Solving Systems Of Equations By Elimination A Step-by-Step Guide

Jul 10, 2025 by ADMIN 65 views

Solve Systems of Equations by Elimination Method

Introduction to Solving Systems of Equations

In mathematics, a system of equations is a set of two or more equations containing the same variables. The solution to a system of equations is the set of values for the variables that satisfy all equations simultaneously. There are several methods to solve systems of equations, including graphing, substitution, and elimination. This article will focus on the elimination method, a powerful technique that allows us to systematically eliminate one variable to solve for the other. This approach is particularly useful when dealing with equations in standard form (Ax + By = C). Understanding the elimination method is crucial for various applications in mathematics, science, and engineering, where systems of equations frequently arise in modeling real-world problems.

The elimination method hinges on the principle that we can add or subtract equations without changing the solution set, provided we perform the same operation on both sides of the equations. The goal is to manipulate the equations in such a way that, when added or subtracted, one of the variables cancels out, leaving us with a single equation in one variable. Once we solve for this variable, we can substitute its value back into one of the original equations to find the value of the other variable. This step-by-step process ensures a systematic and accurate approach to solving systems of equations. The beauty of the elimination method lies in its efficiency and versatility, making it a preferred choice for many mathematicians and problem solvers. It allows for a straightforward solution, especially when dealing with linear equations, which are the most common type encountered in various fields.

The application of the elimination method is not just limited to academic exercises. In practical scenarios, systems of equations are used to model diverse situations, such as determining the equilibrium point in economics, calculating the forces in a structural engineering problem, or even optimizing resource allocation in business. By mastering the elimination method, one can effectively tackle these real-world problems, gaining valuable insights and making informed decisions. For example, consider a scenario where a company needs to determine the optimal production levels for two products, given certain constraints on resources and demand. By formulating the problem as a system of equations and applying the elimination method, the company can find the exact production levels that maximize profit while adhering to the limitations. This highlights the significance of understanding and applying the elimination method in various professional domains, ensuring its continued relevance in the modern world.

Problem Statement

We are given the following system of equations:

-2x + 3y = 13
x + y = 11

Our task is to find the values of x and y that satisfy both equations simultaneously. We will use the elimination method to solve this system. The elimination method involves manipulating the equations so that either the x or y coefficients are opposites, allowing us to eliminate one variable by adding the equations together. This simplifies the system into a single equation with one unknown, which can then be easily solved. By strategically choosing which variable to eliminate, we can streamline the solution process and reduce the complexity of the calculations.

Before we dive into the steps, it’s essential to understand the underlying principle of the elimination method. When we add or subtract equations, we are essentially combining the relationships they represent. If the coefficients of one variable are opposites, their sum will be zero, effectively eliminating that variable from the resulting equation. This allows us to isolate the remaining variable and solve for its value. The success of the elimination method relies on our ability to carefully manipulate the equations, often involving multiplication by constants, to create these opposing coefficients. Once we find the value of one variable, we can easily substitute it back into one of the original equations to solve for the other variable, completing the solution of the system.

In the context of real-world applications, solving systems of equations like this one is crucial for various scenarios. For instance, this system might represent the constraints in a resource allocation problem, where x and y represent the quantities of two different resources, and the equations describe how these resources are utilized. Alternatively, the equations could represent the supply and demand curves in economics, where the solution (x, y) would represent the equilibrium point where supply equals demand. Therefore, mastering the technique of solving systems of equations, particularly the elimination method, equips us with a powerful tool to tackle a wide range of practical problems across various disciplines. The elegance of the elimination method lies in its ability to reduce a complex system into simpler, manageable components, making it an indispensable technique in mathematical problem-solving.

Step-by-Step Solution Using Elimination

Identify the Variable to Eliminate: Examine the given equations:
```
-2x + 3y = 13
x + y = 11
```
We can choose to eliminate either x or y. In this case, it appears easier to eliminate x because the coefficients of x are -2 and 1. By multiplying the second equation by 2, we can make the coefficients of x opposites. The strategic choice of which variable to eliminate can significantly impact the complexity of the calculations. For instance, if the coefficients of one variable are already close to being opposites, it might be more efficient to target that variable. On the other hand, if one equation has a variable with a coefficient of 1 or -1, it might be simpler to manipulate the other equation to match it. This initial assessment is crucial for optimizing the solution process.

When deciding which variable to eliminate, it's also important to consider the overall structure of the equations. Sometimes, the coefficients might have a common factor, which can be leveraged to simplify the multiplication process. For example, if we had coefficients of 4 and 6, we could find the least common multiple (LCM) and multiply the equations accordingly to achieve the desired elimination. This methodical approach not only reduces the chances of errors but also enhances the efficiency of the solution. The ability to quickly identify the most efficient path is a hallmark of a proficient problem solver, and it comes with practice and a deep understanding of the underlying mathematical principles.

Furthermore, the elimination method is particularly advantageous when dealing with systems of linear equations, where the relationships between the variables are straightforward and predictable. This allows for a clear and systematic approach to finding the solution. However, for non-linear systems or systems with more complex relationships, other methods, such as substitution or numerical methods, might be more appropriate. Therefore, understanding the characteristics of the system of equations is essential for selecting the most effective solution technique, making the elimination method a key tool in the mathematician's arsenal for tackling a wide array of problems.
Multiply the Second Equation by 2: This will make the coefficient of x in the second equation equal to 2, which is the opposite of -2 in the first equation.
```
2 * (x + y) = 2 * 11
2x + 2y = 22
```
The multiplication step is a fundamental aspect of the elimination method. By multiplying an equation by a constant, we are essentially scaling the entire relationship represented by the equation. This scaling does not alter the solution set, as long as we perform the same operation on both sides of the equation. The key is to choose the appropriate constant that will create opposing coefficients for the variable we want to eliminate. In this case, multiplying the second equation by 2 achieves the desired outcome of making the x coefficients opposites.

It's crucial to distribute the multiplication to every term in the equation. Overlooking even a single term can lead to an incorrect result and invalidate the entire solution process. Therefore, a careful and meticulous approach is essential during this step. The ability to accurately perform algebraic manipulations is a cornerstone of mathematical problem-solving, and the multiplication step in the elimination method provides a practical application of this skill. By mastering this step, we can confidently proceed to the next stage of the elimination process, knowing that the foundation for our solution is solid.

Moreover, the multiplication step highlights the power of algebraic transformations in simplifying complex problems. By strategically manipulating the equations, we can create a scenario where the elimination of a variable becomes straightforward. This showcases the elegance and efficiency of the elimination method as a problem-solving technique. The careful selection of the multiplying factor is an art in itself, often requiring a bit of foresight and an understanding of the relationships between the coefficients. With practice, this skill becomes intuitive, allowing us to quickly identify the optimal path to solving the system of equations.
Add the Modified Equations: Now, add the first equation to the modified second equation:
```
(-2x + 3y) + (2x + 2y) = 13 + 22
```
When we add the equations together, the x terms cancel out, as intended:
```
(-2x + 2x) + (3y + 2y) = 35
0x + 5y = 35
5y = 35
```
The addition of equations is the heart of the elimination method. This step leverages the principle that adding equal quantities to both sides of an equation maintains the equality. By carefully manipulating the equations in the previous steps to create opposing coefficients for one variable, we set the stage for this critical step. When we add the equations, the targeted variable disappears, leaving us with a single equation in one unknown. This transformation significantly simplifies the problem, allowing us to isolate and solve for the remaining variable.

The cancellation of terms is a direct consequence of the strategic preparation in the earlier steps. It's a satisfying moment in the solution process, as it validates our approach and brings us closer to the answer. However, it's crucial to ensure that all terms are correctly combined during the addition. A small error in arithmetic can derail the entire process, so attention to detail is paramount. The ability to accurately perform algebraic operations, such as addition and simplification, is essential for success in the elimination method.

Furthermore, the addition step demonstrates the power of algebraic manipulation in reducing complexity. By combining the equations in a specific way, we effectively distill the system into its essential components. This highlights the elegance and efficiency of the elimination method as a problem-solving technique. The resulting single equation is much easier to solve than the original system, showcasing the transformative power of this method. With the value of one variable now within reach, we can confidently proceed to the final steps of the solution.
Solve for y: Divide both sides of the equation by 5:
```
5y / 5 = 35 / 5
y = 7
```
Now that we have the value of y, we have made significant progress towards solving the system of equations. The division operation is a straightforward step, but it's crucial to perform it accurately to ensure we obtain the correct value for y. The goal is to isolate y on one side of the equation, and dividing both sides by the coefficient of y accomplishes this. This step demonstrates the fundamental principle of maintaining equality in an equation by performing the same operation on both sides.

The value of y = 7 represents one component of the solution to the system. It tells us the y-coordinate of the point where the two lines represented by the equations intersect. This is a significant piece of information, but we still need to find the value of x to complete the solution. The next step involves substituting this value of y back into one of the original equations to solve for x. This substitution step is a common technique in solving systems of equations, and it highlights the interconnectedness of the variables in the system.

Moreover, the solution for y provides a concrete numerical value, making the abstract algebraic manipulations more tangible. It serves as a checkpoint in the solution process, allowing us to verify our work and ensure that we are on the right track. The ability to translate abstract equations into concrete numerical values is a key skill in mathematical problem-solving, and it underscores the practical applications of the elimination method. With the value of y in hand, we can confidently move on to the final stage of the solution, knowing that we have a solid foundation for finding the value of x.
Substitute y = 7 into One of the Original Equations: Let's use the second equation:
```
x + y = 11
x + 7 = 11
```
The substitution step is a powerful technique in solving systems of equations. Now that we have the value of y, we can substitute it back into one of the original equations to solve for x. The choice of which equation to use is often a matter of convenience; we can select the equation that appears simpler or easier to manipulate. In this case, the second equation, x + y = 11, seems straightforward to work with. Substituting y = 7 into this equation allows us to create a new equation with only x as the unknown.

The substitution process involves replacing the variable y with its numerical value, 7. This transformation effectively reduces the equation to a single variable equation, which we can then solve using basic algebraic techniques. The ability to accurately perform substitutions is a fundamental skill in mathematics, and it's used extensively in various problem-solving scenarios. It's crucial to ensure that the substitution is done correctly, as any error in this step will propagate through the rest of the solution.

Furthermore, the substitution step highlights the interconnectedness of the variables in a system of equations. The value of y is directly linked to the value of x, and by knowing one, we can determine the other. This interdependence is a key characteristic of systems of equations, and it's what allows us to solve them. The substitution step effectively leverages this relationship to find the remaining unknown, bringing us closer to the complete solution of the system.
Solve for x: Subtract 7 from both sides:
```
x + 7 - 7 = 11 - 7
x = 4
```
With the value of x now determined, we have completed the solution for the system of equations. The subtraction operation is the final step in isolating x and finding its numerical value. By subtracting 7 from both sides of the equation, we maintain the equality while effectively removing the constant term from the left side. This leaves us with x isolated on one side, and its value on the other.

The value of x = 4 represents the x-coordinate of the point where the two lines represented by the equations intersect. Together with the previously found value of y = 7, we have the complete solution to the system. This solution, (4, 7), is the ordered pair that satisfies both equations simultaneously. It's a crucial piece of information, as it represents the point of intersection, which is often the key to solving real-world problems modeled by systems of equations.

Moreover, the solution for x provides a sense of closure to the problem-solving process. It confirms that our efforts in applying the elimination method have been successful, and we have found the values that make the equations true. This sense of accomplishment is important in building confidence in our mathematical abilities. The ability to systematically solve systems of equations is a valuable skill, and the final step of finding x is the culmination of that process.

Final Answer

The solution to the system of equations is (x, y) = (4, 7). Therefore, the correct answer is:

D. (4, 7)

This solution can be verified by substituting the values of x and y back into the original equations to ensure they hold true:

-2(4) + 3(7) = -8 + 21 = 13 (Correct)
4 + 7 = 11 (Correct)

The verification step is a crucial component of the problem-solving process. After obtaining a solution, it's essential to check its validity by substituting the values back into the original equations. This step serves as a safeguard against errors that might have occurred during the solution process. If the values satisfy both equations, we can confidently conclude that our solution is correct. However, if the values do not satisfy the equations, it indicates that there is an error somewhere, and we need to revisit our steps to identify and correct it.

The **substitution of values** into the original equations is a straightforward process, but it requires careful attention to detail. We need to ensure that we are substituting the values correctly and performing the arithmetic operations accurately. This step provides a concrete check on our work, as it allows us to see if the equations hold true with the obtained values. The ability to verify solutions is a valuable skill in mathematics, as it promotes accuracy and confidence in our problem-solving abilities.

Moreover, the **verification step** underscores the importance of a systematic approach to problem-solving. It's not enough to simply arrive at an answer; we need to ensure that the answer is correct. This step reinforces the idea that mathematical problem-solving is not just about finding the solution, but also about verifying its validity. By including the verification step in our problem-solving routine, we develop a more rigorous and reliable approach to mathematics, enhancing our overall problem-solving skills and ensuring that our solutions are accurate and dependable.

Conclusion

In conclusion, the elimination method is a powerful and efficient technique for solving systems of equations. By strategically manipulating the equations to eliminate one variable, we can simplify the system and solve for the remaining variable. This method is particularly useful for linear systems, and it provides a systematic approach to finding the solution. The key steps involve identifying the variable to eliminate, multiplying the equations by appropriate constants to create opposing coefficients, adding the modified equations to eliminate the variable, solving for the remaining variable, and substituting the value back into one of the original equations to find the other variable. The ability to apply the elimination method is a valuable skill in mathematics, with applications in various fields.

Throughout this article, we have demonstrated the step-by-step process of solving a system of equations using the elimination method. We started by introducing the concept of systems of equations and the importance of finding solutions that satisfy all equations simultaneously. We then presented the problem statement and outlined the specific system of equations we aimed to solve. The core of the article was dedicated to the detailed steps of the elimination method, including multiplying equations, adding equations, solving for variables, and substituting values. Each step was explained thoroughly, with clear examples and justifications, ensuring a comprehensive understanding of the method.

Furthermore, we emphasized the importance of verification as the final step in the problem-solving process. By substituting the obtained values back into the original equations, we can confirm the accuracy of our solution and gain confidence in our work. This systematic approach, from problem statement to verification, is crucial for developing effective problem-solving skills in mathematics. The elimination method, with its clear and logical steps, serves as a prime example of how algebraic techniques can be used to solve complex problems. By mastering this method, students and practitioners alike can enhance their ability to tackle a wide range of mathematical challenges, making it an indispensable tool in the field of mathematics and beyond.