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Relevant wiki: System of Equations - Problem Solving - Basic

You must like the number 41 :)

Relevant wiki: System of Equations - Problem Solving - Basic

$\dfrac{5}{2x}+\dfrac{9}{y}=2 \implies 5y+18x=4xy \\ \dfrac{15}{x}-\dfrac{12}{y}=1 \implies 15y-12x=xy \\ \text{ Divide Both the equations } \\ 5y+18x=60y-48x \implies y=\dfrac{6}{5}x \quad \quad \color{#3D99F6}{\text{Put this value in any of the two equations}} \\ 15\cdot \dfrac{6}{5}x-12x=x\cdot \dfrac{6}{5}x \implies x=5 \text{ and } y=6 \\ \boxed{x+y+xy=5+6+30 = 41}$

$\begin{cases} \dfrac 5{2x} + \dfrac 9y = 2 & \times 4 \implies \dfrac {10}x + \dfrac {36}y = 8 & ...(1) \\ \dfrac {15}x - \dfrac {12}y = 1 & \times 3 \implies \dfrac {45}x - \dfrac {36}y = 3 & ...(2) \end{cases}$

$\begin{aligned} (1)+(2): \quad \frac {10}x + \frac {45}x & = 11 \\ \frac {55}x & = 11 \\ \implies x & = 5 \end{aligned}$

$\begin{aligned} (2): \quad \frac {15}5 - \frac {12}y & = 1 \\ \frac {12}y & = 2 \\ \implies y & = 6 \end{aligned}$

$\implies x + y + xy = 5 + 6 + 5\times 6 = \boxed{41}$

Relevant wiki: System of Equations - Problem Solving - Basic

$\large \begin{aligned} & & \frac { 5 }{ 2x } +\frac { 9 }{ y } & =2 & \longrightarrow (1) \\ & & \frac { 15 }{ x } -\frac { 12 }{ y } & =1 & \longrightarrow (2) \\ From\, \, \, \, (1) & & \frac { 5y }{ 2xy } +\frac { 18x }{ 2xy } & =2 & \\ & \Rightarrow & \frac { 5y+18x }{ 2xy } & =2 & \\ & \Rightarrow & 5y+18x & =4xy & \longrightarrow (3) \\ From\, \, \, \, (2) & & \frac { 15y }{ xy } -\frac { 12x }{ xy } & =1 & \\ & \Rightarrow & \frac { 15y-12x }{ xy } & =1 & \\ & \Rightarrow & 15y-12x & =xy & \longrightarrow (5) \\ & \Rightarrow & 60y-48x & =4xy & \longrightarrow (4) \\ From\, \, \, \, (3)\, \, \, and\, \, \, (4) & & 5y+18x & =60y-48x & \\ & \Rightarrow & 66x & =55y & \\ & \Rightarrow & x & =\frac { 5y }{ 6 } & \longrightarrow (6) \\ (6)\rightarrow (5) & & 15y-12x & =xy & \\ & \Rightarrow & 15y & =xy+12x & \\ & \Rightarrow & 15y & =x(y+12) & \\ & \Rightarrow & 15y & =\frac { 5y(y+12) }{ 6 } & \\ & \Rightarrow & 90y & =5y^{ 2 }+60y & \\ & \Rightarrow & 5{ y }^{ 2 } & =30y & \\ & \Rightarrow & { y }^{ 2 } & =6y & \\ & \therefore & y & =6 & \longrightarrow (7) \\ (7)\rightarrow (6) & & x & =\frac { 5y }{ 6 } & \\ & \Rightarrow & x & =\frac { 5(6) }{ 6 } & \\ & \therefore & x & =5 & \\ & \therefore & x+y+xy & =6+5+(6)(5) & \\ & & & =11+30 & \\ & & & =\boxed{41} & \end{aligned}$