Solve For (x, Y): System Of Equations Explained
Hey guys! Let's dive into solving systems of equations. This is a super important concept in math, and once you get the hang of it, you'll be cruising through these problems. We're going to tackle a specific problem today, walking through each step so you understand exactly how to find the solution. So, buckle up and let's get started!
The Problem: Finding the Ordered Pair (x, y)
Our mission, should we choose to accept it, is to find the value of the ordered pair (x, y) that represents the solution to the following system of equations:
y = 3x - 2
y = 2 - x
We have a few options for the answer:
A) (0, -2) B) (1, 1) C) (2, 4) D) (3, 7)
Now, let's figure out which one is the correct answer and, more importantly, why. We're not just about getting the answer; we're about understanding the process. Finding the correct ordered pair (x, y) involves a methodical approach to solving systems of equations, and the options provided give us a clear target to aim for. This is like having a treasure map, where each (x, y) pair is a potential location, and our job is to find the one where the treasure (the solution) is buried. The beauty of this problem lies in its straightforwardness: we have two equations and two unknowns, which means we can find a unique solution if the equations are consistent and independent. Each equation represents a line on a graph, and the solution we seek is the point where these lines intersect. This intersection point is the only (x, y) pair that satisfies both equations simultaneously. To get there, we can use several methods, each with its own charm and efficiency. We could graph the lines and visually identify the intersection, but for precise solutions, algebraic methods like substitution or elimination are our best bet. By understanding these methods, we not only solve this problem but also equip ourselves with tools to tackle more complex systems of equations. So, let’s roll up our sleeves and dive into the solution!
Understanding Systems of Equations
First, let's break down what a system of equations actually is. Think of it as a set of two or more equations that share the same variables. In our case, we have two equations, and both use the variables 'x' and 'y'. The solution to a system of equations is the set of values for the variables that make all the equations true at the same time. For a system with two variables, like ours, the solution is typically an ordered pair (x, y), which represents a point on a graph where the lines represented by the equations intersect. This point of intersection is crucial; it's the one and only place where both equations hold true simultaneously. Visualizing this on a graph can be incredibly helpful. Each equation represents a line, and unless the lines are parallel (meaning they never intersect) or the same line (meaning they intersect at every point), there will be exactly one point where they cross. This unique point is the solution we're after. Now, why is this important? Systems of equations pop up everywhere in real life, from calculating the break-even point for a business to determining the trajectory of a rocket. They allow us to model situations with multiple constraints and find the sweet spot that satisfies all conditions. Mastering the art of solving these systems opens doors to a wide range of applications and problem-solving scenarios. So, understanding the fundamental concept of what a system of equations represents is the first step towards conquering these types of problems. With this foundation in place, we can now explore the methods we can use to find the elusive (x, y) pair that solves our system.
Method 1: Substitution
The substitution method is a powerful technique for solving systems of equations. The basic idea is to solve one equation for one variable, and then substitute that expression into the other equation. This eliminates one variable, leaving us with a single equation in one variable, which we can easily solve. Let's see how it works with our system:
y = 3x - 2
y = 2 - x
Notice that both equations are already solved for 'y'. This is perfect for the substitution method! We can take the expression for 'y' from the first equation (3x - 2) and substitute it into the second equation in place of 'y'. This gives us:
3x - 2 = 2 - x
Now we have a single equation with just 'x' as the variable. Let's solve for 'x'. First, we'll add 'x' to both sides:
3x + x - 2 = 2 - x + x
4x - 2 = 2
Next, we'll add 2 to both sides:
4x - 2 + 2 = 2 + 2
4x = 4
Finally, we'll divide both sides by 4:
4x / 4 = 4 / 4
x = 1
Great! We've found that x = 1. Now we need to find the value of 'y'. We can substitute this value of 'x' into either of our original equations. Let's use the second equation, y = 2 - x, because it looks a bit simpler:
y = 2 - 1
y = 1
So, we've found that y = 1. This means our solution, the ordered pair (x, y), is (1, 1). The beauty of the substitution method is its directness. By cleverly replacing one variable with an equivalent expression, we transform the problem into a simpler form that we can easily handle. This method is particularly effective when one of the equations is already solved for a variable, as was the case in our problem. But even if the equations aren't initially in this form, we can often manipulate them to make substitution a viable strategy. This flexibility makes substitution a valuable tool in our problem-solving arsenal. And the best part? We're not just finding numbers; we're uncovering the hidden relationship between the equations, the point where they harmoniously meet. Now that we've conquered the problem using substitution, let's explore another powerful method: elimination.
Method 2: Elimination
The elimination method, also known as the addition method, is another fantastic way to solve systems of equations. The core idea here is to manipulate the equations so that when you add them together, one of the variables cancels out (is eliminated). This leaves you with a single equation in one variable, which you can solve. Let's apply this to our system:
y = 3x - 2
y = 2 - x
To make the elimination method work smoothly, we want to line up the 'x' and 'y' terms in both equations. Let's rearrange the second equation to look more like the first one:
y = 2 - x
y + x = 2
x + y = 2
Now our system looks like this:
y = 3x - 2
x + y = 2
To eliminate a variable, we need the coefficients (the numbers in front of the variables) of either 'x' or 'y' to be opposites (like 3 and -3, or 1 and -1). Let's focus on eliminating 'y'. The coefficient of 'y' in both equations is already 1, but we need one of them to be -1. We can achieve this by multiplying the second equation by -1:
-1 * (x + y) = -1 * 2
-x - y = -2
Now our system is:
y = 3x - 2
-x - y = -2
Now, we can add the two equations together. Notice what happens to the 'y' terms:
y + (-x - y) = (3x - 2) + (-2)
y - x - y = 3x - 2 - 2
-x = 3x - 4
Adding the equations eliminated 'y', leaving us with an equation in just 'x'. Let's solve for 'x'. First, subtract 3x from both sides:
-x - 3x = 3x - 4 - 3x
-4x = -4
Then, divide both sides by -4:
-4x / -4 = -4 / -4
x = 1
We found x = 1, just like with the substitution method! Now, we need to find 'y'. We can plug this value of 'x' into any of our original equations. Let's use x + y = 2:
1 + y = 2
Subtract 1 from both sides:
y = 2 - 1
y = 1
Again, we find y = 1. So, our solution is (1, 1). The elegance of the elimination method lies in its strategic manipulation of equations. By carefully choosing what to multiply by, we can make variables vanish, simplifying the problem and leading us to the solution. This method is particularly useful when the equations are in a standard form (Ax + By = C), but it's a versatile technique that can be adapted to various situations. Just like substitution, elimination empowers us to see the underlying structure of the system and systematically unravel its secrets. By mastering both substitution and elimination, we equip ourselves with a powerful toolkit for tackling a wide range of systems of equations. Now, let's check our answer and make sure we've got it right!
Checking Our Answer
It's always a good idea to check your answer, guys! This ensures that you haven't made any sneaky mistakes along the way. To check our solution (1, 1), we'll plug x = 1 and y = 1 into both of our original equations:
Equation 1: y = 3x - 2
1 = 3(1) - 2
1 = 3 - 2
1 = 1 ✅
Equation 2: y = 2 - x
1 = 2 - 1
1 = 1 ✅
Our solution (1, 1) satisfies both equations! This confirms that we've found the correct answer. Checking our answer isn't just about getting a gold star; it's about building confidence in our problem-solving abilities. It's a way of saying, “I’ve done the work, and I know my solution is solid.” This step is especially crucial in exams or real-world applications where errors can have significant consequences. By plugging our solution back into the original equations, we're essentially performing a quality control check, ensuring that our calculations align with the initial conditions of the problem. It’s like a detective double-checking their evidence to make sure everything fits the case. And in the world of mathematics, where precision is paramount, this final check is an indispensable part of the process. So, never skip this step! It's the ultimate safeguard against careless mistakes and the key to confidently declaring victory over the problem. With our answer verified, we can now confidently select the correct option from the choices provided.
The Correct Answer
Looking back at our answer choices:
A) (0, -2) B) (1, 1) C) (2, 4) D) (3, 7)
The correct answer is B) (1, 1). We've not only found the solution, but we've also justified our answer by showing our work and verifying the solution in both equations. Knowing that we've arrived at the correct answer through a clear and logical process is incredibly rewarding. It's not just about the answer itself; it's about the journey we've taken to get there. We've explored two powerful methods for solving systems of equations, substitution and elimination, and we've seen how each method can lead us to the same solution. We've also emphasized the importance of checking our answer, a critical step that ensures accuracy and builds confidence. This problem serves as a microcosm of the broader world of mathematics, where understanding the process is just as important as finding the solution. Each step we take, each technique we learn, adds to our toolkit and empowers us to tackle more complex challenges. So, congratulations on reaching the end of this mathematical journey! You've not only solved a problem but also strengthened your problem-solving skills, which will serve you well in countless situations. Keep exploring, keep questioning, and keep solving! The world of mathematics is vast and fascinating, and there's always something new to discover.
Conclusion
Solving systems of equations is a fundamental skill in mathematics. By understanding the concepts and practicing different methods, you can confidently tackle these problems. Remember to always check your answers to ensure accuracy. You've got this, guys! Keep practicing, and you'll become a system-of-equations-solving superstar in no time!
