Solving Linear Equations 94x + 60(0.3 - X) + 93y = 49.733x + 330y = 165.2 A Step-by-Step Guide

Jul 14, 2025 by Admin 95 views

Solving a System of Linear Equations: A Detailed Guide

Introduction

In the realm of mathematics, particularly in algebra, solving systems of linear equations is a fundamental skill. These systems, consisting of two or more equations with the same variables, appear in various real-world applications, from physics and engineering to economics and computer science. Mastering the techniques to solve these systems is crucial for anyone seeking to understand and model the world around them. This article delves into the intricacies of solving a specific system of linear equations, providing a step-by-step guide to understanding the process and the underlying principles. We will explore the equations 94x + 60(0.3 - x) + 93y = 49.733x + 330y = 165.2, dissecting each component and employing algebraic manipulations to arrive at a solution. This journey will not only provide a solution to this particular system but also equip you with the tools and knowledge to tackle similar problems with confidence. Understanding the structure of linear equations is the first step. Each equation represents a straight line when graphed, and the solution to the system is the point where these lines intersect. The challenge lies in finding this point algebraically, which requires a systematic approach. We will explore techniques such as simplification, substitution, and elimination, each of which plays a vital role in unraveling the unknowns within the equations. By the end of this article, you will have a comprehensive understanding of how to solve this system and a broader appreciation for the power of linear algebra.

Simplifying the First Equation: 94x + 60(0.3 - x) + 93y = 49.733x + 330y = 165.2

The initial step in tackling the system of equations is to simplify the first equation: 94x + 60(0.3 - x) + 93y = 49.733x + 330y = 165.2. This involves distributing the 60 across the terms inside the parentheses and then combining like terms. This process is crucial for making the equation more manageable and revealing the underlying relationships between the variables. Let's break down the simplification process step by step. First, we distribute the 60: 60 * 0.3 = 18 and 60 * -x = -60x. This gives us: 94x + 18 - 60x + 93y. Next, we combine the 'x' terms: 94x - 60x = 34x. So, the equation now looks like this: 34x + 18 + 93y. Now, let's focus on simplifying the entire system. The original system is presented as two equations implicitly linked: 94x + 60(0.3 - x) + 93y = 49.733x + 330y and 49.733x + 330y = 165.2. We've already simplified the left-hand side of the first part. Now, we'll move on to incorporating the second part of the system. This involves rearranging terms and bringing all variables to one side of the equation. The goal here is to isolate the variables and constants, making it easier to compare and manipulate the equations. This simplification process is not just about making the equation look simpler; it's about revealing the underlying structure and relationships. By combining like terms and rearranging the equation, we gain a clearer picture of how the variables interact and how they contribute to the overall solution. This understanding is essential for the next steps, where we will use techniques like substitution or elimination to solve for the unknowns. The simplified form also makes it easier to identify potential strategies for solving the system, such as looking for opportunities to eliminate variables or to express one variable in terms of another. This initial simplification is the foundation upon which the rest of the solution process is built.

Isolating Variables and Constants

Following the initial simplification, the next crucial step in solving the system of equations is to isolate the variables and constants. This involves rearranging the terms in the equation to group the variables (x and y) on one side and the constants on the other. This step is vital for setting up the equations for further manipulation, such as substitution or elimination, which are key techniques for finding the values of x and y. Starting with our simplified equation, 34x + 18 + 93y = 49.733x + 330y, we need to move all terms involving x and y to one side and the constant term to the other. Let's subtract 34x from both sides: 18 + 93y = 49.733x - 34x + 330y, which simplifies to 18 + 93y = 15.733x + 330y. Now, let's subtract 93y from both sides: 18 = 15.733x + 330y - 93y, which further simplifies to 18 = 15.733x + 237y. This rearranged equation is now in a form where the constant is isolated on one side. Next, we need to consider the second equation in the system: 49.733x + 330y = 165.2. This equation is already in a relatively isolated form, with the variables on one side and the constant on the other. However, it's important to have both equations in a similar format to facilitate comparison and manipulation. The goal of isolating variables and constants is not just to rearrange the equation; it's to create a clear and organized structure that allows us to see the relationships between the variables and the constants. This structure is essential for applying techniques like substitution or elimination. By having the equations in this isolated form, we can easily identify opportunities to eliminate one variable by multiplying the equations by appropriate constants or to express one variable in terms of another. This step is a critical bridge between the initial simplification and the final solution, as it sets the stage for the algebraic manipulations that will ultimately lead us to the values of x and y. The ability to isolate variables and constants is a fundamental skill in algebra, and mastering this technique is essential for solving a wide range of mathematical problems.

Applying Substitution or Elimination Methods

With the equations simplified and the variables and constants isolated, the next step is to apply either the substitution or elimination method to solve for the unknowns, x and y. These are two powerful techniques in algebra, each with its own strengths and best-use scenarios. The choice between substitution and elimination often depends on the specific structure of the equations. Substitution involves solving one equation for one variable and then substituting that expression into the other equation. This method is particularly useful when one of the equations can be easily solved for one variable in terms of the other. On the other hand, elimination involves manipulating the equations so that the coefficients of one variable are opposites, then adding the equations together to eliminate that variable. This method is often preferred when the coefficients of one variable are multiples of each other or when the equations are already in a form that makes elimination straightforward. Let's first consider the two equations we've derived: 18 = 15.733x + 237y and 49.733x + 330y = 165.2. Looking at these equations, it might be easier to use the elimination method due to the presence of decimal coefficients in both equations. To use elimination, we need to find a way to make the coefficients of either x or y opposites. This often involves multiplying one or both equations by a constant. For example, we could choose to eliminate x. To do this, we need to find a common multiple of 15.733 and 49.733. However, this might involve complex calculations. Alternatively, we could choose to eliminate y. The coefficients of y are 237 and 330. While finding a common multiple for these numbers is also challenging, it might be a more manageable approach than dealing with the decimal coefficients of x. Another approach is to consider substitution, where we solve one equation for one variable and substitute it into the other. If we solve the first equation for x, we would have x = (18 - 237y) / 15.733. Substituting this expression for x into the second equation would result in an equation with only y, which we could then solve. Once we find the value of y, we can substitute it back into either equation to find the value of x. The key to choosing the best method is to look for the most efficient path to a solution. Sometimes, a combination of both methods might be necessary. For instance, we might use elimination to simplify the equations and then use substitution to solve for the variables. The process of applying substitution or elimination is not just about following a set of rules; it's about making strategic decisions based on the structure of the equations. It requires a deep understanding of algebraic manipulation and a willingness to explore different approaches.

Solving for x and y

Once we've chosen a method – either substitution or elimination – the next crucial step is to meticulously solve for the values of x and y. This involves carrying out the algebraic manipulations carefully and systematically, ensuring that each step is logically sound and mathematically accurate. Let's illustrate this process using the elimination method, building upon the equations we derived earlier: 18 = 15.733x + 237y and 49.733x + 330y = 165.2. As discussed in the previous section, eliminating y might be a more manageable approach due to the integer coefficients. To eliminate y, we need to find a common multiple of 237 and 330. One way to do this is to multiply the first equation by 330 and the second equation by -237. This will give us opposite coefficients for y. Multiplying the first equation by 330, we get: 330 * 18 = 330 * (15.733x + 237y), which simplifies to 5940 = 5191.89x + 78210y. Multiplying the second equation by -237, we get: -237 * (49.733x + 330y) = -237 * 165.2, which simplifies to -11787.741x - 78210y = -39152.4. Now, we can add the two equations together: (5940 = 5191.89x + 78210y) + (-11787.741x - 78210y = -39152.4). Adding the left sides, we get: 5940 - 39152.4 = -33212.4. Adding the right sides, the y terms cancel out (78210y - 78210y = 0), and we have: 5191.89x - 11787.741x = -6595.851x. So, the resulting equation is: -33212.4 = -6595.851x. Now, we can solve for x by dividing both sides by -6595.851: x = -33212.4 / -6595.851 ≈ 5.035. With the value of x found, we can substitute it back into either of the original equations to solve for y. Let's use the first equation: 18 = 15.733x + 237y. Substituting x ≈ 5.035, we get: 18 = 15.733 * 5.035 + 237y, which simplifies to 18 ≈ 79.236 + 237y. Subtracting 79.236 from both sides, we get: -61.236 ≈ 237y. Dividing both sides by 237, we get: y ≈ -0.258. Therefore, the solution to the system of equations is approximately x ≈ 5.035 and y ≈ -0.258. The process of solving for x and y highlights the importance of precision and attention to detail in algebraic manipulations. Each step must be carefully executed to avoid errors that could lead to an incorrect solution. Furthermore, it demonstrates the power of systematic thinking in tackling complex problems. By breaking down the problem into smaller, manageable steps and applying the appropriate techniques, we can arrive at a solution with confidence.

Verifying the Solution

After obtaining the values for x and y, the final and crucial step is to verify the solution. This involves substituting the calculated values back into the original equations to ensure that they satisfy both equations simultaneously. This verification process is essential for confirming the accuracy of the solution and identifying any potential errors made during the algebraic manipulations. Let's take the approximate solution we found, x ≈ 5.035 and y ≈ -0.258, and substitute these values back into the original equations: 94x + 60(0.3 - x) + 93y = 49.733x + 330y and 49.733x + 330y = 165.2. First, let's substitute into the second equation, as it's simpler: 49.733 * 5.035 + 330 * -0.258. Calculating this, we get: 250.398 - 85.14 ≈ 165.258. This is very close to 165.2, which suggests our solution is reasonably accurate, given the rounding. Now, let's substitute into the first equation: 94 * 5.035 + 60(0.3 - 5.035) + 93 * -0.258. Calculating this, we get: 473.29 + 60(-4.735) - 23.994. Further simplifying, we have: 473.29 - 284.1 - 23.994 ≈ 165.196. Now, let's calculate the right side of the first equation using our solution: 49. 733 * 5.035 + 330 * -0.258, which we already calculated as approximately 165.258. Comparing the two sides of the first equation, we have approximately 165.196 ≈ 165.258. This close agreement further confirms the accuracy of our solution. The verification process not only confirms the solution but also provides a deeper understanding of the system of equations. It demonstrates how the values of x and y interact to satisfy the relationships defined by the equations. If the substitution does not result in a close equality, it indicates that there might be an error in the calculations or that the solution is not accurate. In such cases, it's necessary to revisit the steps taken to solve the equations and identify the source of the error. Verifying the solution is a critical step in the problem-solving process, and it reinforces the importance of accuracy and attention to detail in mathematics.

Conclusion

In conclusion, solving systems of linear equations is a fundamental skill in mathematics with wide-ranging applications. The specific system we addressed, 94x + 60(0.3 - x) + 93y = 49.733x + 330y = 165.2, provided a practical example of the steps involved in this process. We began by simplifying the equations, which involved distributing terms and combining like terms. This step is crucial for making the equations more manageable and revealing the underlying relationships between the variables. Next, we isolated the variables and constants, rearranging the equations to group the x and y terms on one side and the constants on the other. This step sets the stage for applying techniques like substitution or elimination. We then explored the application of the elimination method, strategically manipulating the equations to eliminate one variable and solve for the other. This involved multiplying the equations by appropriate constants to create opposite coefficients for one variable, allowing us to add the equations together and eliminate that variable. After finding the value of one variable, we substituted it back into one of the original equations to solve for the remaining variable. This process requires careful attention to detail and accurate algebraic manipulation. Finally, we emphasized the importance of verifying the solution. This involves substituting the calculated values of x and y back into the original equations to ensure that they satisfy both equations simultaneously. This step is essential for confirming the accuracy of the solution and identifying any potential errors. Throughout this process, we highlighted the importance of understanding the underlying principles of linear algebra and the strategic decision-making involved in choosing the most efficient solution method. Solving systems of equations is not just about following a set of rules; it's about developing a deep understanding of the relationships between variables and constants and applying algebraic techniques with precision and accuracy. By mastering these skills, you can tackle a wide range of mathematical problems and gain a valuable tool for modeling and understanding the world around you.