find x and y
x/y - x/y+4 = 1/3
x/y-1 - x/y = 13/15

Answer :

We have our equations , As  :

$\frac{x}{y} - \frac{x}{y + 4} = \frac{1}{3}$                     ----- ( A )
And
$\frac{x}{y - 1} - \frac{x}{y} = \frac{13}{15}$                     ----- ( B )

We simplify our equation  " A " , As :

$$\begin{gathered} \Rightarrow \frac{x\left(y + 4\right) - xy}{y\left(y + 4\right)} = \frac{1}{3} \\ \Rightarrow \frac{xy + 4x - xy}{y^2 + 4y} = \frac{1}{3} \\ \Rightarrow \frac{4x}{y^2 + 4y} = \frac{1}{3} \\ \Rightarrow 12x = y^2 + 4y , So, \\ \Rightarrow x = \frac{ y^2 + 4y}{12} \\ \Rightarrow x = \frac{{\mathbf{y}}^{\mathbf{2}}}{\mathbf{12}}\mathbf{ }\mathbf{+}\mathbf{ }\frac{\mathbf{y}}{\mathbf{3}}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{-}\mathbf{ }\mathbf{\hspace{0.17em}}\mathbf{(}\mathbf{\hspace{0.17em}}\mathbf{1}\mathbf{ }\mathbf{)} \end{gathered}$$  

We simplify our equation  " B " , As :

$$\begin{gathered} \Rightarrow \frac{xy - x\left( y - 1\right)}{y\left(y - 1\right)}= \frac{13}{15} \\ \Rightarrow \frac{xy - xy + x}{y^2 - y}= \frac{13}{15} \\ \Rightarrow \frac{ x}{y^2 - y}= \frac{13}{15} \\ \Rightarrow 15 x = 13y^2 - 13y , \mathrm{Substitute} \mathrm{value} \mathrm{from} \mathrm{equation} 1 , \mathrm{We} \mathrm{get} \\ \Rightarrow 15 \left(\frac{y^2}{12} +\frac{y}{4}\right)= 13y^2 - 13y \\ \Rightarrow \frac{15y^2}{12} +\frac{15y}{4}= 13y^2 - 13y \\ \Rightarrow \frac{15y^2 + 45y}{12}= 13y^2 - 13y \\ \Rightarrow 15y^2 + 45y= 156y^2 - 156y \\ \Rightarrow 201y= 141y^2 \\ \Rightarrow 201 = 141y \\ \Rightarrow y = \frac{201}{141} \\ \Rightarrow y = \frac{67}{47} , \mathrm{Substitute} \mathrm{value} \mathrm{from} \mathrm{equation} 1 , \mathrm{We} \mathrm{ge}t \\ x = \frac{{\left(\frac{67}{47}\right)}^2}{12} +\frac{\frac{67}{47}}{4} \\ \Rightarrow x = \frac{{\left(\frac{67}{47}\right)}^2 +\hspace{0.17em}4\left(\frac{67}{47}\right)}{12} \\ \Rightarrow x = \frac{\frac{4489 + 12596}{2209}}{12} \\ \Rightarrow x = \frac{17085}{2209\times 12} \\ \Rightarrow x = \frac{5695}{2209\times 4} \\ \Rightarrow x = \frac{5695}{8836} \\ So, \\ \mathbf{\Rightarrow }\mathit{x}\mathbf{ }\mathbf{=}\mathbf{ }\frac{\mathbf{5695}}{\mathbf{8836}}\mathbf{ }\mathbf{ }\mathit{a}\mathit{n}\mathit{d}\mathbf{ }\mathbf{ }\mathbf{ }\mathit{y}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{ }\frac{\mathbf{67}}{\mathbf{47}}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{(}\mathbf{ }\mathit{A}\mathit{n}\mathit{s}\mathbf{ }\mathbf{)} \end{gathered}$$