Mathematical Formulas for Industrial and Mechanical Engineering

3.11 Methods of Solution

I. Row-equivalent matrix method: According to this method, to solve a system of linear

equations in two variables, say,

a

1

x 1 b

1

y 5 c

1

a

2

x 1 b

2

y 5 c

2



, we form the following matrix, called

augmented matrix:

a

1

b

1

: c

1

a

2

b

2

: c

2



.

Then, we use row operations to change this matrix into row-equivalent matrices. Some

of the elementary row operations are as follows:

i. Interchange of any two rows, e.g., R

1

2R

2

.

ii. Multiplication of each element of a row by a non-zero number, e.g., R

1

! 3R

1

.

iii. Multiple of a row added to (or subtracted from) any other row, e.g.,

R

2

! R

2

1 2R

1

; R

2

! R

2

2 3R

1

.

In this way, row operations are performed until