The age difference between two siblings is 3 years. One year ago, the father's age was double the sum of the siblings' ages, and in 20 years, the father's age will be the sum of the ages of these two children. What is the age of each?

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Let $x, y$ be the current ages of the two siblings, and $z$ the age of the father. We have:

$$
\left\{\begin{array} { l } 
{ x - y = 3 } \\
{ z - 1 = 2 [ ( x - 1 ) + ( y - 1 ) ] } \\
{ z + 20 = ( x + 20 ) + ( y + 20 ) }
\end{array} \Rightarrow \left\{\begin{array}{rl}
x-y & =3 \\
z-1 & =2 x+2 y-4 \\
z+20 & =x+y+40
\end{array}\right.\right.
$$

A simple way to obtain $z$ is to multiply the 3rd equation by 2 and subtract the 2nd equation from the result: $2 z+40-(z-1)=80-(-4)$, which implies $z=43$.

Now let's calculate the ages of the children using the first two equations:

![](https://cdn.mathpix.com/cropped/2024_05_01_a444074f1f8ae50444d5g-27.jpg?height=140&width=731&top_left_y=1513&top_left_x=659)

We get $2 x=26$, hence $x=13$ and $y=10$.