Solving 5x + 4y = -14 And 3x + 6y = 6 A Comprehensive Guide

Hey guys! Let's dive into the fascinating world of solving systems of linear equations. Today, we're tackling a specific problem: solving the system:

5x + 4y = -14
3x + 6y = 6

This is a classic example, and we're going to explore different methods to crack this nut. We will also be looking at the significance of these methods in various real-world applications. So, buckle up, and let's get started!

Understanding Systems of Linear Equations

Before we jump into the solution, let's quickly recap what systems of linear equations are all about. Essentially, we're dealing with two or more linear equations that share the same variables. A linear equation, in its simplest form, represents a straight line on a graph. When we have a system, we're looking for the point (or points) where these lines intersect. This intersection represents the solution that satisfies all equations in the system simultaneously. Think of it like finding a common ground where all the equations agree. This understanding forms the bedrock upon which our solution strategies will be built. It also lays the groundwork for appreciating the diverse applications of linear equation systems in everyday problem-solving. The concept of a solution satisfying multiple conditions is not just confined to mathematics; it permeates various fields, from engineering to economics. To further illustrate, imagine designing a bridge. The structural integrity hinges on ensuring that multiple load-bearing equations are satisfied concurrently. Similarly, in economics, market equilibrium is determined by finding the price and quantity that satisfy both supply and demand equations. Thus, grasping the fundamental nature of linear equation systems provides a powerful analytical tool for navigating a wide spectrum of real-world scenarios. We're not just solving equations; we're learning to model and solve complex interconnected problems, which is a crucial skill in both academic and professional pursuits.

Methods to Solve Systems of Equations

There are several ways to solve systems of equations, and each method has its own strengths. We will focus on two popular methods today: the substitution method and the elimination method. Understanding these methods will equip you with a versatile toolkit for tackling various linear equation systems. The substitution method involves solving one equation for one variable and substituting that expression into the other equation. This reduces the system to a single equation with one variable, making it easier to solve. The elimination method, on the other hand, aims to eliminate one of the variables by adding or subtracting the equations. This often involves multiplying one or both equations by constants to ensure that the coefficients of one variable are opposites. By strategically applying these methods, we can systematically unravel the solutions of complex systems. Moreover, each method provides unique insights into the structure of the equations and the relationships between the variables. For instance, the substitution method highlights the interdependence of variables, showing how one variable's value is directly influenced by the other. The elimination method, conversely, emphasizes the overall balance of the equations, revealing how combining equations can lead to simplification. Therefore, mastering both methods not only enhances our problem-solving capabilities but also deepens our understanding of the underlying mathematical principles. We gain a more holistic perspective on linear equation systems, empowering us to approach diverse problems with confidence and flexibility. Whether we are dealing with two equations or a larger system, the core principles remain the same, making these methods universally applicable and incredibly valuable.

Solving with the Substitution Method

Let's start with the substitution method. The idea here is to solve one of the equations for one variable and then substitute that expression into the other equation. Looking at our system:

5x + 4y = -14  (Equation 1)
3x + 6y = 6   (Equation 2)

Equation 2 looks simpler for solving for x. So, let's do that:

3x + 6y = 6
3x = 6 - 6y
x = (6 - 6y) / 3
x = 2 - 2y

Now we have x in terms of y. We can substitute this expression for x into Equation 1:

5(2 - 2y) + 4y = -14
10 - 10y + 4y = -14
-6y = -24
y = 4

Great! We've found the value of y. Now, we can substitute y = 4 back into our expression for x:

x = 2 - 2(4)
x = 2 - 8
x = -6

So, the solution is x = -6 and y = 4. This means the point of intersection of the two lines represented by our equations is (-6, 4). The substitution method is particularly effective when one of the equations can be easily solved for one variable. This approach strategically simplifies the system, allowing us to systematically unravel the values of each unknown. The beauty of this method lies in its ability to reduce a complex problem into simpler, manageable steps. By isolating one variable, we create a pathway to solve for the other, and then we circle back to find the initial variable. This iterative process not only provides the solution but also offers a clear understanding of the relationship between the variables. The substitution method also lends itself well to visual representation. Imagine tracing the lines on a graph; the point where they intersect is precisely the solution we've calculated. This visual confirmation adds another layer of understanding and reinforces the mathematical concepts at play. Furthermore, the substitution method serves as a foundational technique for solving more advanced mathematical problems, including systems of nonlinear equations and optimization problems. The core principle of expressing one variable in terms of others is a versatile tool that extends far beyond basic linear systems. Therefore, mastering this method is not just about solving equations; it's about developing a strategic approach to problem-solving that will be invaluable in various mathematical and real-world contexts. It empowers us to think systematically, break down complexity, and arrive at solutions with confidence.

Solving with the Elimination Method

Next up, let's use the elimination method to solve the same system. This method involves manipulating the equations so that when we add or subtract them, one of the variables cancels out. Again, our system is:

5x + 4y = -14  (Equation 1)
3x + 6y = 6   (Equation 2)

To eliminate y, we can multiply Equation 1 by 3 and Equation 2 by -2. This will give us 12y and -12y, which will cancel out when added:

3 * (5x + 4y) = 3 * (-14)  =>  15x + 12y = -42  (Equation 3)
-2 * (3x + 6y) = -2 * (6)   =>  -6x - 12y = -12   (Equation 4)

Now, let's add Equation 3 and Equation 4:

(15x + 12y) + (-6x - 12y) = -42 + (-12)
9x = -54
x = -6

We got x = -6, which matches our solution from the substitution method. Now, we can substitute this value back into either Equation 1 or Equation 2 to solve for y. Let's use Equation 2:

3(-6) + 6y = 6
-18 + 6y = 6
6y = 24
y = 4

Again, we find y = 4. So, the solution is x = -6 and y = 4, confirming our previous result. The elimination method shines when the coefficients of one variable are easily made opposites through multiplication. This method skillfully balances the equations, allowing us to selectively eliminate variables and simplify the system. The elegance of the elimination method lies in its ability to orchestrate the equations, transforming them into a form where variables neatly cancel out. By carefully choosing the multipliers, we create a mathematical symphony where terms harmonize and solutions emerge. This process not only solves the system but also provides a deep appreciation for the interconnectedness of the equations. Imagine the equations as two sides of a scale; the elimination method is like adding or removing weights in a balanced way to isolate the unknowns. This intuitive analogy helps us grasp the fundamental principle behind the method and its effectiveness in untangling complex relationships. Moreover, the elimination method is a cornerstone technique for solving larger systems of linear equations with multiple variables. The same principles apply, and by systematically eliminating variables one by one, we can reduce the system to a manageable size and arrive at the solution. This scalability makes the elimination method an indispensable tool in various fields, including engineering, physics, and economics, where large systems of equations often arise. Therefore, mastering this method not only equips us with a powerful problem-solving technique but also prepares us to tackle more intricate mathematical challenges in both academic and professional pursuits. It empowers us to think strategically, manipulate equations with confidence, and extract solutions from seemingly complex systems.

Solution and Verification

Both methods led us to the same solution: x = -6 and y = 4. To be absolutely sure, let's verify this solution by plugging these values back into our original equations:

5(-6) + 4(4) = -30 + 16 = -14  (Equation 1 - Correct!)
3(-6) + 6(4) = -18 + 24 = 6   (Equation 2 - Correct!)

The solution checks out! We've successfully solved the system of equations. Verifying our solution is a crucial step in the problem-solving process. It serves as a final check, ensuring that our calculations are accurate and our solution satisfies all the given conditions. By plugging the values back into the original equations, we're essentially testing whether our solution is a harmonious fit for the system as a whole. This step not only confirms our answer but also deepens our understanding of the equations and their relationships. Imagine our solution as a key that unlocks the system; verification is the act of inserting the key and confirming that it turns the lock smoothly. This analogy highlights the importance of ensuring that our solution is a perfect match for the problem at hand. Moreover, the verification process instills confidence in our problem-solving abilities. It provides a sense of closure, knowing that we've not only arrived at an answer but also validated its correctness. This confidence is invaluable, especially when tackling more complex problems where the stakes are higher. Furthermore, the habit of verifying solutions is a transferable skill that extends far beyond mathematics. In any field, whether it's engineering, science, or even everyday decision-making, the ability to check and validate our results is essential for ensuring accuracy and avoiding costly mistakes. Therefore, making verification a standard part of our problem-solving routine not only enhances our mathematical proficiency but also cultivates a valuable mindset that will serve us well in various aspects of life. It empowers us to approach problems with diligence, attention to detail, and a commitment to accuracy.

Real-World Applications

Systems of linear equations aren't just abstract mathematical concepts; they have tons of real-world applications. For example, they're used in:

• Economics: Determining market equilibrium (supply and demand).
• Engineering: Designing structures and circuits.
• Computer Graphics: Creating 3D models and animations.
• Navigation: Calculating routes and distances.

These are just a few examples, but the possibilities are endless. The power of systems of linear equations lies in their ability to model and solve problems involving multiple variables and constraints. The real-world applicability of systems of linear equations stems from their ability to represent and solve problems involving multiple interconnected variables. They provide a mathematical framework for modeling complex relationships and finding solutions that satisfy a set of conditions. This capability makes them indispensable tools in a wide range of fields, from the sciences and engineering to economics and computer science. In economics, for instance, systems of linear equations are used to model market equilibrium, where the supply and demand curves intersect. By solving the system, economists can determine the equilibrium price and quantity of a product, which are crucial for understanding market dynamics. In engineering, systems of linear equations are essential for designing structures, circuits, and control systems. Engineers use these equations to analyze forces, currents, and feedback loops, ensuring that their designs are safe, efficient, and reliable. In computer graphics, systems of linear equations play a vital role in creating 3D models and animations. They are used to transform objects, calculate lighting effects, and simulate realistic movements, bringing virtual worlds to life. In navigation, systems of linear equations are used to calculate routes and distances. GPS systems, for example, rely on solving systems of equations to determine a user's location and provide directions. Beyond these specific examples, systems of linear equations are used in countless other applications, from optimizing logistics and resource allocation to analyzing data and making predictions. Their versatility and power make them a fundamental tool for problem-solving in the modern world. Learning to solve these systems is not just about mastering a mathematical technique; it's about gaining the ability to model and solve real-world problems that impact our lives in numerous ways. Therefore, the investment in understanding these concepts is an investment in our ability to navigate and shape the world around us.

Conclusion

So, we've successfully solved the system of equations 5x + 4y = -14 and 3x + 6y = 6 using both the substitution and elimination methods. We found that x = -6 and y = 4. Remember, practicing these methods is key to mastering them. Keep solving problems, and you'll become a pro in no time! Understanding the methods for solving systems of equations, as we have demonstrated with the example 5x + 4y = -14 and 3x + 6y = 6, is a crucial skill in mathematics and its applications. We successfully employed both the substitution and elimination methods to arrive at the solution x = -6 and y = 4, highlighting the versatility of these approaches. The substitution method, where we solved for one variable in terms of the other and substituted it into the second equation, showcased its effectiveness in simplifying the system. On the other hand, the elimination method, which involved manipulating the equations to eliminate one variable, demonstrated a different yet equally powerful strategy. Both methods, when applied correctly, led us to the same solution, reinforcing the consistency and reliability of these techniques. Moreover, we emphasized the importance of verifying the solution by plugging the values back into the original equations, ensuring the accuracy of our calculations. Beyond the mechanics of solving, we explored the real-world applications of systems of equations, ranging from economics and engineering to computer graphics and navigation. This underscored the relevance of these mathematical concepts in various fields and their ability to model and solve complex problems. The ability to solve systems of equations is not just an academic exercise; it's a valuable skill that empowers us to analyze, model, and solve real-world challenges. The process of mastering these methods involves practice, perseverance, and a willingness to explore different approaches. Each problem we solve strengthens our understanding and builds our confidence in tackling more complex challenges. Therefore, we encourage continued practice and exploration of these techniques, as they form a cornerstone of mathematical problem-solving and have far-reaching applications in our increasingly quantitative world. The journey of learning mathematics is a continuous one, and the ability to solve systems of equations is a significant milestone along this path. It is a skill that opens doors to new possibilities and empowers us to think critically and creatively about the world around us. So, let's embrace the challenge and continue to hone our problem-solving abilities, unlocking the power of mathematics to shape our future.