If the lines 3y + 4x = 1, y = x + 5 and 5y +bx = 3 are concurrent, then what is the value of b?

Concept:

If a₁x + b₁y + c₁ = 0, a₂x + b₂y + c₂ = 0 and a₃x + b₃y + c₃ = 0 are 3 lines then these lines are said to be concurrent if:

Calculation:

Given: the lines 3y + 4x = 1, y = x + 5 and 5y + bx = 3 are concurrent.

Now, by comparing the three lines with the standard equation of line ax + by + c = 0 we get:

⇒ a₁ = 4, b₁ = 3, c₁ = -1, a₂ = 1, b₂ = -1, c₂ = 5, a₃ = b, b₃ = 5 and c₃ = -3.

As we know that, if a₁x + b₁y + c₁ = 0, a₂x + b₂y + c₂ = 0 and a₃x + b₃y + c₃ = 0 are 3 lines then these lines are said to be concurrent if:

⇒ b = 6.