Solving Systems Of Equations 2x + 5y = 31 And 2x + 3y = 20 A Step-by-Step Guide
Introduction to Solving Linear Equations
In mathematics, solving systems of linear equations is a fundamental skill with applications across various fields, including engineering, economics, and computer science. A system of linear equations involves two or more equations with the same variables, and the goal is to find values for these variables that satisfy all equations simultaneously. This article delves into a step-by-step method for solving the system of equations: 2x + 5y = 31 and 2x + 3y = 20. Understanding how to solve such systems is crucial for anyone studying algebra or related disciplines. The process involves manipulating the equations to isolate variables and ultimately find their values. Mastering this technique opens doors to solving more complex problems and understanding mathematical models used in real-world scenarios. This introduction serves as a foundation for exploring the intricacies of linear equations and their solutions. Let’s embark on this mathematical journey, unraveling the methods to tackle these equations effectively. We will cover the elimination method, which is particularly useful when coefficients of one variable are the same or can be easily made the same through multiplication. By the end of this guide, you'll have a clear understanding of how to approach similar problems and solve them with confidence. Remember, practice is key, so work through examples and apply the techniques you learn here to various systems of equations. This article aims to be your comprehensive resource for mastering the solution of linear systems.
Understanding the Equations: 2x + 5y = 31 and 2x + 3y = 20
Before we dive into the solution, let's first understand the equations we're dealing with: 2x + 5y = 31 and 2x + 3y = 20. These are two linear equations, each representing a straight line when graphed on a coordinate plane. The 'x' and 'y' are variables, and our task is to find the values of 'x' and 'y' that satisfy both equations simultaneously. Each equation has two terms involving the variables 'x' and 'y', along with a constant term. The coefficients of 'x' and 'y' (the numbers multiplying them) determine the slope and position of the line. In this case, the coefficient of 'x' is 2 in both equations, which will be crucial for our solution method. The constants (31 and 20) represent the y-intercept if the equation were rearranged into slope-intercept form (y = mx + b). To solve this system, we need to find a pair of (x, y) values that, when substituted into both equations, make both equations true. This point (x, y) represents the intersection of the two lines on the graph. Understanding the structure of these equations is the first step towards solving them. We can use different methods, such as substitution, elimination, or graphing, to find the solution. In this article, we will focus on the elimination method, which is particularly efficient when the coefficients of one variable are the same or multiples of each other. Grasping these foundational concepts is essential for navigating the solution process effectively.
The Elimination Method: A Step-by-Step Approach
The elimination method is a powerful technique for solving systems of linear equations. It involves manipulating the equations to eliminate one variable, allowing us to solve for the other. In our system, we have:
- 2x + 5y = 31
- 2x + 3y = 20
Notice that the coefficient of 'x' is the same in both equations (2). This makes the elimination method particularly convenient. The first step is to subtract one equation from the other. Let's subtract equation (2) from equation (1): (2x + 5y) - (2x + 3y) = 31 - 20. This simplifies to 2y = 11. Now, we can solve for 'y' by dividing both sides by 2: y = 11/2 or y = 5.5. We have now found the value of 'y'. The next step is to substitute this value back into one of the original equations to solve for 'x'. Let's use equation (2): 2x + 3(5.5) = 20. This simplifies to 2x + 16.5 = 20. Subtracting 16.5 from both sides gives 2x = 3.5. Finally, dividing by 2 gives x = 3.5 / 2 or x = 1.75. So, the solution to the system of equations is x = 1.75 and y = 5.5. The elimination method works by creating a situation where adding or subtracting equations cancels out one variable, leaving a single equation with one variable. This method is especially efficient when coefficients match or can be easily matched through multiplication. Understanding this method provides a clear path to solving various systems of equations.
Detailed Solution: Eliminating 'x' to Solve for 'y'
To delve deeper into the solution, let's break down the process of eliminating 'x' to solve for 'y'. We start with our system of equations:
- 2x + 5y = 31
- 2x + 3y = 20
As we noted, the coefficients of 'x' in both equations are the same, which is ideal for elimination. Our goal is to subtract one equation from the other in a way that the 'x' terms cancel out. To achieve this, we subtract equation (2) from equation (1). This ensures that the 2x terms will eliminate each other:
(2x + 5y) - (2x + 3y) = 31 - 20
Now, we distribute the subtraction across the terms in the parentheses:
2x + 5y - 2x - 3y = 11
Combine like terms on the left side of the equation. The 2x and -2x cancel each other out:
(2x - 2x) + (5y - 3y) = 11
This simplifies to:
2y = 11
Now, we have a simple equation with just one variable, 'y'. To solve for 'y', we divide both sides of the equation by 2:
y = 11 / 2
So, y = 5.5. This step-by-step breakdown clarifies how subtracting the equations leads to the elimination of 'x' and allows us to isolate 'y'. The key is to perform the subtraction carefully, ensuring that each term is correctly accounted for. Once we have the value of 'y', we can proceed to substitute it back into one of the original equations to find 'x'. This methodical approach is the cornerstone of the elimination method.
Substituting 'y' to Find 'x': A Comprehensive Explanation
Now that we have found the value of 'y' (y = 5.5), the next step is to substitute this value into one of the original equations to solve for 'x'. We can choose either equation (1) or equation (2); the result will be the same. For this explanation, let's use equation (2) as it appears simpler:
2x + 3y = 20
Substitute y = 5.5 into the equation:
2x + 3(5.5) = 20
Now, we simplify the equation by multiplying 3 by 5.5:
2x + 16.5 = 20
Our goal is to isolate 'x'. To do this, we subtract 16.5 from both sides of the equation:
2x = 20 - 16.5
This simplifies to:
2x = 3.5
Finally, to solve for 'x', we divide both sides of the equation by 2:
x = 3.5 / 2
So, x = 1.75. This process of substitution involves replacing the variable 'y' with its numerical value, simplifying the equation, and then using basic algebraic operations to isolate 'x'. It's a crucial step in solving systems of equations, as it allows us to find the value of the remaining variable once one variable is known. By carefully following these steps, we can confidently determine the value of 'x'. This completes the solution process, giving us the values of both 'x' and 'y' that satisfy the system of equations.
Verifying the Solution: Ensuring Accuracy
After finding the values of x and y, it's essential to verify the solution to ensure accuracy. This involves substituting the values we found (x = 1.75 and y = 5.5) back into both of the original equations to see if they hold true. Let's start with the first equation:
2x + 5y = 31
Substitute the values of x and y:
2(1.75) + 5(5.5) = 31
Now, perform the multiplication:
- 5 + 27.5 = 31
Add the numbers on the left side:
31 = 31
The equation holds true. Now let's check the second equation:
2x + 3y = 20
Substitute the values of x and y:
2(1.75) + 3(5.5) = 20
Perform the multiplication:
- 5 + 16.5 = 20
Add the numbers on the left side:
20 = 20
This equation also holds true. Since the values of x and y satisfy both equations, we can confidently say that our solution is correct. Verifying the solution is a crucial step in the problem-solving process. It helps prevent errors and ensures that the values we found are indeed the correct solution to the system of equations. This step provides peace of mind and reinforces the accuracy of our work. It's a good practice to always verify your solutions, especially in mathematics, to maintain precision and correctness.
Conclusion: Mastering Systems of Linear Equations
In conclusion, we have successfully navigated the process of solving the system of linear equations 2x + 5y = 31 and 2x + 3y = 20. Through the elimination method, we systematically eliminated one variable, solved for the other, and then substituted back to find the remaining variable. Our solution is x = 1.75 and y = 5.5. We also emphasized the importance of verifying the solution to ensure accuracy, which confirmed that our values satisfy both original equations. This journey through the solution process highlights the power and elegance of algebraic methods in solving mathematical problems. Mastering systems of linear equations is a fundamental skill in mathematics with wide-ranging applications in various fields. The ability to solve such systems enables us to model and analyze real-world scenarios, make predictions, and solve complex problems. The elimination method, as demonstrated in this article, is a versatile and efficient technique that can be applied to many systems of equations. By understanding the underlying principles and practicing regularly, one can become proficient in solving linear systems. This article aimed to provide a clear and comprehensive guide, empowering you to tackle similar problems with confidence. Remember, mathematics is a journey of discovery, and each solved problem is a step forward in your understanding.
