Question

1. the perimeter of a rectangular garden is 48 feet. one side is four feet longer than the other side. which system of equations can be used to find x, the first side length of the garden, and y, the the second side length of the garden? a. $x + y = 48$ $x = 4y$ b. $x + y = 48$ $x = y + 4$ c. $2x + 2y = 48$ $x = 4y$ d. $2x + 2y = 48$ $x = y + 4$ 8. the perimeter of a rectangular garden is 48 feet. one side is four feet longer than the other side. find the dimensions of the garden. 9. alex and taylor bought a total of 12 concert tickets for their friends. alex bought 3 times as many tickets as taylor. which system of equations can be used to find x, the number of tickets alex bought, and y, the number of tickets taylor bought? a. $x + y = 12$ $x = 3y$ b. $x + y = 12$ $x = y + 3$ c. $x + y = 12$ $x + y = 3$ d. $x + y = 12$ $3x + y = 12$ 10. alex and taylor bought a total of 12 concert tickets for their friends. alex bought 3 times as many tickets as taylor. how many tickets did each person buy? 11. lily bought 5 vip tickets and 3 regular tickets for a total of $135. sarah bought 2 vip tickets and 4 regular tickets for $82. which system of equations can be used to find x, the number of vip tickets, and y, the number of regular tickets? a. $5x + 3y = 82$ $2x + 4y = 135$ b. $5x + 3y = 82$ $4x + 2y = 135$ c. $5x + 3y = 135$ $2x + 4y = 82$ d. $5x + 3y = 135$ $4x + 2y = 82$ 12. lily bought 5 vip tickets and 3 regular tickets for a total of $135. sarah bought 2 vip tickets and 4 regular tickets for $82. what is the cost of each vip ticket and each regular ticket?

Explanation:

Question 7

Step1: Recall perimeter formula for rectangle

The perimeter \( P \) of a rectangle is given by \( P = 2x + 2y \), where \( x \) and \( y \) are the length and width. Here, perimeter is 48, so \( 2x + 2y = 48 \).

Step2: Analyze the side length relationship

One side is 4 feet longer than the other, so if \( x \) is the longer side, \( x = y + 4 \).

Step1: Use the system from Q7

From Q7, the system is \( 2x + 2y = 48 \) and \( x = y + 4 \).

Step2: Substitute \( x \) in perimeter equation

Substitute \( x = y + 4 \) into \( 2x + 2y = 48 \): \( 2(y + 4) + 2y = 48 \).

Step3: Simplify and solve for \( y \)

Expand: \( 2y + 8 + 2y = 48 \) → \( 4y + 8 = 48 \) → \( 4y = 40 \) → \( y = 10 \).

Step4: Find \( x \) using \( x = y + 4 \)

\( x = 10 + 4 = 14 \).

Step1: Total tickets equation

Alex (\( x \)) and Taylor (\( y \)) bought 12 tickets, so \( x + y = 12 \).

Step2: Ticket quantity relationship

Alex bought 3 times as many as Taylor, so \( x = 3y \).

Answer:

D. \( 2x + 2y = 48 \); \( x = y + 4 \)

Question 8