Graphing Linear Equations and Finding Feasible Regions

A farmer has 100 meters of fencing to create a rectangular pen for his sheep. If the length of the pen is represented by the equation y = 50 - x, where x is the width, graph the equation and identify the feasible region for the dimensions of the pen.

• The feasible region is the area under the line x + y = 50 in the first quadrant, where x >= 0 and y >= 0.

• The pen can only have a width of 25 meters and a length of 25 meters.

• The feasible region includes negative values for x and y.

• The feasible region is the area above the line x + y = 50 in the first quadrant.

A school is planning to build a new playground. They have a budget of $20,000. The cost of equipment is represented by the equation y = 2000x, where x is the number of equipment pieces. Graph this equation and identify the feasible region for the number of equipment pieces they can buy.

• The feasible region for the number of equipment pieces is 0 <= x <= 10.

• The feasible region for the number of equipment pieces is 0 <= x <= 20.

• The feasible region for the number of equipment pieces is 0 <= x <= 15.

• The feasible region for the number of equipment pieces is 0 <= x <= 5.

A company produces two types of gadgets, A and B. Each gadget A requires 3 hours of labor and gadget B requires 2 hours. If the company has 24 hours of labor available, graph the system of equations and identify the feasible region for the number of gadgets they can produce.

• The feasible region is defined by the inequalities 4x + 2y ≤ 24, x ≥ 0, and y ≥ 0.

• The feasible region is defined by the inequalities 2x + 3y ≤ 24, x ≥ 0, and y ≥ 0.

• The feasible region is defined by the inequalities 3x + 2y ≤ 24, x ≥ 0, and y ≥ 0.

• The feasible region is defined by the inequalities 3x + y ≤ 24, x ≥ 0, and y ≥ 0.

A local bakery sells cakes and cookies. Each cake costs $15 and each cookie costs $2. If the bakery wants to make at least $100 in sales, graph the inequality and identify the feasible region for the number of cakes and cookies they can sell.

• The feasible region is defined by the inequality 15x + 2y ≤ 100, with x ≥ 0 and y ≥ 0.

• The feasible region is defined by the inequality 15x + 2y ≥ 100, with x ≥ 0 and y ≥ 0.

• The feasible region is defined by the inequality 15x + 2y ≥ 50, with x ≥ 0 and y ≥ 0.

• The feasible region is defined by the inequality 15x + 2y = 100, with x ≥ 0 and y ≥ 0.

A car rental company charges a flat fee of $30 plus $0.50 per mile driven. If a customer has a budget of $60, graph the equation and identify the feasible region for the number of miles they can drive.

• The customer can drive between 30 and 90 miles.

• The customer can drive between 0 and 60 miles.

• The customer can drive between 0 and 45 miles.

• The customer can drive between 0 and 100 miles.

A gym offers two types of memberships: a monthly membership for $40 and a yearly membership for $400. If a customer wants to spend no more than $500, graph the inequality and identify the feasible region for the number of months they can maintain a monthly membership.

• 3 months

• 4 months

• 5 months

• The feasible region for the number of months is 0, 1, or 2 months.

A concert venue has a seating capacity of 500. If tickets for the front row cost $50 and tickets for the back row cost $20, and the venue wants to make at least $10,000, graph the system of equations and identify the feasible region for the number of tickets sold in each row.

• The feasible region is defined by the inequalities x + y <= 400 and 50x + 20y >= 8000.

• The feasible region is defined by the inequalities x + y <= 600 and 50x + 20y >= 12000.

• The feasible region is defined by the inequalities x + y >= 500 and 50x + 20y <= 10000.

• The feasible region is defined by the inequalities x + y <= 500 and 50x + 20y >= 10000.

A clothing store sells shirts for $25 and pants for $40. If the store wants to make at least $1,000 in sales, graph the inequality and identify the feasible region for the number of shirts and pants they can sell.

• The feasible region is the area below the x-axis and above the y-axis.

• The feasible region is the area to the left of the line 25x + 40y = 1000 in the first quadrant.

• The feasible region is the area above the line 25x + 40y = 1000 in the first quadrant.

• The feasible region is below the line 25x + 40y = 1000 in the first quadrant.

A charity event is selling tickets for $10 each and food for $5 each. If they want to raise at least $500, graph the inequality and identify the feasible region for the number of tickets and food items sold.

• The feasible region is defined by the inequality 10x + 5y ≥ 500, graphed as y ≥ -2x + 100 in the first quadrant.

• The graph shows y ≤ -2x + 100 in the first quadrant.

• The feasible region is defined by the inequality 10x + 5y ≤ 500.

• The charity needs to sell at least 50 tickets to raise $500.

A tech company has a budget of $15,000 for new computers. Each computer costs $1,200 and each tablet costs $300. Graph the system of equations and identify the feasible region for the number of computers and tablets they can purchase.

• The feasible region is defined by the inequalities 0 ≤ x ≤ 15 and 0 ≤ y ≤ 30.

• The feasible region is defined by the inequalities 0 ≤ x ≤ 8 and 0 ≤ y ≤ 60.

• The feasible region is defined by the inequalities 0 ≤ x ≤ 10 and 0 ≤ y ≤ 40.

• The feasible region is defined by the inequalities 0 ≤ x ≤ 12 and 0 ≤ y ≤ 50.