Exploring Intersection Points in Systems of Equations

A farmer has 100 meters of fencing to create two adjacent rectangular pens. If the length of the first pen is represented by x and the length of the second pen is represented by y, write the system of equations and graph it to find the intersection point that represents the maximum area of both pens combined.

• The maximum area occurs at (30, 20) for x and y respectively.

• The maximum area is found at the point (10, 40) for x and y respectively.

• The maximum area occurs at the intersection point of the equations, which can be found to be approximately (20, 26.67) for x and y respectively.

• The maximum area can be calculated as 500 square meters at the point (50, 0).

A school is planning a field trip and has a budget of $300. The cost per student is $15 for transportation and $10 for admission. Write a system of equations to represent the total cost and graph it to find how many students can attend if they want to stay within budget.

• 10

• 15

• 20

• 12

Two friends are selling lemonade and cookies. The lemonade costs $2 per cup and the cookies cost $1 each. If they want to make a total of $50, write the system of equations and graph it to find the number of cups of lemonade and cookies they need to sell to reach their goal.

• 20 cups of lemonade and 10 cookies

• 15 cups of lemonade and 5 cookies

• 30 cups of lemonade and 5 cookies

• They need to sell 25 cups of lemonade and 0 cookies, or any combination that satisfies 2x + y = 50 with non-negative integers.

A concert venue has a seating capacity of 500. If tickets for the front row are $50 each and tickets for the back row are $30 each, write a system of equations to represent the total revenue and graph it to find the combination of tickets sold that maximizes revenue.

• 100 front row tickets and 400 back row tickets.

• 250 front row tickets and 250 back row tickets.

• 0 front row tickets and 500 back row tickets.

• The combination of tickets sold that maximizes revenue is 500 front row tickets and 0 back row tickets.

A local gym offers two membership plans. Plan A costs $30 per month and Plan B costs $20 per month plus a one-time fee of $100. Write a system of equations to represent the total cost over time and graph it to find when both plans cost the same.

• 10 months

• 5 months

• 8 months

• 12 months

A bookstore sells novels for $12 each and textbooks for $25 each. If they want to make $600 in sales, write the system of equations and graph it to find how many novels and textbooks they need to sell to meet their sales goal.

• 30 novels and 10 textbooks

• They need to sell 50 novels and 0 textbooks, or combinations like 25 novels and 12 textbooks, etc., to meet the sales goal.

• 10 novels and 20 textbooks

• 20 novels and 15 textbooks

A car rental company charges a flat fee of $50 plus $0.20 per mile driven. Another company charges $30 plus $0.25 per mile. Write a system of equations to represent the total cost and graph it to find when both companies charge the same amount.

• 250 miles

• 500 miles

• 400 miles

• 300 miles

A charity event is selling tickets for $10 each and donations are being collected at $5 each. If they want to raise $500, write the system of equations and graph it to find how many tickets and donations they need to sell to reach their goal.

• They need to sell 25 tickets and 50 donations.

• They need to sell 50 tickets and 0 donations, or any combination that satisfies 10x + 5y = 500 with non-negative integers.

• They need to sell 100 tickets and 0 donations.

• They need to sell 0 tickets and 100 donations.

A bakery sells muffins for $3 each and cupcakes for $4 each. If they want to make $120 in one day, write the system of equations and graph it to find how many muffins and cupcakes they need to sell to meet their sales target.

• 50 muffins and 10 cupcakes

• 20 muffins and 20 cupcakes

• They need to sell 40 muffins and 0 cupcakes, or combinations like 36 muffins and 6 cupcakes, etc., to meet the target.

• 30 muffins and 15 cupcakes