Hey everyone! Today, we're diving into the fascinating world of linear equations and how to solve them. Specifically, we're tackling a system of two linear equations. Don't worry if that sounds intimidating – we'll break it down into easy-to-follow steps. Our mission is to find the values of x and y that satisfy both equations simultaneously. Think of it like finding the sweet spot where two lines intersect on a graph. That point of intersection represents the solution to our system. So, grab your thinking caps, and let's get started!
Understanding the Problem
Before we jump into solving, let's make sure we understand what we're dealing with. We have two equations:
- x - 3y = -2
- x + 3y = 16
Each of these equations represents a straight line on a graph. The solution to the system is the point (x, y) that lies on both lines. In other words, it's the point where the lines cross. There are several methods to solve systems of linear equations, but we're going to focus on the elimination method in this case, as it's particularly well-suited for this problem. The elimination method works by adding or subtracting the equations to eliminate one of the variables, making it easier to solve for the other. This method leverages the additive property of equality, which states that adding equal quantities to both sides of an equation does not change the equality. By strategically adding or subtracting multiples of equations, we can cancel out one variable, leading to a single equation in one variable that can be easily solved. The key is to identify a variable with coefficients that are either the same or opposites in the two equations. If the coefficients are opposites, as in our case with the y terms, we can simply add the equations together. If the coefficients are the same, we can subtract one equation from the other. Once we've solved for one variable, we can substitute that value back into either of the original equations to solve for the remaining variable. This process of elimination and substitution allows us to systematically find the solution to the system of equations.
The Elimination Method: Our Key to Success
The elimination method is our secret weapon here. Notice that the y terms in our equations have opposite coefficients (-3 and +3). This is perfect! When we add the two equations together, the y terms will cancel out, leaving us with an equation in just x. Let's do it:
(x - 3y) + (x + 3y) = -2 + 16
Simplifying this, we get:
2x = 14
Now, we can easily solve for x by dividing both sides by 2:
x = 7
Woohoo! We've found the value of x. But we're not done yet. We still need to find the value of y. The elimination method, at its core, is about strategic manipulation of equations to simplify the system. It's not just about blindly adding or subtracting equations; it's about identifying the structure of the equations and using that structure to our advantage. In this case, the opposite coefficients of the y terms were the key. But what if the coefficients weren't opposites? What if they were, say, 2 and 3? In that case, we would need to multiply each equation by a suitable constant so that the coefficients of one variable become opposites. For example, we could multiply the first equation by 3 and the second equation by -2. This would give us coefficients of 6 and -6 for the chosen variable, allowing us to eliminate it by addition. The choice of which variable to eliminate is often a matter of convenience. We look for the variable whose coefficients are easiest to manipulate. Sometimes, this might mean multiplying only one equation by a constant. Other times, it might mean multiplying both equations. The goal is always the same: to create a situation where adding or subtracting the equations will eliminate one variable, simplifying the system and allowing us to solve for the remaining variables.
Finding the Value of y
Now that we know x = 7, we can substitute this value into either of our original equations to solve for y. Let's use the first equation, x - 3y = -2:
7 - 3y = -2
To isolate the y term, we subtract 7 from both sides:
-3y = -9
Finally, we divide both sides by -3:
y = 3
Awesome! We've found y = 3. So, our solution is the point (7, 3). Remember, substituting the known value back into the original equation is a crucial step. It allows us to reduce a two-variable equation to a single-variable equation, which is much easier to solve. This process of substitution is not limited to the elimination method; it's a fundamental technique in algebra and is used in various contexts. For instance, it's used in solving inequalities, evaluating functions, and even in calculus. The key is to replace a variable with its known value, thereby simplifying the expression or equation. When choosing which equation to substitute into, it's often helpful to pick the simpler one. This can reduce the chances of making arithmetic errors and make the algebra cleaner. In our case, both equations were relatively simple, so it didn't make a huge difference. However, in more complex systems, choosing the right equation can save you a significant amount of time and effort. The beauty of solving systems of equations lies in the flexibility of the methods. We've used the elimination method here, but we could have also used the substitution method. The choice of method often depends on the specific equations in the system and which method seems most efficient.
Checking Our Solution: Double the Fun!
It's always a good idea to check our solution to make sure we didn't make any mistakes. We can do this by substituting x = 7 and y = 3 into both original equations:
- x - 3y = -2 => 7 - 3(3) = 7 - 9 = -2 (Correct!)
- x + 3y = 16 => 7 + 3(3) = 7 + 9 = 16 (Correct!)
Our solution (7, 3) works in both equations, so we're confident we've got the right answer. Checking our solution is not just a matter of verifying our calculations; it's a way to deepen our understanding of the problem. By plugging the values back into the original equations, we're essentially confirming that the point we found lies on both lines. This reinforces the geometric interpretation of the solution as the intersection point of the lines. Moreover, checking our solution can help us identify common errors, such as sign errors or arithmetic mistakes. It's a valuable habit to develop, especially in more complex problems where the chances of making a mistake are higher. In addition to checking by substitution, we could also check our solution graphically. We could plot the two lines on a graph and visually verify that they intersect at the point (7, 3). This graphical approach provides a different perspective on the solution and can be particularly helpful for students who are visual learners. The act of checking our solution also promotes a sense of self-reliance and confidence in our mathematical abilities. It allows us to take ownership of our work and verify its correctness without relying on external sources. This is an important skill to develop, as it fosters critical thinking and problem-solving abilities.
The Answer and Why It's Correct
The solution to the system of equations is A. (7, 3). We arrived at this answer by using the elimination method to solve for x and y. We then checked our solution by substituting the values back into the original equations. This systematic approach ensures that we have a correct and reliable solution. The key to solving systems of linear equations is to be organized and methodical. Each step builds upon the previous one, and it's important to keep track of what you're doing. Whether you're using the elimination method, the substitution method, or a graphical method, the underlying principle is the same: to find the values of the variables that satisfy all the equations in the system. The ability to solve systems of equations is a fundamental skill in mathematics and has wide-ranging applications in various fields, including physics, engineering, economics, and computer science. It's a skill that will serve you well throughout your academic and professional life. So, keep practicing, keep exploring different methods, and keep challenging yourself with increasingly complex problems. The more you work with systems of equations, the more comfortable and confident you'll become in your ability to solve them.
Practice Makes Perfect
Solving systems of linear equations is a fundamental skill in algebra. The more you practice, the better you'll become at it. So, keep solving those equations, and don't be afraid to try different methods. You've got this! Remember, guys, math can be fun, especially when you break it down step by step. Keep practicing, and you'll be a pro in no time!