Solving 2x + 3y = 11 And X + Y = 8 Using Elimination Method A Step-by-Step Guide

Introduction to Solving Systems of Equations

In the realm of mathematics, particularly in algebra, solving systems of equations is a fundamental skill. 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 that, when substituted for the variables, make all the equations true simultaneously. There are several methods to solve systems of equations, including graphing, substitution, and elimination. This comprehensive guide will delve into the elimination method, a powerful technique for solving linear systems. We will use the specific system of equations:

1. 2x + 3y = 11
2. x + y = 8

as an example to illustrate the step-by-step process. Understanding the elimination method is crucial for various applications in mathematics, science, and engineering, where problems often involve multiple variables and constraints. Mastering this method allows for a more efficient and accurate approach to problem-solving. The elimination method is particularly useful when dealing with systems where the coefficients of one variable are easily made opposites, allowing for the elimination of that variable through addition or subtraction. This method not only simplifies the process of finding solutions but also enhances algebraic manipulation skills, which are essential for more advanced mathematical concepts. This guide aims to provide a clear, concise, and thorough understanding of the elimination method, ensuring that readers can confidently apply it to solve a variety of systems of equations.

Understanding the Elimination Method

The elimination method, also known as the addition method, is an algebraic technique used to solve systems of linear equations. The core principle behind this method is to manipulate the equations in the system so that, when added together, one of the variables is eliminated. This leaves us with a single equation in one variable, which can then be easily solved. Once the value of one variable is found, it can be substituted back into one of the original equations to find the value of the other variable. This process efficiently reduces a two-variable problem to a single-variable problem, simplifying the solution process. The elimination method is particularly effective when the coefficients of one variable in the two equations are either the same or easily made opposites by multiplying one or both equations by a constant. The method involves several key steps, including identifying the variable to eliminate, manipulating the equations to create opposite coefficients for that variable, adding the equations to eliminate the variable, solving for the remaining variable, and substituting the solved value back into one of the original equations to find the value of the other variable. Understanding these steps and the underlying logic is crucial for successfully applying the elimination method to solve a wide range of systems of equations. The elimination method is a versatile and widely used technique in algebra, offering a systematic approach to solving systems of equations and providing a foundation for more advanced mathematical concepts and applications.

Step-by-Step Solution: 2x + 3y = 11 and x + y = 8

To effectively demonstrate the elimination method, let's apply it to our system of equations:

1. 2x + 3y = 11
2. x + y = 8

Our first goal is to choose a variable to eliminate. Looking at the coefficients, it seems easier to eliminate x. To do this, we need to make the coefficients of x in both equations opposites. The coefficient of x in the first equation is 2, and in the second equation, it is 1. We can multiply the second equation by -2 to make the coefficients of x opposites. This step is crucial as it sets the stage for the elimination of one variable, simplifying the system and allowing us to solve for the remaining variable. Multiplying the equation by a constant ensures that we maintain the equality, a fundamental principle in algebraic manipulations. The choice of which variable to eliminate often depends on the specific coefficients in the system; in this case, multiplying the second equation is a straightforward way to create opposite coefficients for x. This strategic decision streamlines the solution process and makes the elimination method efficient. Once we have opposite coefficients, we can proceed to add the equations, which will eliminate one variable and allow us to solve for the other. This step-by-step approach is characteristic of the elimination method, providing a clear and organized path to the solution.

Step 1: Manipulate the Equations

Multiply the second equation (x + y = 8) by -2. This gives us a new equation:

-2(x + y) = -2(8) -2x - 2y = -16

Now we have the following system:

1. 2x + 3y = 11
2. -2x - 2y = -16

This manipulation is a critical step in the elimination method. By multiplying the second equation by -2, we have successfully created opposite coefficients for the variable x. This sets the stage for the next step, where we will add the two equations together, and the x terms will cancel out, leaving us with an equation in terms of y only. The choice of -2 as the multiplier was strategic, as it directly creates the opposite coefficient needed for elimination. This demonstrates the core idea behind the elimination method: to strategically manipulate equations so that adding them eliminates one variable, simplifying the problem. Ensuring accuracy during this step is crucial, as any error in multiplication will propagate through the rest of the solution. The result of this manipulation is a new system of equations that is equivalent to the original but is now set up for the easy elimination of one variable, making the problem significantly simpler to solve. This step highlights the power and elegance of the elimination method in solving systems of equations.

Step 2: Eliminate a Variable

Add the two equations together:

(2x + 3y) + (-2x - 2y) = 11 + (-16) 2x + 3y - 2x - 2y = -5 y = -5

By adding the modified equations, the x terms cancel each other out (2x - 2x = 0), as planned. This is the heart of the elimination method: strategically manipulating equations so that one variable disappears upon addition. This step leaves us with a single equation in one variable (y), which can be easily solved. The simplicity of this step highlights the efficiency of the elimination method, transforming a system of two equations into a single equation that can be solved directly. The accuracy of this step depends on the correct manipulation in the previous step and the careful addition of the equations. The result, y = -5, is a significant milestone in solving the system, as we have now found the value of one of the variables. This value will be crucial in the next step, where we substitute it back into one of the original equations to find the value of the other variable. This process of elimination and substitution is a hallmark of the elimination method, providing a systematic and effective way to solve systems of equations.

Step 3: Solve for the Remaining Variable

Now that we have found y = -5, substitute this value into one of the original equations. Let's use the second equation (x + y = 8):

x + (-5) = 8 x - 5 = 8 x = 8 + 5 x = 13

This step demonstrates the back-substitution process, a crucial part of the elimination method. Once we have solved for one variable, we substitute its value back into one of the original equations to find the value of the other variable. The choice of which equation to use for substitution is arbitrary; either equation will yield the correct answer. In this case, we chose the second equation (x + y = 8) because it appeared simpler, but the first equation (2x + 3y = 11) would have worked equally well. The key is to carefully substitute the known value and solve the resulting equation for the remaining variable. This step completes the solution process, giving us the value of the second variable, x = 13. With both x and y values determined, we have successfully solved the system of equations using the elimination method. The accuracy of this step depends on the correctness of the previously found value and the correct substitution and algebraic manipulation. The result provides a complete solution to the system, demonstrating the effectiveness of the elimination method.

Step 4: Verify the Solution

To ensure our solution is correct, substitute the values x = 13 and y = -5 back into both original equations:

Equation 1: 2x + 3y = 11 2(13) + 3(-5) = 26 - 15 = 11 (Correct)

Equation 2: x + y = 8 13 + (-5) = 13 - 5 = 8 (Correct)

Since the values satisfy both equations, our solution is correct.

This verification step is an essential part of solving any system of equations, including those solved using the elimination method. It provides a crucial check to ensure that the values we found for x and y are indeed the correct solution. By substituting the values back into the original equations, we can confirm that they satisfy both equations simultaneously. This step helps to catch any errors that may have occurred during the elimination, substitution, or algebraic manipulation processes. The verification process reinforces the understanding of what it means to solve a system of equations: finding values that make all equations in the system true. Successfully verifying the solution gives confidence in the accuracy of the answer and demonstrates a thorough understanding of the problem-solving process. This step is not just a formality; it is a critical step in ensuring the correctness of the solution and developing sound mathematical practices.

Conclusion: Mastering the Elimination Method

In conclusion, the elimination method is a powerful and versatile technique for solving systems of linear equations. By strategically manipulating equations to eliminate one variable, we can simplify the problem and find the solution more efficiently. In our example, we successfully solved the system:

1. 2x + 3y = 11
2. x + y = 8

and found the solution x = 13 and y = -5. The key steps involved manipulating one or both equations to create opposite coefficients for one variable, adding the equations to eliminate that variable, solving for the remaining variable, and substituting the solved value back into one of the original equations to find the value of the eliminated variable. The verification step is crucial to ensure the accuracy of the solution. Mastering the elimination method not only enhances problem-solving skills in algebra but also provides a foundation for more advanced mathematical concepts. The ability to solve systems of equations is essential in various fields, including science, engineering, economics, and computer science. Therefore, understanding and practicing the elimination method is a valuable investment in one's mathematical toolkit. The systematic approach of the elimination method, with its clear steps and logical progression, makes it a reliable and efficient tool for tackling a wide range of systems of equations, reinforcing its importance in mathematics education and practical applications.