Painting a house

Let $x$ be the number of hours it takes Smith to paint the house alone and $y$ be the number of hours it takes John to paint the house alone

Case 1:

$10\left(\dfrac{1}{x}\right)+5\left(\dfrac{1}{y}\right)=1$ , let $a=\dfrac{1}{x}$ and $b=\dfrac{1}{y}$ , then

$10a+5b=1$ $\implies$ $\color{#D61F06}\boxed{1}$

Case 2: $6\left(\dfrac{1}{x}\right)+8\left(\dfrac{1}{y}\right)=1$ , let $a=\dfrac{1}{x}$ and $b=\dfrac{1}{y}$ , then

$6a+8b=1$ $\implies$ $\color{#D61F06}\boxed{2}$

$(-8 \times \color{#D61F06}\boxed{1}$ $)+(5 \times$ $\color{#D61F06}\boxed{2}$ $)$ , we get

$-50a=-3$

$a=\dfrac{3}{50}$

Finally,

$\color{plum}\boxed{x=\dfrac{50}{3}~hours}$