Solve: 5x + 7y = 6, 10x - 3y = 46 (System Of Equations)
Hey guys! Today, we're diving into the fascinating world of solving systems of equations. Specifically, we'll be tackling the system:
This might look intimidating at first, but don't worry! We'll break it down step by step, exploring different methods to find the values of x and y that satisfy both equations simultaneously. Systems of equations are a fundamental concept in mathematics, appearing in various fields like physics, engineering, economics, and computer science. Mastering them opens doors to solving real-world problems involving multiple variables and constraints. So, buckle up, and let's embark on this mathematical journey together!
Understanding systems of equations is crucial for anyone venturing into advanced mathematics or applied sciences. A system of equations, at its core, represents a set of two or more equations with the same variables. The solution to a system is the set of values for the variables that make all the equations true. In simpler terms, it's finding the point(s) where the lines represented by the equations intersect. There are several methods to tackle these systems, and we'll explore the most common ones – substitution, elimination, and graphical methods – in detail. Each method has its strengths and weaknesses, and choosing the right approach can significantly simplify the solution process. This guide aims to provide a comprehensive understanding of how to solve systems of equations, focusing on the given example but also extending the concepts to more general cases. By the end, you'll not only be able to solve the given system but also have a solid foundation for tackling similar problems in your academic or professional pursuits.
The beauty of linear algebra, which systems of equations fall under, lies in its ability to model real-world scenarios. Imagine you're trying to determine the optimal mix of ingredients for a recipe, considering both cost and nutritional value. Or perhaps you're analyzing the flow of traffic in a network, where the number of vehicles entering and exiting certain points must balance. These scenarios can often be expressed as systems of equations, and solving them provides actionable insights. Furthermore, understanding systems of equations is a stepping stone to more advanced topics like matrices, determinants, and linear transformations, which are essential tools in various scientific and engineering disciplines. So, as we delve into the methods for solving the given system, remember that we're not just manipulating numbers; we're developing a skillset that will empower you to analyze and solve complex problems in diverse fields. Let's get started by exploring the first method: substitution.
The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with a single variable, which is much easier to solve. Once we find the value of that variable, we can plug it back into either of the original equations to find the value of the other variable. Let's apply this to our system:
First, we need to choose an equation and a variable to isolate. Looking at the equations, it seems easiest to solve the first equation for x. Here's how we do it:
- Isolate x in the first equation:
5x + 7y = 6 5x = 6 - 7y x = (6 - 7y) / 5 - Substitute this expression for x into the second equation:
10 * ((6 - 7y) / 5) - 3y = 46 - Simplify and solve for y:
2 * (6 - 7y) - 3y = 46 12 - 14y - 3y = 46 12 - 17y = 46 -17y = 34 y = -2 - Substitute the value of y back into the expression for x:
x = (6 - 7 * (-2)) / 5 x = (6 + 14) / 5 x = 20 / 5 x = 4
Therefore, the solution to the system using the substitution method is x = 4 and y = -2. The substitution method shines when one of the equations can be easily solved for one variable in terms of the other. However, it can become cumbersome if the expressions get complicated, involving fractions or radicals. In such cases, another method, like elimination, might be more efficient. The key is to choose the method that best suits the given system of equations. So, what have we learned? The substitution method works by expressing one variable in terms of the other, thereby reducing the system to a single equation. This process is particularly effective when one equation readily lends itself to isolating a variable. However, when dealing with more complex systems, or when neither equation easily isolates a variable, the elimination method provides a powerful alternative. Before we move on, let's take a moment to appreciate the elegance of this method – how it transforms a two-variable problem into a single-variable equation, making it solvable with basic algebraic manipulations.
The elimination method, also known as the addition method, involves manipulating the equations so that the coefficients of one variable are opposites. Then, we add the equations together, eliminating that variable and leaving us with a single equation in the other variable. Let's see how this works with our system:
- Multiply the first equation by -2: This will make the coefficient of x in the first equation -10, which is the opposite of the coefficient of x in the second equation.
-2 * (5x + 7y) = -2 * 6 -10x - 14y = -12 - Add the modified first equation to the second equation:
(-10x - 14y) + (10x - 3y) = -12 + 46 -17y = 34 - Solve for y:
y = -2 - Substitute the value of y back into either of the original equations to solve for x. Let's use the first equation:
5x + 7 * (-2) = 6 5x - 14 = 6 5x = 20 x = 4
Again, we find the solution x = 4 and y = -2. The elimination method is particularly useful when the equations are in standard form (Ax + By = C) and when the coefficients of one variable are easily made opposites by multiplication. It's a powerful technique for solving larger systems of equations as well. The beauty of the elimination method lies in its systematic approach. By strategically multiplying equations, we can create a scenario where adding them together neatly eliminates one variable. This transforms the system into a simpler, single-variable equation, which we can easily solve. This method is especially advantageous when the equations are already in a standard form (Ax + By = C), making it straightforward to identify the coefficients that need manipulation. Moreover, the elimination method scales well to larger systems of equations with three or more variables, making it a versatile tool in the mathematician's arsenal. It's a testament to the power of algebraic manipulation, showcasing how seemingly complex problems can be simplified through strategic transformations.
Consider the alternative. Suppose we didn't strategically multiply the first equation by -2. We would then have to deal with adding equations with different x and y coefficients, making it impossible to isolate either variable. This illustrates the critical role of the multiplication step in the elimination method. It's not just about changing the numbers; it's about creating a specific relationship between the coefficients that allows for the cancellation of one variable upon addition. This elegant maneuver is what makes the elimination method so efficient and powerful. It's a prime example of how careful planning and strategic thinking can simplify complex problems in mathematics. So, as you practice solving systems of equations, remember to focus on this strategic manipulation of coefficients – it's the key to mastering the elimination method. Now, let's move on to the next method for solving systems of equations.
While not always the most precise, the graphical method provides a visual way to understand the solution to a system of equations. Each equation represents a line, and the solution is the point where the lines intersect. To use this method, we graph each equation and find the intersection point.
Let's rewrite our equations in slope-intercept form (y = mx + b):
- Equation 1:
5x + 7y = 6 7y = -5x + 6 y = (-5/7)x + 6/7 - Equation 2:
10x - 3y = 46 -3y = -10x + 46 y = (10/3)x - 46/3
Now, we can graph these two lines. The intersection point appears to be at (4, -2), which confirms our solutions from the previous methods. The graphical method offers an intuitive understanding of what it means to solve a system of equations. Each equation represents a line, and the solution is the point where these lines meet. This visual representation can be particularly helpful for understanding the concept of solutions, especially when dealing with systems of two equations in two variables. However, the graphical method has its limitations. It relies on accurate graphing, which can be challenging, especially when the coefficients are not integers or when the solution involves fractions or decimals. In such cases, the point of intersection might not be easily determined from the graph alone. Therefore, while the graphical method is a valuable tool for visualizing solutions, it's often used in conjunction with algebraic methods like substitution or elimination to obtain precise answers.
Moreover, the graphical method is less practical for systems with more than two variables, as it becomes difficult to visualize the intersections of planes or higher-dimensional objects. In these cases, algebraic methods are generally preferred for their accuracy and efficiency. Despite these limitations, the graphical method remains a powerful pedagogical tool, helping students develop a deeper understanding of the relationship between equations and their solutions. It provides a visual check for algebraic solutions and can reveal insights that might not be immediately apparent from the equations themselves. So, while it might not be the method of choice for all systems, the graphical method holds a significant place in the toolbox of anyone studying systems of equations. In summary, the graphical method works by visually representing each equation as a line and identifying the point of intersection, which corresponds to the solution. While it offers a valuable visual aid, its limitations in accuracy and scalability make algebraic methods often more practical for complex systems. The visual clarity of the graphical method makes it a useful tool for understanding the nature of solutions to systems of equations, complementing the precision of algebraic techniques.
We've successfully solved the system of equations using three different methods: substitution, elimination, and graphing. All methods led us to the same solution: x = 4 and y = -2. This demonstrates that different approaches can be used to solve the same problem, and the best method often depends on the specific equations in the system. Remember, the key is to understand the underlying principles and choose the method that you find most efficient and comfortable. Mastering systems of equations is a fundamental skill in mathematics, and with practice, you'll become proficient at solving them. Solving systems of equations is a journey, not a destination. Each method – substitution, elimination, and graphing – offers a unique perspective and a different set of tools. The ability to choose the most appropriate method for a given problem is a mark of mathematical maturity. As you continue your studies, you'll encounter systems of equations in increasingly complex contexts, from modeling physical phenomena to optimizing business strategies. The skills you've developed here will serve you well in those endeavors. So, keep practicing, keep exploring, and remember that the beauty of mathematics lies in its ability to provide multiple paths to the same truth. Each method we've explored offers a unique pathway to the solution, highlighting the versatility and interconnectedness of mathematical concepts.
As we've seen, understanding systems of equations is not just about finding the right numbers; it's about developing a deeper appreciation for the relationships between variables and the power of algebraic manipulation. The techniques we've covered are not just tools for solving textbook problems; they are fundamental skills that can be applied to a wide range of real-world situations. Whether you're analyzing data, designing a bridge, or managing a budget, the ability to solve systems of equations will be a valuable asset. So, embrace the challenge, hone your skills, and remember that every problem you solve is a step towards greater mathematical fluency. Keep exploring different methods, and don't be afraid to experiment. The more you practice, the more comfortable and confident you'll become in your ability to tackle any system of equations that comes your way. The journey of mathematical discovery is ongoing, and the skills you've acquired here are just the beginning.
So, congratulations on mastering these techniques for solving systems of equations! You've not only learned how to find the numerical solutions but also gained a deeper understanding of the underlying principles and the interconnectedness of mathematical concepts. Remember, practice is key to solidifying your understanding and building confidence. Don't hesitate to revisit these methods and apply them to new problems. The world is full of situations that can be modeled and solved using systems of equations, so keep your eyes open for opportunities to use your newfound skills. And remember, the journey of learning mathematics is a continuous one, filled with exciting discoveries and challenges. Embrace the process, and enjoy the satisfaction of solving complex problems. Now, go forth and conquer more mathematical frontiers!
