$95$ passengers rode in a train from City A to City B. Tickets for regular coach seats cost $\$116$ . Tickets for sleeper seats cost $\$286.00$ . The receipts for the trip totaled $\$19,520$ . How many passengers purchased each type of ticket?

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95

passengers rode in a train from City A to City B. Tickets for regular coach seats cost

\$116

. Tickets for sleeper seats cost

\$286.00

. The receipts for the trip totaled

\$19,520

. How many passengers purchased each type of ticket?

Define Unknowns: Let's denote the number of regular coach seats sold as $x$ and the number of sleeper seats sold as $y$ . We have two unknowns and will need two equations to solve for them. The first equation comes from the total number of passengers, and the second equation comes from the total amount of money collected.
First Equation: The first equation represents the total number of passengers: $x + y = 95$ . This is because the sum of passengers in regular coach seats and sleeper seats must equal the total number of passengers.
Second Equation: The second equation represents the total amount of money collected: $116x + 286y = 19,520$ . This is because each regular coach seat costs $\$116$ and each sleeper seat costs $\$286$ , and the total receipts from all the tickets sold must equal $\$19,520$ .
Solve for y: Now we have a system of two equations: $\newline$ $1$ ) $x + y = 95$ $\newline$ $2$ ) $116x + 286y = 19,520$ $\newline$ We can solve this system using the method of substitution or elimination. Let's use the substitution method. We can express $y$ in terms of $x$ from the first equation: $y = 95 - x$ .
Substitute $y$ into Second Equation: Substitute $y = 95 - x$ into the second equation: $\newline$ $116x + 286(95 - x) = 19,520$ $\newline$ Now, let's distribute and simplify the equation.
Simplify Equation: $116x + 27110 - 286x = 19,520$ Combine like terms: $-170x + 27110 = 19,520$ Now, let's solve for $x$ .
Solve for x: Subtract $27110$ from both sides of the equation: $\newline$ $-170x = 19,520 - 27110$ $\newline$ $-170x = -7,590$ $\newline$ Now, divide both sides by $-170$ to find $x$ .
Check Calculation: $x = -7,590 / -170$ $\newline$ $x = 44.6470588235$ $\newline$ Since the number of passengers must be a whole number, we round $x$ to the nearest whole number. However, this is a mistake because the number of passengers cannot be a fraction. We need to recheck our calculations.