If the system of equations 2x+3y=7 …

The given system of equations may be written as
{tex}2x + 3y - 7 = 0{/tex}
{tex}2ax + ay + by - 28 = 0{/tex} {tex}\Rightarrow{/tex}{tex} 2ax + (a + b)y - 28 = 0{/tex}
This system of equations is of the form
{tex}a_1x + b_1y + c_1 = 0{/tex}
{tex}a_2x + b_2y + c_2 = 0{/tex}
where, a₁ = 2, b₁ = 3, c₁ = -7
And, a₂ = 2a, b₂ = a + b, c₂ = -28
For the system of equations to have infinite solutions,
{tex}\frac{{{a_1}}}{{{a_2}}} = \frac{{{b_1}}}{{{b_2}}} = \frac{{{c_1}}}{{{c_2}}}{/tex}
{tex} \Rightarrow \frac{2}{{2a}} = \frac{3}{{a + b}} = \frac{{ - 7}}{{ - 28}}{/tex}
{tex}\Rightarrow \frac{1}{a} = \frac{3}{{a + b}} = \frac{1}{4}{/tex}
Now, {tex}\frac{1}{a} = \frac{1}{4}{/tex}
{tex}\Rightarrow{/tex} a = 4
And, {tex}\frac{1}{a} = \frac{3}{{a + b}}{/tex}
{tex}\Rightarrow{/tex}{tex} a + b = 3a{/tex}
{tex}\Rightarrow{/tex} {tex}2a = b{/tex}
{tex}\Rightarrow{/tex} 2 {tex}\times{/tex} 4 = b
{tex}\Rightarrow{/tex}{tex} b = 8{/tex}
Hence, the given system of equations will have infinite number of solutions for {tex}a = 4\ and\ b = 8.{/tex}