Elementary Row Operations | Simplifying Linear Systems of Equations for Easy Solving

Elementary row ops permit one equation in a linear system to be subtracted from another.

In linear algebra, a linear system of equations can be represented as a set of equations with variables

In linear algebra, a linear system of equations can be represented as a set of equations with variables. The goal is to find values for these variables that satisfy all the equations simultaneously.

Elementary row operations, also known as Gaussian elimination or row operations, are steps that can be performed on the equations of a linear system without changing the solutions. These operations include:

1. Interchanging two equations: This operation involves swapping the positions of two equations in the linear system.

2. Multiplying an equation by a constant: In this operation, all terms of an equation are multiplied by a constant.

3. Adding a multiple of one equation to another: This operation allows adding or subtracting a multiple of one equation to or from another equation.

By performing these elementary row operations, we can transform the system of equations into an equivalent system with simpler equations that are easier to solve. Subtracting one equation from another is one such operation.

When we subtract one equation from another, we subtract corresponding terms of both equations. This operation is useful for eliminating a variable or simplifying the system of equations. For example, consider the following linear system:

Equation 1: 3x + 2y = 10
Equation 2: 2x – 3y = 4

To solve this system using elementary row operations, we can multiply Equation 1 by 2 and then subtract Equation 2 from the resulting equation. This will eliminate the variable ‘x’ and simplify the system.

By performing the subtraction, we get:
(2) * Equation 1: 6x + 4y = 20
Equation 2: 2x – 3y = 4

Subtracted Equation: 4x + 7y = 16

Now, we have a new system of equations with only two variables. We can continue applying elementary row operations until we obtain a row echelon form or reduced row echelon form, which allows us to easily solve the system and find the values of the variables.

So, in summary, elementary row operations, including subtracting one equation from another, are mathematical techniques used to transform a linear system of equations into an equivalent form that is easier to solve and find the solution to the system.

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