Substitution Method: Solving System Equations

Hey math enthusiasts! Let's dive into the substitution method – a fantastic way to tackle those tricky systems of equations. We're going to break down how to solve the system: $2x + 3y = 4$ and $-4x - 6y = -1$. Get ready, because by the end of this, you'll be a substitution pro! This method is a cornerstone in algebra, and understanding it well opens doors to solving a huge variety of problems. It's all about strategic manipulation and, of course, a little bit of algebraic finesse. So grab your pencils, and let's get started. We'll explore each step with crystal clarity, ensuring you grasp not just the how but also the why behind each move. Trust me, it's easier than it looks, and the satisfaction of finding the solution is totally worth it. Now, let’s be real – systems of equations might seem a bit daunting at first, but with the substitution method, we can systematically break them down into manageable steps. This process will enable us to isolate variables and find the exact values that satisfy all the equations in the system simultaneously. This approach offers a clear and organized path toward the solution, which is why it's such a fundamental tool in mathematics. It is also important to remember that this method is not only useful for linear equations. It can be applied in various contexts and different types of equations as well. Understanding this concept is the key to unlocking the true potential of solving various mathematical problems. Let's make sure that you are comfortable with the steps. Let's unravel this mystery together!

Understanding the Substitution Method

The substitution method is like a strategic maneuver in a math game. The main idea is to isolate one variable in one equation and then substitute that expression into the other equation. This process effectively reduces the system to a single equation with only one variable, which you can easily solve. Once you find the value of that variable, you can plug it back into either of the original equations to find the value of the other variable. Think of it as a mathematical puzzle where you're trying to find the values of x and y that satisfy both equations simultaneously. The key is to eliminate one variable at a time until you are left with just one, then back-substitute to find the other. Pretty neat, right? This is a fundamental concept in algebra, and it forms the basis for many other more complex problem-solving techniques. By mastering this method, you're not just solving equations; you're building a strong foundation for future mathematical endeavors. Remember, practice makes perfect. The more you work through different examples, the more confident you'll become in applying this method. It is the cornerstone for more advanced algebraic concepts. So keep practicing, and don't be afraid to make mistakes – that's how we learn!

In our particular problem, we have two linear equations. When we're using the substitution method, it doesn't matter which equation we start with or which variable we choose to isolate. But, for simplicity, it's often easiest to start with the equation where one of the variables has a coefficient of 1 or -1. This makes it easier to isolate that variable without dealing with fractions. However, with our system of equations, neither equation readily offers an obvious choice for isolation. Let's start with the first equation: $2x + 3y = 4$. Our goal is to solve the system, and that is to find the values of x and y that satisfy both the given equations simultaneously. This is the heart of the substitution method. Now, let's take a closer look at the step-by-step process of how we can apply the method to solve such a system. The goal is to make sure we understand the technique and can solve the problem easily.

Step-by-Step Solution

Alright, let's get down to the nitty-gritty and solve the system of equations using the substitution method. Remember our equations? Here they are again:

1. $2x + 3y = 4$
2. $-4x - 6y = -1$

Step 1: Isolate a Variable

Let's start by trying to isolate x in the first equation. We have $2x + 3y = 4$. To isolate x, we can subtract $3y$ from both sides: $2x = 4 - 3y$. Then, we can divide both sides by 2: $x = (4 - 3y) / 2$. Cool, we've expressed x in terms of y. However, it could lead to fractions, which we want to avoid for now. Let’s try a different approach. Looking at the second equation, $-4x - 6y = -1$, we can try to isolate x here instead. Add $6y$ to both sides, so we get $-4x = -1 + 6y$. Divide both sides by -4: $x = (-1 + 6y) / -4$ or $x = (1 - 6y) / 4$. Again, this could lead to fractions. In this case, we have to look for a better option. Let's analyze the given equations closely. Notice anything special? The second equation, $-4x - 6y = -1$, can be simplified by dividing the entire equation by -2. This gives us $2x + 3y = 1/2$. Now, compare this to the first equation $2x + 3y = 4$. Do you see the problem? If $2x + 3y$ can equal both 4 and 1/2, then there is no solution!

Step 2: Substitute and Solve

Since we've realized in Step 1 that the equations are inconsistent and will not lead to a solution, we don't even need to proceed to the substitution step. But, for educational purposes, let's pretend we didn't notice this. Now, let's take the expression we found for x (although, technically, we didn't find a convenient one) and substitute it into the other equation. If we use the original first equation: $2x + 3y = 4$, our substitution won't work, because it's the same equation. However, if we take the second equation: $-4x - 6y = -1$, and substitute, we will get: $-4 * ((1 - 6y) / 4) - 6y = -1$. Simplifying this gives us: $-(1 - 6y) - 6y = -1$. Further simplifying, we get: $-1 + 6y - 6y = -1$. This simplifies to $-1 = -1$, which is always true. But it doesn't give us a unique solution. It tells us that the lines are parallel and, therefore, there is no solution. In situations like this, you will know you have parallel lines.

Step 3: Solve for the Remaining Variable

Since we’ve already determined that there is no solution, we don't need to proceed to this step.

Step 4: Check Your Solution

As we concluded in the previous steps, we don't have a specific point (x, y) that satisfies both equations. The lines represented by the equations are parallel, so they never intersect. Thus, there is no solution.

Analyzing the Results and Conclusion

After working through the steps, we found that the system of equations has no solution. This means the lines represented by the equations are parallel and never intersect. This is why when you use the substitution method, and you end up with a statement like $-1 = -1$, you will know that the lines are parallel. So, the correct answer is: A. no solution. Always remember to check your work. Reviewing your steps helps you catch any mistakes you might have made and solidifies your understanding of the concepts. Keep practicing, and you'll become a substitution pro in no time! Keep in mind that the substitution method is versatile. By mastering this method, you'll be well-equipped to tackle various algebraic challenges. Understanding this method is fundamental, so you can solve many problems in algebra.