Solving Systems Of Equations By Elimination A Step By Step Guide

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In the realm of mathematics, solving systems of equations is a fundamental skill. When faced with a system of equations, various methods can be employed to find the solution, one of which is the elimination method. This method proves particularly useful when dealing with linear equations. In this comprehensive guide, we will delve into the intricacies of solving a system of equations using the elimination method, while also addressing the scenario where no solution exists. We will use the example you provided, which is:

4x - 13y = 13
2x - 6.5y = -17

to illustrate the process. By the end of this guide, you will have a firm grasp of how to effectively apply the elimination method and interpret the results.

Understanding the Elimination Method

The elimination method hinges on the principle of manipulating equations to eliminate one variable, thereby simplifying the system and allowing us to solve for the remaining variable. This manipulation typically involves multiplying one or both equations by a constant so that the coefficients of one variable are opposites. When the equations are added together, this variable is eliminated, leaving us with a single equation in one variable. Let's break down the steps involved in the elimination method:

  1. Align the Equations: Ensure that the like terms (terms with the same variable) are aligned vertically.
  2. Multiply to Create Opposing Coefficients: Identify a variable you want to eliminate. Multiply one or both equations by a constant so that the coefficients of the chosen variable are opposites (e.g., 2 and -2).
  3. Add the Equations: Add the equations together. The variable with opposing coefficients should be eliminated.
  4. Solve for the Remaining Variable: Solve the resulting equation for the remaining variable.
  5. Substitute and Solve: Substitute the value obtained in step 4 back into one of the original equations and solve for the other variable.
  6. Check the Solution: Verify your solution by substituting the values of both variables into both original equations. If both equations hold true, your solution is correct.

Applying Elimination to the Given System

Let's apply the elimination method to the system of equations:

4x - 13y = 13   (Equation 1)
2x - 6.5y = -17  (Equation 2)

Our goal is to eliminate one of the variables, either x or y. Observing the equations, we notice that the coefficient of x in Equation 2 (2) is half the coefficient of x in Equation 1 (4). This suggests that we can easily eliminate x. To do this, we can multiply Equation 2 by -2:

-2 * (2x - 6.5y) = -2 * (-17)
-4x + 13y = 34   (Modified Equation 2)

Now we have:

4x - 13y = 13   (Equation 1)
-4x + 13y = 34  (Modified Equation 2)

Adding Equation 1 and Modified Equation 2, we get:

(4x - 13y) + (-4x + 13y) = 13 + 34
0 = 47

Interpreting the Result: No Solution

The result, 0 = 47, is a contradiction. This indicates that the system of equations has no solution. Geometrically, this means that the two lines represented by the equations are parallel and never intersect. When the elimination method leads to a contradiction, such as 0 = 47, it signifies that the system is inconsistent and no values of x and y can simultaneously satisfy both equations. Understanding how to solve systems of equations is critical in mathematics, and the elimination method is a powerful tool in this regard. However, it’s also important to understand the different outcomes possible and what they mean.

Why No Solution? A Deeper Dive

To further understand why this system has no solution, let's analyze the equations more closely. We can rewrite the equations in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. This form provides valuable insights into the relationship between the lines represented by the equations.

First, let’s rewrite Equation 1:

4x - 13y = 13
-13y = -4x + 13
y = (4/13)x - 1

Now, let’s rewrite Equation 2:

2x - 6.5y = -17
-6.5y = -2x - 17
y = (2/6.5)x + (17/6.5)
y = (4/13)x + (34/13)

Notice that both equations have the same slope (4/13) but different y-intercepts (-1 and 34/13). This confirms that the lines are parallel and will never intersect. Therefore, there is no solution to the system of equations.

In summary, when applying the elimination method or any other method to solve a system of equations, a contradictory result (such as 0 = 47) indicates that the system is inconsistent and has no solution. This often corresponds to a situation where the equations represent parallel lines.

Other Scenarios in Solving Systems of Equations

While our example resulted in no solution, it's important to be aware of other possible outcomes when solving systems of equations:

  1. Unique Solution: This is the most common scenario, where the system has exactly one solution. The lines represented by the equations intersect at a single point. The elimination method will lead to unique values for both variables.
  2. Infinitely Many Solutions: This occurs when the equations are dependent, meaning one equation is a multiple of the other. The lines represented by the equations are the same line, and any point on the line is a solution. In this case, the elimination method will result in an identity, such as 0 = 0.

Understanding these different scenarios is crucial for effectively solving systems of equations and interpreting the results.

Best Practices for Using the Elimination Method

To ensure accuracy and efficiency when using the elimination method, consider the following best practices:

  • Careful Arithmetic: Pay close attention to arithmetic operations, especially when multiplying equations and adding them together. A single error can lead to an incorrect solution.
  • Choose the Easiest Variable to Eliminate: Look for variables with coefficients that are easy to make opposites. This can simplify the process and reduce the risk of errors.
  • Double-Check Your Solution: After finding a solution, substitute the values back into both original equations to verify that they hold true. This is an essential step to catch any mistakes.
  • Be Aware of Special Cases: Recognize when the system has no solution (contradiction) or infinitely many solutions (identity). These cases require special attention and interpretation.

Conclusion

In this guide, we have explored the elimination method for solving systems of equations. We have demonstrated how to apply the method, interpret the results, and handle the case where no solution exists. By understanding the principles behind the elimination method and practicing consistently, you can confidently solve a wide range of systems of equations. Remember to always check your solutions and be mindful of special cases. Mastering the elimination method is a valuable asset in your mathematical toolkit.

Through this example, we've learned that the elimination method is not just about finding solutions; it's also about understanding the nature of the equations themselves. The absence of a solution tells us something important about the relationship between the lines, highlighting the power of mathematical tools to reveal underlying structures and connections. The elimination method helps solve systems of equations as well as helps us understand the nature of the equation itself. The ability to solve systems of equations is a cornerstone of numerous fields, including engineering, economics, computer science, and physics. As you continue your mathematical journey, the skills you've gained here will undoubtedly prove invaluable. The elimination method is an efficient way to address many such problems. By mastering the elimination method, you're not just learning a technique; you're developing a way of thinking that will serve you well in countless situations.