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1) 

The school that Carlos goes to is selling tickets to a choral performance.

On the first day of ticket sales the school sold 14 senior citizen tickets and 6 student tickets for a total of $180.

The school took in $117 on the second day by selling 12 senior citizen tickets and 1 student ticket.

Find the price of a senior citizen ticket and the price of a student ticket. 

x = senior citizen ticket price
y = student ticket price

\(\begin{array}{|lrcll|} \hline (1) & 12x+1y &=& $117 \\ & y &=& 117 - 12x \\\\ (2) & 14x+6y &=& $180 \\ & 14x+6(117 - 12x) &=& $180 \quad &| \quad :2 \\ & 7x+3(117 - 12x) &=& 90 \\ & 7x+ 351 - 36x &=& 90 \\ & 351 - 29x &=& 90 \quad &| \quad + 29x \\ & 351 &=& 90 + 29x \quad &| \quad - 90 \\ & 261 &=& 29x \quad &| \quad : 29 \\ & 9 &=& x \\ & \mathbf{x} & \mathbf{=} & \mathbf{$9} \\\\ & y &=& 117 - 12x \quad &| \quad x=9 \\ & y &=& 117 - 12(9) \\ & y &=& 117 - 108 \\ & y &=& 9 \\ & \mathbf{y} & \mathbf{=} & \mathbf{$9} \\ \hline \end{array}\)

The price of a senior citizen ticket is $9.
The price of a student ticket is $9.

2)
A boat traveled 60 miles downstream and back. The trip downstream took 3 hours.
The trip back took 6 hours.
Find the speed of the boat in still water and the speed of the current.

x = speed of the boat in still water.

y = speed of the current.

s = 60 miles.

\(\begin{array}{|lrcll|} \hline (1) \text{ downstream} & s &=& (x+y)\cdot 3\ h \\ (2) \text{ back} & s &=& (x-y)\cdot 6\ h \\ \hline & s = (x+y)\cdot 3 &=& (x-y)\cdot 6\\ & (x+y)\cdot 3 &=& (x-y)\cdot 6 \quad & | \quad : 3 \\ & x+y &=& 2(x-y) \\ & x+y &=& 2x-2y \quad & | \quad -2x \\ & -x+y &=& -2y \quad & | \quad -y \\ & -x &=& -3y \\ & \mathbf{x} & \mathbf{=} & \mathbf{3y} \\\\ & s &=& (x+y)\cdot 3\ h \quad & | \quad s = 60\ \text{miles} \\ & 60 &=& (3y+y)\cdot 3 \\ & 60 &=& 4y\cdot 3 \quad & | \quad : 3 \\ & 20 &=& 4y \quad & | \quad : 4 \\ & 5 &=& y \\ & \mathbf{y} & \mathbf{=} & \mathbf{5} \\\\ & x & = & 3y \quad & | \quad y = 5 \\ & x & = & 3\cdot 5 \\ & \mathbf{x} & \mathbf{=} & \mathbf{15} \\ \hline \end{array}\)

The speed of the boat in still water is  \(\mathbf{15\ \dfrac{\text{miles}}{\text{hours}}}\)
The speed of the current is  \(\mathbf{5\ \dfrac{\text{miles}}{\text{hours}}}\)

3)
Sumalee's school is selling tickets to a spring musical.
On the first day of ticket  sales the school sold 6 adult tickets and 1 student ticket for a total of $23.
The school took in $106 on the second day by selling 12 adult tickets and 14 student tickets.
Find the price of an adult ticket and the price of a student ticket.

x = adult ticket price
y = student ticket price

\(\begin{array}{|lrcll|} \hline (1) & 6x+1y &=& $23 \\ & y &=& 23 - 6x \\\\ (2) & 12x+14y &=& $106 \\ & 12x+14(23 - 6x) &=& 106 \quad &| \quad :2 \\ & 6x+7(23 - 6x) &=& 53 \\ & 6x +161 - 42x &=& 53 \\ & 161 - 36x &=& 53 \quad &| \quad + 36x \\ & 161 &=& 53 + 36x \quad &| \quad - 53 \\ & 108 &=& 36x \quad &| \quad : 36 \\ & 3 &=& x \\ & \mathbf{x} & \mathbf{=} & \mathbf{$3} \\\\ & y &=& 23 - 6x \quad &| \quad x=3 \\ & y &=& 23 - 6(3) \\ & y &=& 23 - 18 \\ & y &=& 5 \\ & \mathbf{y} & \mathbf{=} & \mathbf{$5} \\ \hline \end{array} \)

The price of a adult ticket is $3.
The price of a student ticket is $5.