Determine a & b for which …

2x - (a - 4)y = 2b + 1 ........ (i)
4x - (a - 1)y = 5b - 1 ....... (ii)
Compare the equations with form of equations
a₁x + b₁y = c₁
a₂x + b₂y = c₂
We get, a₁ = 2, b₁ = - (a - 4) and c₁ =  (2b + 1)
a₂ = 4, b₂ = - (a - 1) and c₂ =   (5b - 1)
Equations has infinite number of solutions, if
{tex} \frac { a _ { 1 } } { a _ { 2 } } = \frac { b _ { 1 } } { b _ { 2 } } = \frac { c _ { 1 } } { c _ { 2 } }{/tex}
Therefore, the given system of equations will have infinite number of solutions, if
{tex}\frac { 2 } { 4 } = \frac { - ( a - 4 ) } { - ( a - 1 ) } = \frac { 2 b + 1 } { 5 b - 1 }{/tex}
{tex}\Rightarrow \quad \frac { 1 } { 2 } = \frac { a - 4 } { a - 1 } = \frac { 2 b + 1 } { 5 b - 1 }{/tex}
{tex}\Rightarrow \quad \frac { 1 } { 2 } = \frac { a - 4 } { a - 1 } \text { and } \frac { 1 } { 2 } = \frac { 2 b + 1 } { 5 b - 1 }{/tex}
{tex}\Rightarrow{/tex} a - 1= 2a - 8 and 5b -1 = 4b + 2
{tex}\Rightarrow{/tex} a = 7 and b = 3