Solving Systems Of Equations By Elimination Method Consistent Vs Inconsistent Systems

Solving systems of equations is a fundamental concept in algebra, and the elimination method is a powerful technique for finding solutions. In this comprehensive guide, we will delve into the elimination method, providing a step-by-step explanation with examples and exploring the concepts of consistent and inconsistent systems. If you're grappling with systems of equations, particularly solving by the elimination method, or need to understand the nature of solutions (or the lack thereof) in consistent or inconsistent systems, this article is your go-to resource. We'll break down the process, ensuring you grasp the nuances and can confidently apply the elimination method to various problems.

Understanding the Elimination Method

The elimination method, also known as the addition method, is a technique used to solve systems of linear equations by eliminating one of the variables. The primary goal is to manipulate the equations in such a way that when they are added together, one variable cancels out, leaving a single equation with one variable. This resulting equation can then be easily solved, and the value of the eliminated variable can be found by substitution. This method is particularly effective when the coefficients of one of the variables are multiples of each other or have opposite signs.

Step-by-Step Guide to the Elimination Method

1. Align the Equations: Ensure that the equations are written in the standard form, which is Ax + By = C, where A, B, and C are constants, and x and y are variables. This alignment ensures that like terms are vertically aligned, making the elimination process smoother.

2. Multiply Equations (if needed): Look for the variable you want to eliminate. Multiply one or both equations by a constant so that the coefficients of the chosen variable are either the same or opposite in sign. The goal here is to create coefficients that will cancel each other out when the equations are added. For instance, if you have a system where the x coefficients are 2 and 3, you might multiply the first equation by 3 and the second by -2 to get coefficients of 6 and -6, respectively. This prepares the equations for the next step.

3. Add the Equations: Add the equations together. This step should eliminate one of the variables, resulting in a single equation with one variable. If done correctly, the coefficients of the chosen variable will cancel each other out, leaving an equation that you can solve for the remaining variable. For example, if you have the equations 6x + 3y = 12 and -6x - 2y = -10, adding them together will eliminate x, resulting in y = 2.

4. Solve for the Remaining Variable: Solve the resulting equation for the remaining variable. This is typically a straightforward algebraic step. Once you've isolated the variable, you'll have its value, which is a crucial part of the solution to the system.

5. Substitute Back: Substitute the value you found in the previous step back into one of the original equations to solve for the other variable. Choose the equation that seems easier to work with. This substitution will give you the value of the variable that was initially eliminated, completing the solution.

6. Check the Solution: Substitute both values into both original equations to check your solution. This is a vital step to ensure that your solution is correct and that no errors were made during the process. If the solution satisfies both equations, you've successfully solved the system.

Example Application of the Elimination Method

Consider the following system of equations:

$egin{cases} 2x + 3y = 7 \ 4x - y = 2

\end{cases}$

1. Align Equations: The equations are already aligned in standard form.

2. Multiply Equations: To eliminate y, multiply the second equation by 3:

   $egin{cases} 2x + 3y = 7 \ 3(4x - y) = 3(2)

   \end{cases}$

   Which simplifies to:

   $egin{cases} 2x + 3y = 7 \ 12x - 3y = 6

   \end{cases}$

3. Add the Equations: Add the two equations together:

   $(2x + 3y) + (12x - 3y) = 7 + 6$

   $14x = 13$

4. Solve for x: Divide both sides by 14:

   x = rac{13}{14}

5. Substitute Back: Substitute the value of x into the first original equation:

   2(rac{13}{14}) + 3y = 7

   rac{13}{7} + 3y = 7

   3y = 7 - rac{13}{7}

   3y = rac{49 - 13}{7}

   3y = rac{36}{7}

   y = rac{12}{7}

6. Check the Solution: Substitute x = 13/14 and y = 12/7 into both original equations to verify the solution.

Consistent and Inconsistent Systems

When solving systems of equations, it's crucial to understand the concepts of consistent and inconsistent systems. These terms describe the nature of the solutions that a system of equations can have.

Consistent Systems

A system of equations is considered consistent if it has at least one solution. This means there is at least one set of values for the variables that satisfies all equations in the system. Consistent systems can be further classified into two types:

• Independent Systems: These systems have exactly one solution. The equations represent distinct lines that intersect at a single point. This intersection point is the unique solution to the system. Graphically, independent systems are represented by two lines that cross each other at one point.

• Dependent Systems: These systems have infinitely many solutions. The equations represent the same line or lines that overlap entirely. Any point on the line is a solution to the system. Graphically, dependent systems are represented by two lines that coincide with each other.

Inconsistent Systems

A system of equations is considered inconsistent if it has no solution. This means there is no set of values for the variables that can satisfy all equations in the system simultaneously. Inconsistent systems typically arise when the equations represent parallel lines that never intersect. Graphically, inconsistent systems are represented by two parallel lines that never meet.

Determining Consistency and Consistency Using Elimination Method

The elimination method can help determine whether a system is consistent or inconsistent. Here’s how:

• Unique Solution (Consistent and Independent): If the elimination method leads to a unique solution for the variables, the system is consistent and independent. This means there is one point of intersection between the lines represented by the equations.

• Infinite Solutions (Consistent and Dependent): If the elimination method results in an identity (e.g., 0 = 0), the system is consistent and dependent. This indicates that the equations represent the same line, and there are infinitely many solutions.

• No Solution (Inconsistent): If the elimination method leads to a contradiction (e.g., 0 = 5), the system is inconsistent. This implies that the lines represented by the equations are parallel and do not intersect.

Applying the Concepts to the Given System

Now, let's apply the elimination method to the given system of equations:

$egin{cases} 3x - 6y = 12 \ 2x - 4y = 8

\end{cases}$

1. Align Equations: The equations are already aligned in standard form.

2. Multiply Equations: To eliminate x, multiply the first equation by 2 and the second equation by -3:

   $egin{cases} 2(3x - 6y) = 2(12) \ -3(2x - 4y) = -3(8)

   \end{cases}$

   Which simplifies to:

   $egin{cases} 6x - 12y = 24 \ -6x + 12y = -24

   \end{cases}$

3. Add the Equations: Add the two equations together:

   $(6x - 12y) + (-6x + 12y) = 24 + (-24)$

   $0 = 0$

4. Interpret the Result: The result is an identity (0 = 0), which means the system is consistent and dependent. There are infinitely many solutions.

5. Express the Solution: Since the system is dependent, the equations represent the same line. To express the solution, we can solve one of the equations for y in terms of x (or vice versa). Let's solve the first equation for y:

   $3x - 6y = 12$

   $-6y = -3x + 12$

   y = rac{1}{2}x - 2

   Thus, the solution can be expressed as all points on the line y = (1/2)x - 2.

Conclusion

The elimination method is a robust technique for solving systems of equations, particularly when the goal is to eliminate one variable and simplify the problem. Understanding the concepts of consistent and inconsistent systems is crucial for interpreting the nature of solutions. Inconsistent systems have no solutions, consistent systems can have one solution (independent) or infinite solutions (dependent). By mastering these concepts and techniques, you'll be well-equipped to tackle a wide range of algebraic problems involving systems of equations. If the system is consistent or inconsistent can be determined by the results of the elimination method, where a unique solution indicates consistency, an identity indicates dependency, and a contradiction indicates inconsistency.