+383 In an election, $1200$ total votes were cast for the $3$ candidates. The second-place candidate received $125$ fewer votes than the winner and $200$ votes more than the third-place candidate. How many votes did the winner receive?
+26367 In an election, 1200 total votes were cast for the 3 candidates.
The second-place candidate received 125 fewer votes than the winner and 200 votes more than the third-place candidate.
How many votes did the winner receive?
\(\text{Let winner candidate = c_1 } \\ \text{Let second-place candidate = c_2 } \\ \text{Let third-place candidate = c_3 } \)
\(\begin{array}{|lrcll|} \hline (1) & \mathbf{c_2} &=& \mathbf{c_1 - 125} \\ \hline (2) & c_2 &=& c_3 + 200 \\ & c_3 &=& c_2 - 200 \quad | \quad \mathbf{c_2=c_1 - 125} \\ & c_3 &=& c_1 - 125 - 200 \\ & \mathbf{c_3} &=& \mathbf{c_1 - 325} \\ \hline (3) &c_1+ c_2+c_3 &=& 1200 \quad | \quad \mathbf{c_2=c_1 - 125},\ \mathbf{c_3=c_1 - 325} \\ &c_1+ (c_1 - 125) + (c_1 - 325) &=& 1200 \\ &3c_1- 125 - 325 &=& 1200 \\ &3c_1- 450 &=& 1200 \\ &3c_1 &=& 1200 + 450 \\ &3c_1 &=& 1650 \quad | \quad : 3 \\ & \mathbf{c_1} &=& \mathbf{550} \\ \hline \end{array}\)
The winner did receive 550 votes?
