**Step 1 :**

Find the augmented matrix [A, B] of the system of equations.

**Step 2 :**

Find the rank of A and rank of [A, B] by applying only elementary row operations.

Note :

Column operations should not be applied.

**Step 3 :**

**Case 1 :**

If there are n unknowns in the system of equations and

ρ(A) = ρ([A|B]) = n

then the system AX = B, is consistent and has a unique solution.

**Case 2 :**

If there are n unknowns in the system AX = B

ρ(A) = ρ([A| B]) < n

then the system is consistent and has infinitely many solutions and these solutions.

**Case 3 :**

If ρ(A) ≠ ρ([A| B])

then the system AX = B is inconsistent and has no solution.

**Example 1 :**

Solve the following linear equation by rank method

2x + 5y + 7z = 52, x + y + z = 9, 2x + y - z = 0

**Solution :**

Number of non zero rows are 3.

ρ(A) = ρ([A|B]) = 3. The system is consistent and it has unique solution.

From 1^{st} row,

x+y+z = 9 -----(1)

From 2^{nd} row,

3y+5z = 34 -----(2)

From 3^{rd} row,

-4z = -20 -----(3)

From (3)

z = 5

By applying the value of z in (2), we get

3y+5(5) = 34

3y + 25 = 34

3y = 34-25

3y = 9

y = 3

By applying the value of y and z in (1), we get

x+3+5 = 9

x+8 = 9

x = 1

So, the solution is (1, 3, 5).

**Example 2 :**

Solve the following linear equation by rank method

4x - 2y + 5z = 6, 3x + 3y + 8z = 4, x - 5y - 3z = 5

**Solution :**

ρ(A) = 2 and ρ([A|B]) = 3. The system is inconsistent and it has no solution.

**Example 3 :**

Solve the following linear equation by rank method

x+9y-z = 27, x-8y+16z = 10, 2x+y+15z = 37

**Solution :**

Here ρ(A) = ρ([A|B]) = 2 < 3, then the system is consistent and it has infinitely many solution.

From the 1^{st} row,

x + 9y-z = 27 ---(1)

From the 2^{nd} row,

17y + 17z = -17 ---(2)

Dividing by 17, we get

y + z = -1

Put z = t

y = -1 - t

By applying the value of y and z in (1), we get

x + 9(-1-t)-t = 27

x - 9+9t-t = 27

x = 27+9-8t

x = 36-8t

Solution :

x = 36-8t, y = -1-t and z = t where t ∈ Real numbers.

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