elimination method

(noun)

Process of solving a system of equations by eliminating one variable in order to more simply solve for the remaining variable.

Related Terms

Examples of elimination method in the following topics:

  • The Elimination Method

    • The elimination method is used to eliminate a variable in order to more simply solve for the remaining variable(s) in a system of equations.
    • The elimination method for solving systems of equations, also known as elimination by addition, is a way to eliminate one of the variables in the system in order to more simply evaluate the remaining variable.
    • The elimination method can be demonstrated by using a simple example:
    • In this example, the variable y can be eliminated if we multiply the top equation by $-2$ and then add the equations together.

  • Inconsistent and Dependent Systems in Two Variables

    • We can apply the substitution or elimination methods for solving systems of equations to identify dependent systems.
    • We can apply the elimination method to evaluate these.
    • We can also apply methods for solving systems of equations to identify inconsistent systems.
    • We can apply the elimination method to attempt to solve this system.
    • Subtracting the first equation from the second one, both variables are eliminated and we get $0 = 6$.

  • Solving Systems of Equations in Three Variables

    • The graphical method of solving a system of equations in three variables involves plotting the planes that are formed when graphing each equation in the system and then finding the intersection point of all three planes.
    • After that smaller system has been solved, whether by further application of the substitution method or by other methods, substitute the solutions found for the variables back into the first right-hand side expression.
    • Elimination by judicious multiplication is the other commonly-used method to solve simultaneous linear equations.
    • As the equations grow simpler through the elimination of some variables, a variable will eventually appear in fully solvable form, and this value can then be "back-substituted" into previously derived equations by plugging this value in for the variable.
    • Using the elimination method, begin by subtracting the first equation from the second and simplifying:

  • Matrices and Row Operations

    • Also, when solving a system of linear equations by the elimination method, row multiplication would be the same as multiplying the whole equation by a number to obtain additive inverses so that a variable cancels.  
    • Finally, row addition is also the same as the elimination method, when one chooses to add or subtract the like terms of the equations to obtain the variable.  
    • Therefore, row operations preserve the matrix and can be used as an alternative method to solve a system of equations.
    • There are several specific algorithms to row-reduce an augmented matrix, the simplest of which are Gaussian elimination and Gauss-Jordan elimination.  

  • Solving Systems Graphically

    • This is the graphical method.

  • Inconsistent and Dependent Systems in Three Variables

    • Solving the system by elimination results in a single ordered triple $(x, y, z)$.
    • The process of elimination will result in a false statement, such as $3 = 7$, or some other contradiction.
    • Using the elimination method for solving a system of equation in three variables, notice that we can add the first and second equations to cancel $x$:

  • Matrix Equations

    • It is possible to solve this system using the elimination or substitution method, but it is also possible to do it with a matrix operation.

  • Further Simplification of Matrices

    • Gauss–Jordan elimination is an algorithm for getting matrices in reduced row echelon form using elementary row operations.
    • Gauss-Jordan elimination, like Gaussian elimination, is used for inverting matrices and solving systems of linear equations.
    • However, the result of Gauss-Jordan elimination (reduced row echelon form) may be retrieved from the result of Gaussian elimination (row echelon form) in arithmetic operations by proceeding from the last pivot to the first one.
    • Thus the needed number of operations has the same order of magnitude for both eliminations.
    • The steps of Gauss-Jordan elimination are very similar to that of Gaussian elimination, the main difference being that we will work in diagonal form instead of putting the augmented matrix into upper triangle form.

  • Simplifying Matrices With Row Operations

    • Using elementary operations, Gaussian elimination reduces matrices to row echelon form.
    • By means of a finite sequence of elementary row operations, called Gaussian elimination, any matrix can be transformed to a row echelon form.  
    • Using elementary row operations at the end of the first part (Gaussian elimination, zeros only under the leading 1) of the algorithm:
    • At the end of the algorithm, if the Gauss–Jordan elimination (zeros under and above the leading 1) is applied:

  • The Inverse of a Matrix

    • The method for finding an inverse matrix comes directly from the definition, along with a little algebra.
    • Solve the first two equations for $a$ and $c$ and the second two equations for $b$ and $d$ by using either elimination or substitution.