Solving The System: 2x + 3y = 3, 7x - 3y = 24
Hey guys! Let's dive into solving a system of linear equations. We've got two equations here, and our mission, should we choose to accept it (and we do!), is to find the values of x and y that make both equations true. This might sound like a tricky puzzle, but don't worry, we'll break it down step by step. Let's get started and make math a little less intimidating and a lot more fun!
Understanding Systems of Linear Equations
Before we jump into solving, let's take a moment to understand what a system of linear equations really is. At its core, a system of linear equations is simply a set of two or more linear equations that we're looking at together. Each equation represents a straight line when graphed, and the solution to the system is the point (or points) where these lines intersect. Think of it like a roadmap where each equation is a different route, and the solution is the destination where they meet.
Now, why do we care about these intersections? Well, in real-world scenarios, systems of equations can model all sorts of things – from balancing chemical equations to planning business budgets. For example, one equation might represent the cost of materials, while another represents the revenue from sales. Finding the solution helps us figure out the point where costs equal revenue, which is super important for any business. So, understanding how to solve these systems isn't just about math class; it's about tackling practical problems in everyday life.
There are several methods we can use to solve these systems, each with its own set of advantages. We could graph the equations and visually find the intersection point, which is great for a quick estimate. Or, we can use algebraic methods like substitution or elimination, which give us precise answers. In our specific problem, we'll use the elimination method because it looks like a particularly good fit, but we'll touch on the other methods as well so you get the full picture. Remember, the goal is to find the method that clicks best for you and the problem at hand.
The Equations at Hand
Okay, let's get down to business and take a good look at the system of equations we're dealing with. We have:
- 2x + 3y = 3
- 7x - 3y = 24
These equations are the bread and butter of our problem, and understanding their structure is key to cracking the solution. Notice anything interesting about them? Specifically, look at the terms involving y. In the first equation, we have +3y, and in the second, we have -3y. This is a huge hint that the elimination method is going to be our best friend here. Why? Because when we add these equations together, the y terms will conveniently cancel each other out, leaving us with an equation that only involves x. It’s like magic, but it’s actually just clever algebra!
But before we jump to adding them, let's appreciate what each equation represents on its own. Each of these equations is a linear equation, meaning that if we were to graph them, we'd get a straight line. The coefficients (the numbers in front of the variables) and the constants (the numbers on their own) determine the slope and position of these lines on the coordinate plane. The solution to the system, as we discussed earlier, is the point where these lines intersect. So, we're not just solving equations; we're finding the exact spot where these two lines cross paths.
Understanding this visual representation can be super helpful, especially if you're a visual learner. You can even use graphing software or a good old-fashioned piece of graph paper to plot these lines and see where they intersect. But for now, let's stick with the algebra and see how the elimination method can make our lives easier. Ready to dive into the solving process? Let's do it!
Applying the Elimination Method
Alright, let's get our hands dirty with the elimination method. As we spotted earlier, the beauty of these equations is the +3y in the first and the -3y in the second. They're practically begging us to add the equations together! When terms have opposite signs and the same coefficient, it sets the stage for eliminating a variable through addition. This is a classic technique in solving systems of equations, and it can save you a lot of time and effort compared to other methods like substitution, especially when the setup is this favorable.
So, what happens when we add the equations? Let's write it out:
(2x + 3y) + (7x - 3y) = 3 + 24
Now, let's simplify. Combine the x terms (2x + 7x), combine the y terms (3y - 3y), and add the constants (3 + 24). What we get is:
9x + 0y = 27
See how the y terms vanished? Poof! That's the magic of elimination. We're now left with a simple equation: 9x = 27. This is much easier to handle, right? To solve for x, we just need to isolate x by dividing both sides of the equation by 9. This gives us:
x = 27 / 9
x = 3
Boom! We've found the value of x. But hold on, our mission isn't complete yet. We've only found half the solution. We still need to find the value of y. But don't worry, now that we know x, finding y is a piece of cake. We'll just substitute this value back into one of our original equations and solve for y. Which equation should we choose? Let's go through the final steps to nail down the solution.
Solving for y
Now that we've triumphantly found x = 3, it’s time to hunt down the value of y. This is where substitution comes into play, a handy technique where we plug the value we've found for one variable into one of the original equations to solve for the other. The key here is choosing the equation that looks easiest to work with. Both equations will work, but one might have smaller numbers or a simpler structure, making it less prone to errors.
Looking at our original equations:
- 2x + 3y = 3
- 7x - 3y = 24
The first equation, 2x + 3y = 3, seems a bit simpler due to the smaller coefficients. So, let's substitute x = 3 into this equation:
2(3) + 3y = 3
Now, let’s simplify. Multiply 2 by 3 to get 6:
6 + 3y = 3
Our next goal is to isolate the term with y. To do this, we'll subtract 6 from both sides of the equation:
3y = 3 - 6
3y = -3
Almost there! Now, to solve for y, we divide both sides by 3:
y = -3 / 3
y = -1
Fantastic! We've found y = -1. So, we now have both the x and y values that satisfy our system of equations. But before we declare victory, it's always a good idea to check our solution. Let's make sure our values work in both original equations to ensure we haven't made any sneaky mistakes along the way.
Checking the Solution
Okay, we've got our candidate solution: x = 3 and y = -1. But before we shout from the rooftops that we've cracked the code, it’s super important to check our solution. Why? Because mistakes can happen, even to the best of us! Checking our solution ensures that our values for x and y actually satisfy both original equations. It’s like double-checking your GPS coordinates before you set off on a road trip – it could save you from a wrong turn!
Let's start with the first equation:
2x + 3y = 3
Substitute x = 3 and y = -1:
2(3) + 3(-1) = 3
Simplify:
6 - 3 = 3
3 = 3
Hooray! Our solution works for the first equation. But we're not done yet – we need to make sure it works for the second equation too:
7x - 3y = 24
Substitute x = 3 and y = -1:
7(3) - 3(-1) = 24
Simplify:
21 + 3 = 24
24 = 24
Double hooray! Our solution also works for the second equation. This means we’ve nailed it! Both equations are happy with our values for x and y. We can confidently say that we've found the solution to the system of equations.
So, what's our final answer? It's not just enough to say x = 3 and y = -1; we usually write the solution as an ordered pair (x, y). This makes it clear that we're talking about a point on the coordinate plane where the two lines intersect. So, let’s present our solution in its full glory.
The Final Answer
Drumroll, please! After our mathematical adventure, carefully applying the elimination method, substituting to find the other variable, and diligently checking our solution, we've arrived at the final destination. The solution to the system of equations:
2x + 3y = 3
7x - 3y = 24
is the ordered pair:
(3, -1)
This means that the point (3, -1) is where the two lines represented by our equations intersect on the coordinate plane. It's the one and only pair of values for x and y that makes both equations true simultaneously. Think of it as the secret handshake that unlocks both equations!
But what does this solution really tell us? Well, in a real-world context, this point could represent anything from the equilibrium price in a supply and demand model to the optimal combination of ingredients in a recipe. Systems of equations are used to model and solve problems in all sorts of fields, from economics and engineering to computer science and even video game design. So, understanding how to solve them is a valuable skill that can open doors to many different areas.
So there you have it! We've successfully navigated the world of systems of linear equations, conquered the elimination method, and emerged victorious with our solution. Give yourselves a pat on the back – you've earned it!