In maths, a system of the linear system is a set of two or more linear equation involving the same set of variables. For example : 2x – y = 1, 3x + 2y = 12 . It is a system of two equation in the two variables that is x and y which is called a two linear equation in two unknown x and y and solution to a linear equation is the value to the variables such that all the equations are fulfilled.

In the matrix, every equation in the system becomes a row and each variable in the system becomes a column and the variables are dropped and the coefficients are placed into a matrix.

A system of two linear equations in two unknown x and y are as follows:

Let , , .

Then system of equation can be written in matrix form as:

= i.e. AX = B and X = .

If the R.H.S., namely B is 0 then the system is homogeneous, otherwise non-homogeneous.

is a homogeneous system of two eqations in two unknowns x and y.

is a non-homogenoeus system of equations.

A system of three linear equations in three unknown x, y, z are as follows:

.

Let , , .

Then system of equation can be written in matrix form as:

= i.e. AX = B and X = .

## Algorithm to solve the Linear Equation via Matrix

- Write the given system in the form of matrix equation as AX = B.
- Find the determinant of the matrix. If determinant |A| = 0, then does not exist so that solution does not exist. Write “System is not consistent”.
- If the determinant exist then find the inverse of the matrix i.e. .
- Find where is the inverse of the matrix.
- Solve the equation by the matrix method of linear equation with the formula and find the values of x,y,z.

**Example 1: **Solve the equation: 4x+7y-9 = 0 , 5x-8y+15 = 0

**Solution: **Given equation can be written in matrix form as : , ,

Given system can be written as : AX = B , where .

Let us find determinant : |A| = 4*(-8) – 5*7 = -32-35 = -67 So, solution exist.

Minor and Cofactor of matrix A are : = -8 = -8, = 5 = -5, = 7 = -7, = 4 = 4.

Cofactor matrix = and Adj A =

.

= = =

x = and y =

**Example 2: **Solve the equation: 2x+y+3z = 1, x+z = 2, 2x+y+z = 3

**Solution: **Given equation can be written in matrix form as : , , .

Given system can be written as : AX = B , where .

Let us find determinant : |A| = 2(0-1) – 1(1-2) + 3(1-0) = -2+1+3 = 2. So, solution exist.

Minor and Cofactor of matrix A are : = -1 = -1, = -1 = 1, = 1 = 1, = -2 = 2, = -4 = -4, = 0 = 0 = 1 = -1, = -1 = -1, = -1 = 1.

and

.

= = = = .

x = 3, y = -2, z = -1.

## Exercise

Solve the following equations:

- 2x+3y=9, -x+y=-2.
- x+3y=-2, 3x+5y=4.
- x+y=1, 3y+3z=5, 3z+3x=4.
- x+y+z=1, 2x+y+2z=3, 3x+3y+4z=4.
- x+y+z=6, 3x-y+3z=10, 5x+5y-4z=3.

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