (24 - 4x) / 6.

Now, we have an expression for y that we can plug into the second equation. Here’s the thing: it’s all about substituting that value into 3x - 4y = 2. When we do this, we get:

3x - 4(24 - 4x) / 6 = 2.

After doing a little math (and who doesn’t love a bit of a math challenge?), we find x = 12. But hold on, we can’t stop there! We need to plug this value back into our first equation to find y. This gives us:

y = -1.

And voilà! We’ve found that the solution to our system of equations is (12, -1). Doesn’t that feel good?

**Checking the Other Options:**

You know what’s even more satisfying? Ensuring that our answer is correct. It’s crucial to check how other options stack up against our solution. For instance, if we were to test option B (5, -6), it fails right off the bat as it doesn't satisfy both equations. 

- For the first equation:
  4(5) + 6(-6) = 20 - 36 = -16 (not equal to 24).
- And the second equation:
  3(5) - 4(-6) = 15 + 24 = 39 (not equal to 2).

So, that option is definitely out. By the same token, options C and D will also fail the test, making our initial solution (12, -1) not just right, but the only right answer.

**Why Does This Matter?**

Understanding how to solve systems of equations is invaluable for the College Algebra CLEP exam. It’s not just about passing the exam; it’s about building a solid mathematical foundation that will aid you in every math-related endeavor in the future. Mathematics is not just a series of numbers and letters—it's a language of logic and reasoning.

And hey, if this system of equations sparked your interest, there’s a whole world of math waiting for you! Whether it’s graphing, quadratic equations, or functions, each topic builds on the last. 

So as you prepare for your College Algebra CLEP exam, remember what you've learned here today. Tackle those systems of equations with confidence, and who knows—you might just find your love for math blossoming!

Happy studying!