6.

The school that Laura goes to is selling tickets to the annual talent show. On the first day of

ticket sales the school sold 4 senior citizen tickets, 2 adult tickets and 5 child tickets for a total of

$55. The school took in $67 on the second day by selling 7 senior citizen tickets, 2 adult tickets

and 5 child tickets. On the third day the show earned $46 when they sold 2 senior citizen tickets,

4 adult tickets and 2 child tickets. What is the price each of one senior citizen ticket, one adult

ticket and one child ticket?

The price of 1 senior citizen ticket is $4

The price of 1 adult ticket is $7

The price of 1 child ticket is $5

Step-by-step explanation:

Assume that the costs of a senior ticket is $x, an adult ticket is $y and

a child ticket is $z

First day:

The school sold 4 senior citizen tickets, 2 adult tickets and 5 child

tickets for a total of $55

∴ 4x + 2y + 5z = 55 ⇒ (1)

Second day:

The school sold 7 senior citizen tickets, 2 adult tickets and 5 child

tickets for $67

∴ 7x + 2y + 5z = 67 ⇒ (2)

Third day:

The school sold 2 senior citizen tickets, 4 adult tickets and 2 child

tickets for $46

∴ 2x + 4y + 2z = 46 ⇒ (3)

The number of adult tickets and the number of child tickets in the first

and second days are equal, then we can subtract equation (1) from

equation (2) to find x

Subtract equation (1) from equation (2)

∴ (7x - 4x) + (2y - 2y) + (5z - 5z) = 67 - 55

∴ 3x = 12

Divide both sides by 3

∴ x = 4

Substitute the value of x in equation (2)

∴ 7 (4) + 2y + 5z = 67

∴ 28 + 2y + 5z = 67

Subtract 28 from both sides

∴ 2y + 5z = 39 ⇒ (4)

Substitute the value of x in equation (3)

∴ 2 (4) + 4y + 2z = 46

∴ 8 + 4y + 2z = 46

Subtract 8 from both sides

∴ 4y + 2z = 38 ⇒ (5)

Now lets solve equations (4) and (5) to find y and z

Multiply equation (4) by - 2 to eliminate y

∴ - 4y - 10z = - 78 ⇒ (6)

Add equations (5) and (6)

∴ - 8z = - 40

Divide both sides by - 8

∴ z = 5

Substitute the value of z in equation (4) or (5)

∴ 2y + 5 (5) = 39

∴ 2y + 25 = 39

Subtract 25 from both sides

∴ 2y = 14

Divide both sides by 2

∴ y = 7

The price of 1 senior citizen ticket is $4

The price of 1 adult ticket is $7

The price of 1 child ticket is $5