Solving Systems Of Equations: A Step-by-Step Guide

Hey guys! Let's dive into solving systems of equations. It might sound intimidating, but trust me, it's like piecing together a puzzle. We're going to tackle this system: $\left\{\begin{aligned} 12x &= 48 - 8y \\ 10x + 8y &= 38 \end{aligned}\right.$ and I'll show you how to not only find the solution but also double-check your work. So, buckle up, and let's get started!

Understanding Systems of Equations

Before we jump into the nitty-gritty, let's quickly recap what a system of equations actually is. Simply put, it's a set of two or more equations that share the same variables. Our goal? To find the values for those variables that make all the equations true simultaneously. Think of it as finding the perfect combination that unlocks all the locks.

In our case, we've got two equations with two variables, x and y. There are several methods to solve these, such as substitution, elimination, or even graphing. We'll focus on the elimination method here because it's super efficient for this particular system. The beauty of the elimination method is that it allows us to strategically manipulate the equations, so when we add them together, one variable disappears, leaving us with a single variable equation that we can easily solve. This strategy involves adding or subtracting multiples of the equations to eliminate one variable, simplifying the problem and paving the way for a straightforward solution.

Why does solving systems of equations matter? Well, they pop up everywhere in real life! From calculating the break-even point for a business to figuring out the optimal mix of ingredients in a recipe, systems of equations are the unsung heroes of problem-solving. They're not just abstract math concepts; they're practical tools that help us make sense of the world around us. The ability to solve systems of equations opens doors to understanding and addressing a wide range of challenges, making it a crucial skill in various fields and daily life scenarios. So, mastering this skill is not just about acing the math test; it's about equipping yourself with a powerful tool for critical thinking and problem-solving in any situation.

Step 1: Rearrange the Equations (if necessary)

First things first, let’s make sure our equations are lined up nicely. We want the x terms, y terms, and constants to be in the same order in both equations. Looking at our system: $\left\{\begin{aligned} 12x &= 48 - 8y \\ 10x + 8y &= 38 \end{aligned}\right.$, we see that the first equation has the y term on the wrong side. No problem! We can easily fix this by adding 8y to both sides. This simple rearrangement is crucial because it sets the stage for the elimination method, ensuring that like terms are aligned and ready for addition or subtraction.

So, let's rewrite the first equation:

$12x + 8y = 48$

Now our system looks like this:

$\left\{\begin{aligned} 12x + 8y &= 48 \\ 10x + 8y &= 38 \end{aligned}\right.$

See? Much neater! This might seem like a small step, but it's a critical step in solving systems of equations. By aligning the variables and constants, we pave the way for a smoother and more accurate solution process. Think of it as organizing your workspace before starting a project – a little preparation can save you a lot of headaches down the road. This initial alignment not only simplifies the subsequent steps but also reduces the chances of making errors, ensuring that we're on the right track from the very beginning.

Step 2: Eliminate a Variable

The heart of the elimination method is, well, eliminating a variable! Notice anything special about our system now? Yep, both equations have an 8y term. This is perfect because we can eliminate y by subtracting one equation from the other. This strategic maneuver is what makes the elimination method so powerful, allowing us to reduce a two-variable problem into a single-variable equation that's much easier to solve.

Let’s subtract the second equation from the first:

$(12x + 8y) - (10x + 8y) = 48 - 38$

Simplify it:

$2x = 10$

Boom! The y terms are gone, and we're left with a simple equation in terms of x. This is the magic of the elimination method – by carefully manipulating the equations, we can isolate one variable and pave the way for a straightforward solution. This step highlights the importance of strategic thinking in math; by identifying the common terms, we can create an opportunity to simplify the system and make it much more manageable. The result is a clear path forward, where solving for the remaining variable becomes a simple and direct process.

Step 3: Solve for the Remaining Variable

Now we have a piece of cake equation: $2x = 10$. To solve for x, we simply divide both sides by 2:

$x = 5$

Awesome! We've found the value of x. This step demonstrates the basic algebraic principle of isolating a variable to find its value. By performing the same operation on both sides of the equation, we maintain the balance and gradually reveal the solution. In this case, dividing both sides by 2 directly leads us to the value of x, showcasing the power of simple algebraic manipulations in solving equations. This process is not only efficient but also fundamental to solving a wide range of mathematical problems, making it a crucial skill to master.

Step 4: Substitute to Find the Other Variable

We're halfway there! Now that we know x = 5, we can substitute this value back into either of the original equations to solve for y. I like the second equation, $10x + 8y = 38$, so let's use that one. The substitution method is a cornerstone technique in algebra, allowing us to leverage known values to uncover the unknowns. By replacing a variable with its numerical value, we transform a multi-variable equation into a single-variable equation, making it solvable.

Substitute x = 5:

$10(5) + 8y = 38$

Simplify:

$50 + 8y = 38$

Subtract 50 from both sides:

$8y = -12$

Divide by 8:

$y = -\frac{12}{8} = -\frac{3}{2}$

Excellent! We've found y = -3/2. This step not only demonstrates the practical application of substitution but also reinforces the importance of simplifying and solving equations methodically. The ability to substitute known values and manipulate equations is a crucial skill for tackling more complex mathematical problems, making this step a key component in the problem-solving process.

Step 5: Verify the Solution

But wait, we're not quite done yet! It's super important to verify our solution to make sure we didn't make any mistakes along the way. The verification step is crucial in any mathematical problem-solving process, acting as a safety net to catch any potential errors. By substituting the values we've found back into the original equations, we can confirm whether our solution satisfies the entire system, ensuring accuracy and building confidence in our answer.

Let's plug x = 5 and y = -3/2 into both original equations:

Equation 1: $12x = 48 - 8y$

$12(5) = 48 - 8(-\frac{3}{2})$

$60 = 48 + 12$

$60 = 60$ (Checks out!)

Equation 2: $10x + 8y = 38$

$10(5) + 8(-\frac{3}{2}) = 38$

$50 - 12 = 38$

$38 = 38$ (Checks out!)

Yay! Our solution works for both equations. This verification process not only confirms the correctness of our solution but also provides a deeper understanding of the system of equations itself. It highlights the interconnectedness of the equations and the importance of finding values that satisfy all conditions simultaneously.

Conclusion

So, the solution to the system of equations is x = 5 and y = -3/2. We did it! Solving systems of equations might seem tricky at first, but with a little practice and a systematic approach, you'll be a pro in no time. Remember, the key is to break down the problem into smaller, manageable steps and always verify your solution. Keep practicing, and you'll conquer any system of equations that comes your way!

And there you have it, guys! We've successfully navigated through this system of equations. Remember, math is all about practice, so keep those pencils moving and those brains churning! You've got this!