Let x, y, z, be the estate received to the eldest, middle, and youndest sibling, respectively.
The eldest sibling receives six times as much as youngest sibling,
[tex]x=6z[/tex]The middle sibling receives $14,000 more than the youngest,
[tex]y=z+14000[/tex]The total estate is worth $503000. So it follows that,
[tex]\begin{gathered} x+y+z=503000 \\ 6z+(z+14000)+z=503000 \\ 8z=503000-14000 \\ 8z=489000 \\ z=61125 \end{gathered}[/tex]Then the corresponding values of x, and y become,
[tex]\begin{gathered} x=6(61125)=366750 \\ y=61125+14000=75125 \end{gathered}[/tex]Thus, the estate received to the siblings are as follows,
[tex]\begin{gathered} \text{Eldest}=366750\text{ dollars} \\ \text{Middle}=75125\text{ dollars} \\ \text{Youngest}=61125\text{ dollars} \end{gathered}[/tex]