Hey guys! Ever been stumped by a system of linear equations? It's like trying to solve a puzzle with multiple pieces, but don't worry, we're here to break it down! Let's dive into a problem Ashur tackled, which involves finding the solution to a system of equations. We'll not only figure out the answer but also understand the why behind it. So, grab your thinking caps, and let's get started!
The Challenge Ashur Faced
Ashur was presented with the following system of linear equations:
-3a + 7b = -16
-9a + 5b = 16
The question is: Which statement accurately describes the solution to this system? To crack this, we need to determine the values of a and b that satisfy both equations simultaneously. There are several ways to solve this, but we'll focus on the elimination method, as it’s super efficient for this type of problem. The core idea behind the elimination method is to manipulate the equations so that when you add or subtract them, one of the variables cancels out. This leaves you with a single equation in one variable, which you can easily solve. Once you've found the value of one variable, you can substitute it back into either of the original equations to find the value of the other variable. This method is particularly useful when the coefficients of one of the variables are multiples of each other, as in this case, where the coefficient of a in the second equation is three times that in the first equation. However, it's also important to be careful with the signs and to ensure that the arithmetic is correct to avoid making mistakes. The final step is to check your solution by substituting the values of both variables back into the original equations to ensure they are satisfied. This ensures that your solution is correct and that you have not made any errors in your calculations.
Diving Deeper into the Elimination Method
The elimination method, at its heart, is about making strategic moves to simplify our equations. Think of it like a game of chess, where each move brings you closer to checkmate (or, in our case, the solution!). The first step usually involves looking at the coefficients of the variables. In our system, we see that the coefficient of a in the second equation (-9) is a multiple of the coefficient of a in the first equation (-3). This is a golden opportunity! We can multiply the first equation by -3, which will make the coefficients of a opposites of each other. When we add the equations together, the a terms will cancel out, leaving us with an equation in just b. But remember, whatever we do to one side of the equation, we must do to the other to keep the equation balanced. This is a fundamental principle of algebra, like the rule of conservation in physics – you can't just create or destroy terms, you can only transform them. Once we have an equation in just b, solving for b is straightforward. We simply isolate b by performing the same operations on both sides of the equation, such as adding or subtracting constants, or multiplying or dividing by coefficients. With the value of b in hand, we're halfway there! The next step is to substitute this value back into one of the original equations. It doesn't matter which one we choose; both will give us the same value for a. This is because the solution to a system of equations is the set of values that satisfy all equations simultaneously. Once we've substituted the value of b, we have a simple equation in a, which we can solve using the same techniques we used to solve for b. And finally, the most important step: always, always, always check your solution! Substitute the values of a and b back into both original equations to make sure they hold true. This is your safety net, catching any errors you might have made along the way.
Cracking the Code: Solving for a and b
To eliminate a, let's multiply the first equation by -3:
-3 * (-3a + 7b) = -3 * (-16)
9a - 21b = 48
Now, we have a modified first equation. Let's add this to the second equation:
(9a - 21b) + (-9a + 5b) = 48 + 16
-16b = 64
Dividing both sides by -16, we get:
b = -4
Awesome! We've found b. Now, let's substitute b = -4 into the first original equation:
-3a + 7(-4) = -16
-3a - 28 = -16
-3a = 12
a = -4
Look at that! We found a too. So, a = -4 and b = -4. But remember, we're not just looking for the values; we need to determine which statement is true about this solution.
The Significance of the Solution
Understanding the solution to a system of equations goes beyond just finding the values of the variables. It's about grasping the bigger picture: what does this solution mean? In the context of linear equations, each equation represents a line on a graph. The solution to the system is the point where these lines intersect. This point satisfies both equations simultaneously, making it the unique solution to the system. But what if the lines are parallel? In that case, they never intersect, and the system has no solution. What if the lines are the same? Then they intersect at every point, and the system has infinitely many solutions. So, when we solve a system of equations, we're not just crunching numbers; we're uncovering the geometric relationship between the lines represented by the equations. This understanding is crucial in many real-world applications, from engineering and physics to economics and computer science. For example, in economics, systems of equations can be used to model supply and demand curves, and the solution represents the equilibrium point in the market. In engineering, systems of equations are used to analyze electrical circuits, structural mechanics, and fluid dynamics. And in computer science, they're used in optimization algorithms, computer graphics, and machine learning. So, the next time you solve a system of equations, remember that you're not just solving a math problem; you're unlocking a powerful tool for understanding the world around you.
Evaluating the Statements
Now that we know a = -4 and b = -4, let's think about what the question might be asking. It could be comparing the values of a and b, or it might involve some other calculation using these values. Without the actual statements, we can't definitively choose the correct one, but we've done the heavy lifting by finding the solution. Remember, the key to tackling these problems is to break them down step by step. Don't get overwhelmed by the entire problem at once. Focus on solving for the variables first, and then you can easily evaluate any statements or answer any questions related to the solution. It's like building a house – you start with the foundation (solving for the variables) and then you can add the walls, roof, and finishing touches (evaluating statements). And just like in construction, accuracy is paramount. Make sure you're careful with your arithmetic and pay attention to signs. A small mistake can throw off the entire solution. And finally, practice makes perfect! The more you solve these types of problems, the more comfortable you'll become with the techniques and the faster you'll be able to solve them. So, keep practicing, keep learning, and you'll become a master of linear equations in no time!
Without the specific options (A, B, C, D), we can't say for sure which statement is true. However, we've successfully solved the system of equations and found that a = -4 and b = -4. Armed with this knowledge, Ashur (and now you!) can confidently choose the correct statement when presented with the options. High five! You've conquered another math challenge. Keep up the awesome work, and remember, every problem you solve is a step closer to mathematical mastery!
