Question

1. $2x - 2y = 18$

$3x + 3y = 9$

1. $-3x + 3y = 0$

$10x - 4y = -18$

1. the senior classes at high school a and high school b planned separate trips to the indoor climbing gym. the senior class at high school a rented and filled 3 vans and 13 buses with 612 students. high school b rented and filled 3 vans and 6 buses with 297 students. each van and each bus carried the same number of students. find the number of students in each van and in each bus.
2. the school that arjun goes to is selling tickets to the annual talent show. on the first day of ticket sales the school sold 8 senior citizen tickets and 2 child tickets for a total of $62. the school took in $80 on the second day by selling 8 senior citizen tickets and 8 child tickets. find the price of a senior citizen ticket and the price of a child ticket.

Explanation:

Problem 38: Solve the system \(

$$\begin{cases}2x - 2y = 18\\3x + 3y = 9\end{cases}$$

\)

Step 1: Simplify the equations

Divide the first equation by 2: \(x - y = 9\) (Equation 1)
Divide the second equation by 3: \(x + y = 3\) (Equation 2)

Step 2: Add Equation 1 and Equation 2

\((x - y) + (x + y) = 9 + 3\)
\(2x = 12\)
\(x = 6\)

Step 3: Substitute \(x = 6\) into Equation 2

\(6 + y = 3\)
\(y = 3 - 6 = -3\)

Problem 39: Solve the system \(

$$\begin{cases}-3x + 3y = 0\\10x - 4y = -18\end{cases}$$

\)

Step 1: Simplify the first equation

Divide by 3: \(-x + y = 0\) → \(y = x\) (Equation 1)

Step 2: Substitute \(y = x\) into the second equation

\(10x - 4x = -18\)
\(6x = -18\)
\(x = -3\)

Step 3: Find \(y\)

Since \(y = x\), \(y = -3\)

Problem 40: Van and Bus Students

Let \(x\) = students per van, \(y\) = students per bus.
System: \(

$$\begin{cases}3x + 13y = 612\\3x + 6y = 297\end{cases}$$

\)

Step 1: Subtract the second equation from the first

\((3x + 13y) - (3x + 6y) = 612 - 297\)
\(7y = 315\)
\(y = 45\)

Step 2: Substitute \(y = 45\) into \(3x + 6y = 297\)

\(3x + 6(45) = 297\)
\(3x + 270 = 297\)
\(3x = 27\)
\(x = 9\)

Problem 41: Ticket Prices

Let \(x\) = senior ticket price, \(y\) = child ticket price.
System: \(

$$\begin{cases}8x + 2y = 62\\8x + 8y = 80\end{cases}$$

\)

Answer:

s:

• Problem 38: \(x = 6\), \(y = -3\)
• Problem 39: \(x = -3\), \(y = -3\)
• Problem 40: Van: \(9\) students, Bus: \(45\) students
• Problem 41: Senior ticket: \(\$7\), Child ticket: \(\$3\)