Solve the following equation:4/x +5y=7 and …

The given system of equations is
{tex}\frac{4}{x} + 5y = 7{/tex} ....(1)
{tex}\frac{3}{x} + 4y = 5{/tex} .....(2)
Put {tex}\frac{1}{x} = X{/tex} ....(3)
Then equations (1) and (2) can be rewritten as
4X + 5y = 7 ....(4)
3X + 4y = 5 .....(5)
{tex}\Rightarrow{/tex} 4X + 5y - 7 = 0 ....(6)
3X + 4y - 5 = 0 .....(7)

Then,
{tex}\frac{X}{{(5)( - 5) - ( 4)( - 7)}} = \frac{y}{{( - 7)(3) - ( - 5)(4)}}{/tex} {tex}= \frac{1}{{(4)(4) - (3)(5)}}{/tex}
{tex}\Rightarrow \;\frac{X}{{ - 25 + 28}} = \frac{y}{{ - 21 + 20}} = \frac{1}{{16 - 15}}{/tex}
{tex}\Rightarrow \;\frac{X}{3} = \frac{y}{-1} = \frac{1}{1}{/tex}
{tex}\Rightarrow{/tex} X = 3 and y = -1
{tex} \Rightarrow \;\frac{1}{x} = 3{/tex} and y = -1 ....using (3)
{tex}\Rightarrow \;x = \frac{1}{3}{/tex} and y = -1
Hence, the solution of the given system of equations is
{tex}x = \frac{1}{3}{/tex}, y = -1
Verification : Substituting {tex}x = \frac{1}{3}{/tex}, y = -1,
We find that both the equations (1) and (2) are satisfied as shown below
{tex}\frac{4}{x} + 5y = \frac{4}{{\left( {\frac{1}{3}} \right)}} + 5( - 1) = 12 - 5 = 7{/tex}
{tex}\frac{3}{x} + 4y = \frac{3}{{\left( {\frac{1}{3}} \right)}} + 4( - 1) = 9 - 4 = 5{/tex}
Hence, the solution of the given system of equations is {tex}x = \frac{1}{3}{/tex}, y = -1