 # Solving Systems of Linear Equations

## In Mathematics, simultaneous equations are a set of equations containing multiple variables. This set of equations is often referred to as a system of equations. A linear system is a set of linear equations and a homogeneous linear system is a set of homogeneous linear equations. Matrices are helpful in rewriting a linear system in a very simple form. The algebraic properties of matrices may then be used to solve systems. A very well-known technique used to solve simultaneous equations is the Gaussian Elimination Method. The Gauss Elimination method eliminates the variables and finally the set of equations is reduced into a lower triangular form. Linear equations are functions that have two variables. For solving simultaneous equations using the ‘elimination’ method, two equations are simplified by adding them or subtracting them. This eliminates one of the variables, so that the other variable can be found. Gauss elimination methods provides an algorithm for solving systems of linear equations. It can also be used to find the rank of a matrix. Elementary row operations are used to reduce a matrix to what is called a triangular form. The Gauss elimination method consists of Forward Elimination of Unknowns method and Back Substitution method to solve the equation. The goal of Forward Elimination is to transform the coefficient matrix into an upper triangular matrix.

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• Q1

What is the first step to start solving the matrix of augmented matrix?

Marks:1

The first step to solve is to divide each element in the first row a1j by the element first element of the row a11.

a1j=a1j/a11

• Q2

What does each row in an augmented matrix represent?

Marks:1

Each row of the matrix represents one equation.

Example: ax+by+cz=1, where a, b, c and 1 are values of one row.

• Q3

Write the augmented matrix of the given linear equations.
x+yz=1 (Eq. 1)
8x+3y−5z=1 (Eq. 2)
−4xy+3z=1 (Eq. 3)

Marks:1

The augmented matrix of the given equations is: • Q4

Write the general equation for the solution of simultaneous linear equations.

Marks:1

The general equation is:
AX=C