Respuesta :
We need to find the base x in the following equation:
[tex]365_7+43_x=217_{10}[/tex]First, lets convert 365 from base 7 to base 10. This is given by
[tex]365_7=3\times7^2+6\times7^1+5\times7^0[/tex]where the upperindex denotes the position of eah number. This gives
[tex]\begin{gathered} 365_7=3\times49+6\times7+5\times1 \\ 365_7=147+42+5 \\ 365_7=194_{10} \end{gathered}[/tex]that is, 365 based 7 is equal to 194 bases 10.
Now, lets do the same for 43 based x. Lets convert 43 based x to base 10:
[tex]43_x=4\times x^1+3\times x^0[/tex]where again, the superindex 0 and 1 denote the position 0 and 1 in the number 43. This gives
[tex]43_x=(4x+3)_{10}[/tex]Now, we have all number in base 10. Then, our first equation can be written in base 10 as
[tex]194_{10}+(4x+3)_{10}=217_{10}[/tex]For simplicity, we can omit the 10 and get
[tex]194+4x+3=217[/tex]so, we can solve this equation for x. By combining similar terms. we have
[tex]197+4x=217[/tex]and by moving 197 to the right hand side, we obtain
[tex]\begin{gathered} 4x=217-197 \\ 4x=20 \end{gathered}[/tex]Finally, we get
[tex]\begin{gathered} x=\frac{20}{4} \\ x=5 \end{gathered}[/tex]Therefore, the solution is x=5
