Question

use the information given to solve for the variables. 13) emilys school is selling tickets to a play. on the first day of ticket sales the school sold 7 senior citizen tickets and 9 child tickets for a total of $208. the school took in $299 on the second day by selling 14 senior citizen tickets and 9 child tickets. find the price of a senior citizen ticket and the price of a child ticket.

Explanation:

Step1: Define variables

Let \( s \) be the price of a senior citizen ticket (in dollars) and \( c \) be the price of a child ticket (in dollars).

Step2: Set up equations

From the first day: \( 7s + 9c = 208 \)
From the second day: \( 14s + 9c = 299 \)

Step3: Subtract equations

Subtract the first equation from the second to eliminate \( c \):
\( (14s + 9c) - (7s + 9c) = 299 - 208 \)
Simplify: \( 7s = 91 \)

Step4: Solve for \( s \)

Divide both sides by 7: \( s = \frac{91}{7} = 13 \)

Step5: Substitute \( s \) into first equation

Substitute \( s = 13 \) into \( 7s + 9c = 208 \):
\( 7(13) + 9c = 208 \)
\( 91 + 9c = 208 \)

Step6: Solve for \( c \)

Subtract 91 from both sides: \( 9c = 208 - 91 = 117 \)
Divide by 9: \( c = \frac{117}{9} = 13 \)? Wait, no—wait, \( 117 \div 9 = 13 \)? Wait, no, 913=117? Wait, 912=108, 913=117. Wait, but let's check again. Wait, 713=91, 208-91=117, 117/9=13? Wait, but let's check the second equation: 1413 + 913 = (14+9)13=2313=299. Yes, that works. Wait, so both tickets are $13? Wait, but let's recheck the first day: 7*13 + 9*13 = (7+9)*13=16*13=208. Yes, 16*13=208 (13*10=130, 13*6=78; 130+78=208). And second day: 14*13 + 9*13=23*13=299 (13*20=260, 13*3=39; 260+39=299). So yes, both are $13? Wait, that's correct? Wait, but the problem says "senior citizen ticket" and "child ticket"—maybe that's possible. Wait, but let's go through the steps again.

Wait, step 3: 14s +9c -7s -9c = 299-208 → 7s=91 → s=13. Then step 5: 7*13=91, 208-91=117, 117/9=13. So c=13. So both tickets are $13. Let's verify with second day: 14*13=182, 9*13=117, 182+117=299. Correct. First day: 7*13=91, 9*13=117, 91+117=208. Correct. So both are $13. Wait, but maybe I made a mistake? Wait, no, the math checks out.

Answer:

Senior citizen ticket: \$13, Child ticket: \$13 (Wait, but that seems unusual, but the numbers work. Wait, let me check again. Wait, 7 senior and 9 child: 7s +9c=208. 14 senior and 9 child:14s +9c=299. Subtract first from second: 7s=91 → s=13. Then 7*13=91, 208-91=117, 117/9=13. So c=13. So both are \$13. So the price of a senior citizen ticket is \$13 and the price of a child ticket is \$13.