Master Solving Systems Of Equations

Hey math wizards and welcome back! Today, we're diving deep into the awesome world of systems of equations. You know, those situations where you've got two (or more!) equations hanging out together, and you need to find that sweet spot, that one solution that makes both of them true? It's like solving a puzzle, and trust me, once you get the hang of it, it's super satisfying. We're going to tackle a specific example today:

$egin{array}{l} -6 x+2 y=-32 \ 4 x+3 y=-9

Now, there are a few ways to go about solving these bad boys. You've got your substitution method, your elimination method, and sometimes, if you're lucky, you can even graph 'em to find the intersection point. For this particular system, the elimination method looks pretty promising, but we'll explore why. The goal is always to isolate one of the variables or to eliminate one of them entirely so we can solve for the other. Think of it as a strategic dance. You're trying to manipulate the equations, multiply them by clever numbers, add or subtract them strategically, all to get closer to that single, beautiful solution (x, y). This problem is a fantastic example because it requires a little bit of multiplication before we can eliminate. We need to make the coefficients of either x or y opposites so that when we add the equations together, that variable disappears. It's a common scenario, so mastering this will set you up for tons of other problems. Don't sweat it if it seems a bit daunting at first; we'll break it down step-by-step, and by the end, you'll be a system-solving pro. Let's get started on untangling this mathematical mystery!

Method 1: The Elimination Approach

The elimination method is often a go-to for solving systems of equations when the variables are nicely lined up, like in our example. The whole idea here is to make the coefficients of either 'x' or 'y' opposites. That way, when you add the two equations together, one of the variables cancels out, leaving you with a simple equation in just one variable. For our system:

$egin{array}{l} -6 x+2 y=-32 4 x+3 y=-9

See how the coefficients for 'x' are -6 and 4, and for 'y' they are 2 and 3? Neither are opposites right now. We need to do a little multiplying to make them so. Let's aim to eliminate 'x'. We need to find a common multiple for 6 and 4. The least common multiple is 12. So, we want one coefficient to be +12x and the other to be -12x. We can achieve this by multiplying the first equation by 2 and the second equation by 3.

• Multiply the first equation ($-6x + 2y = -32$) by 2: $2 * (-6x + 2y) = 2 * (-32)$ This gives us: $-12x + 4y = -64$

• Multiply the second equation ($4x + 3y = -9$) by 3: $3 * (4x + 3y) = 3 * (-9)$ This gives us: $12x + 9y = -27$

Now, check this out! Our 'x' coefficients are -12 and +12. They are opposites! We can now add these two new equations together:

$(\ -12x + 4y = -64$) $+ (12x + 9y = -27$)

$0x + 13y = -91$

Boom! The 'x' terms are gone. We're left with $13y = -91$. Now, solving for 'y' is a piece of cake. Just divide both sides by 13:

$y = -91 / 13$ $y = -7$

Awesome! We've found our 'y' value. But we're not done yet. Remember, a solution to a system of equations is a pair of values (x, y). We need to find 'x' too.

Method 2: Finding the 'x' Value

Now that we've got our 'y' value, which is y = -7, we can plug this back into either of the original equations to solve for 'x'. It doesn't matter which one you choose; you should get the same answer. Let's use the second original equation, as the numbers might be a little smaller:

$4x + 3y = -9$

Substitute $y = -7$ into the equation:

$4x + 3(-7) = -9$

Now, simplify:

$4x - 21 = -9$

To get the '4x' term by itself, we need to add 21 to both sides of the equation:

$4x - 21 + 21 = -9 + 21$ $4x = 12$

Finally, to isolate 'x', divide both sides by 4:

$x = 12 / 4$ $x = 3$

And there you have it! We've found our 'x' value. So, the solution to our system of equations is $x = 3$ and $y = -7$. We can write this as an ordered pair: $(3, -7)$.

Verifying Your Solution

This step is super important, guys, and often overlooked. Always, always, always plug your solution back into both of the original equations to make sure it works. This is your foolproof way to know you've nailed it. Let's check our solution $(3, -7)$ in both equations.

• Check in the first equation: $-6x + 2y = -32$ Substitute $x=3$ and $y=-7$: $-6(3) + 2(-7) = -18 - 14 = -32$ $-32 = -32$. It works!

• Check in the second equation: $4x + 3y = -9$ Substitute $x=3$ and $y=-7$: $4(3) + 3(-7) = 12 - 21 = -9$ $-9 = -9$. It works!

Since our solution $(3, -7)$ satisfies both equations, we know for sure that we've found the correct answer. High five!

Alternative: The Substitution Method

While elimination was great for this problem, let's quickly touch on the substitution method. This method involves solving one of the equations for one variable, and then substituting that expression into the other equation. It's like peeling back layers to get to the core.

Let's try solving the first equation for 'y':

$-6x + 2y = -32$ $2y = 6x - 32$ $y = 3x - 16$

Now, we take this expression for 'y' and substitute it into the second original equation ($4x + 3y = -9$):

$4x + 3(3x - 16) = -9$

Distribute the 3:

$4x + 9x - 48 = -9$

Combine like terms:

$13x - 48 = -9$

Add 48 to both sides:

$13x = -9 + 48$ $13x = 39$

Divide by 13:

$x = 39 / 13$ $x = 3$

See? We got the same 'x' value as before! Now, we plug $x=3$ back into our expression for 'y':

$y = 3x - 16$ $y = 3(3) - 16$ $y = 9 - 16$ $y = -7$

Again, we arrive at the solution $(3, -7)$. Both methods get us to the same correct answer, which is exactly what we want. It's good to have multiple tools in your math toolbox!

Why Systems of Equations Matter

So, why do we even bother learning how to solve systems of equations? Well, these aren't just abstract math problems. They show up everywhere in the real world, guys! Think about business: figuring out where the cost of producing an item meets the revenue from selling it (break-even point). In science: modeling how different forces interact or how populations grow. Even in everyday life, like planning a budget or comparing two different phone plans, you might be implicitly setting up and solving a system of equations. Understanding how to solve them gives you a powerful analytical skill to dissect complex situations and find optimal solutions. It's about making informed decisions by understanding the interplay of different factors. The ability to represent real-world scenarios with mathematical equations and then solve them is a cornerstone of quantitative reasoning and problem-solving. So, the next time you're faced with a system of equations, remember you're not just doing homework; you're practicing a skill that unlocks real-world insights and helps you make sense of a complex world. Keep practicing, and you'll be amazed at how many problems you can untangle!