Find the conditions that the following system is consistent

Find the condition(s) that the following system is consistent x_1 + 4x_2 - 2x_3 = b_1 -3x_1 - 12x_2 + 6x_3 = b_2 5x_1 + 20x_2 - 10x_3 = b_3

Solution

The given equations are as under:

x₁ + 4x₂ - 2x₃ = b₁ ......(1)

-3x₁ - 12x₂ + 6x₃ = b₂ On dividing both the sides by -3, we have x₁ + 4x₂ - 2x₃ = ( - b₂ ) / 3 ...(2)

5x₁ + 20x₂ -10x₃ = b₃  On dividing both the sides by 5, we have x₁ + 4x₂ - 2x₃ = (b₃) / 5 ....(3)

The left hand sides of all the 3 equations are identical, therefore, if the given equations are consistent, then the right hand sides of the 3 equations ashould be equal. Thus, the condition for the given equations to be consistent is

b₁ = ( - b₂ ) / 3 = (b₃) / 5 or, 15b₁ = - 5b₂ = 3b_{3 .}