Elimination Method

Definition

The elimination method, also known as the method of elimination, is a technique used to solve systems of linear equations by systematically eliminating variables to find the unique solution. This method is applicable in the context of various topics, including parametric equations, systems of linear equations in two and three variables, and systems of nonlinear equations and inequalities.

5 Must Know Facts For Your Next Test

1. The elimination method involves adding or subtracting multiples of one equation from another to eliminate a variable, leading to a simpler system of equations that can be solved.
2. In the context of parametric equations, the elimination method can be used to eliminate the parameter and express the equations in terms of the Cartesian coordinates x and y.
3. For systems of linear equations in two variables, the elimination method involves manipulating the equations to obtain a single equation in one variable, which can then be solved.
4. In systems of linear equations with three variables, the elimination method is often used in conjunction with the substitution method to systematically eliminate variables and find the unique solution.
5. The elimination method can also be applied to systems of nonlinear equations and inequalities, where it is used to transform the system into a simpler form that can be more easily solved.

Review Questions

• Explain how the elimination method can be used to solve a system of linear equations in two variables.
  • To solve a system of linear equations in two variables using the elimination method, the goal is to eliminate one of the variables by adding or subtracting multiples of the equations. This is typically done by multiplying one or both equations by a constant to make the coefficients of one variable the same in both equations, and then adding or subtracting the equations to eliminate that variable. Once one variable is eliminated, the remaining equation can be solved for the other variable, and the solution can be substituted back into one of the original equations to find the value of the first variable.
• Describe how the elimination method can be applied to solve a system of linear equations in three variables.
  • Solving a system of linear equations in three variables using the elimination method involves a more complex process. First, two of the variables are eliminated by manipulating the equations, leaving a single equation in one variable. This is typically done by multiplying one or more equations by constants and then adding or subtracting the equations to eliminate two of the variables. Once a single equation in one variable is obtained, it can be solved, and the solution can then be substituted back into one of the original equations to find the values of the other two variables. This process may need to be repeated multiple times to eliminate all but one variable and find the unique solution to the system.
• Analyze how the elimination method can be used to solve a system of nonlinear equations and inequalities in two variables.
  • When dealing with a system of nonlinear equations and inequalities in two variables, the elimination method can still be applied, but the process is more complex. The goal is to transform the system into a simpler form that can be more easily solved. This may involve manipulating the equations to eliminate one of the variables, similar to the linear case. However, the nonlinear nature of the equations means that additional algebraic techniques, such as factoring or completing the square, may be necessary to simplify the equations further. Once the system has been transformed, the remaining equations and inequalities can be solved using a combination of algebraic and graphical methods. The elimination method, in this context, serves as a tool to reduce the complexity of the system and facilitate the overall solution process.

Related terms

Substitution Method: The substitution method is another technique for solving systems of linear equations, where one variable is expressed in terms of the other and then substituted into the other equation to solve for the remaining variable.

Augmented Matrix: An augmented matrix is a matrix that combines the coefficients of the variables and the constants on the right-hand side of a system of linear equations, allowing for efficient manipulation and solution using various methods, including the elimination method.

Gaussian Elimination: Gaussian elimination is a systematic process of transforming a system of linear equations into an equivalent system with an upper triangular coefficient matrix, which can then be solved using the elimination method.