Solving Systems Of Equations A Step By Step Guide
In mathematics, a system of equations is a set of two or more equations that share the same variables. Solving a system of equations means finding the values for the variables that satisfy all equations in the system simultaneously. There are several methods for solving systems of equations, including substitution, elimination, and graphing. This article will delve into the step-by-step process of solving a specific system of equations using the elimination method, providing a comprehensive guide for readers to understand and apply the technique effectively. Understanding how to solve systems of equations is crucial in various fields, including mathematics, physics, engineering, and economics, where multiple variables and constraints interact. Mastering these methods enables us to model and analyze real-world problems more accurately, making informed decisions based on the relationships between different factors. Whether you are a student learning the basics or a professional applying these concepts in your work, this guide will provide a solid foundation for solving systems of equations.
The Given System of Equations
We are presented with the following system of linear equations:
3x + 2y = 10
2x + 4y = 4
Our goal is to find the values of x and y that satisfy both equations. To achieve this, we'll employ the elimination method, which involves manipulating the equations to eliminate one variable, allowing us to solve for the other. In the realm of algebra, systems of equations play a vital role in modeling scenarios with multiple interacting variables. These systems often represent real-world situations where several conditions must be met simultaneously. For instance, in economics, systems of equations can model supply and demand curves, while in physics, they can describe the motion of objects under various forces. The ability to solve these systems accurately is therefore essential for making predictions and informed decisions in various fields.
The Elimination Method
The elimination method is a powerful technique for solving systems of equations. It involves manipulating the equations in such a way that the coefficients of one of the variables become opposites. This allows us to eliminate that variable when we add the equations together. In the context of our system, we can eliminate either x or y. Let's choose to eliminate x in this case. The core principle of the elimination method lies in the addition or subtraction of equations to cancel out one of the variables. This simplifies the system into a single equation with a single variable, which can then be easily solved. The key to successful elimination is to identify the appropriate multipliers for each equation, ensuring that the coefficients of the chosen variable are additive inverses. The method's versatility and efficiency make it a staple in algebraic problem-solving, particularly when dealing with linear systems. Understanding and mastering the elimination method is crucial for anyone looking to solve complex mathematical problems involving multiple variables and constraints.
Step 1: Multiplying the Equations
To eliminate x, we need to make the coefficients of x in both equations opposites. We can achieve this by multiplying the first equation by -2 and the second equation by 3:
-2 * (3x + 2y) = -2 * 10 => -6x - 4y = -20
3 * (2x + 4y) = 3 * 4 => 6x + 12y = 12
Now, the coefficients of x are -6 and 6, which are opposites. The multiplication step is crucial as it sets the stage for the elimination of one variable. By carefully selecting the multipliers, we ensure that the coefficients of the target variable become additive inverses, making the elimination process straightforward. This step often requires careful observation and strategic thinking to identify the most efficient multipliers. The resulting equations are equivalent to the original ones but are now in a form that allows for direct application of the elimination principle. The ability to manipulate equations in this way is a fundamental skill in algebra and is essential for solving systems of equations efficiently and accurately.
Step 2: Adding the Equations
Now, we add the two modified equations together:
(-6x - 4y) + (6x + 12y) = -20 + 12
This simplifies to:
8y = -8
Notice how the x terms have canceled out, leaving us with a single equation in terms of y. The addition step is the heart of the elimination method, where the carefully crafted coefficients allow for the direct removal of one variable. This simplification transforms the system into a solvable equation, making it possible to isolate and determine the value of the remaining variable. The process highlights the elegance and efficiency of the method, demonstrating how strategic manipulation can lead to a straightforward solution. The resulting equation, in this case, 8y = -8, is a simple linear equation that can be easily solved using basic algebraic techniques.
Step 3: Solving for y
To solve for y, we divide both sides of the equation by 8:
y = -8 / 8
y = -1
We have found the value of y to be -1. Solving for y involves isolating the variable by performing the inverse operation on both sides of the equation. In this case, we divide both sides by 8 to undo the multiplication. This step is a fundamental aspect of algebraic manipulation, where the goal is to simplify the equation and reveal the value of the unknown variable. The result, y = -1, provides one piece of the solution to the system of equations. This value will be used in the subsequent step to find the corresponding value of x. The accuracy of this step is critical, as any error will propagate through the rest of the solution process. The process underscores the importance of careful arithmetic and attention to detail in solving algebraic equations.
Step 4: Substituting to Find x
Now that we have the value of y, we can substitute it back into either of the original equations to solve for x. Let's use the first equation:
3x + 2y = 10
3x + 2(-1) = 10
3x - 2 = 10
3x = 12
x = 12 / 3
x = 4
We have found the value of x to be 4. Substituting the value of y back into one of the original equations allows us to solve for x. This step leverages the fact that we now have a known value for one variable, which simplifies the equation and makes it possible to isolate and determine the other. The choice of which equation to use for substitution is arbitrary, as both will yield the same result. However, selecting the equation that appears simpler can sometimes reduce the complexity of the calculations. The process highlights the interconnectedness of the variables in a system of equations, where knowing the value of one variable provides information about the others. The result, x = 4, completes the solution to the system.
The Solution
The solution to the system of equations is x = 4 and y = -1. This can be written as the ordered pair (4, -1). The solution to a system of equations represents the point where the lines represented by the equations intersect on a graph. This point satisfies both equations simultaneously, meaning that when the values of x and y are substituted into the equations, both equations hold true. The ordered pair notation (4, -1) is a standard way of representing this solution, with the x-coordinate listed first and the y-coordinate listed second. Verifying the solution by substituting the values back into the original equations is a crucial step to ensure accuracy. This confirms that the calculated values indeed satisfy all the conditions of the system.
Verification
To verify our solution, we substitute x = 4 and y = -1 into both original equations:
3x + 2y = 10
3(4) + 2(-1) = 12 - 2 = 10 (Correct)
2x + 4y = 4
2(4) + 4(-1) = 8 - 4 = 4 (Correct)
Both equations are satisfied, confirming that our solution is correct. Verification is a critical step in solving any mathematical problem, as it ensures the accuracy of the result. By substituting the calculated values back into the original equations, we can confirm that they satisfy all the conditions of the system. This process helps to catch any errors that may have occurred during the solution process and provides confidence in the final answer. The verification step underscores the importance of precision and attention to detail in mathematics, where even small errors can lead to incorrect results. The confirmation that both equations hold true validates our solution and completes the problem-solving process.
Conclusion
Therefore, the correct answer is A. (4, -1). This step-by-step guide demonstrates the elimination method for solving systems of equations. By multiplying equations, adding them strategically, and substituting values, we can efficiently find the solutions that satisfy multiple equations simultaneously. Mastering these techniques is essential for success in algebra and beyond. The ability to solve systems of equations is a fundamental skill that has wide-ranging applications in various fields. From modeling complex systems in physics and engineering to analyzing economic trends and financial markets, the principles of solving systems of equations are invaluable. This guide has provided a comprehensive overview of the elimination method, illustrating its effectiveness and efficiency in finding solutions. By practicing and applying these techniques, readers can develop a deeper understanding of algebraic concepts and enhance their problem-solving abilities. The solution (4, -1) represents the point of intersection of the two lines, highlighting the geometric interpretation of systems of equations and their solutions.
