Exploring Graph Intersections in Linear Equations

1. A farmer has 100 feet of fencing to create a rectangular pen for his animals. If the length of the pen is represented by the equation y = 50 - x, where x is the width, graph the equations and find the intersection point that represents the dimensions of the pen.

• Width: 30 feet, Length: 20 feet

• Width: 10 feet, Length: 40 feet

• Width: 20 feet, Length: 30 feet

• Width: 25 feet, Length: 25 feet

2. Two friends are selling lemonade and cookies. The equation for lemonade sales is y = 2x + 5 and for cookies is y = 3x + 2. Graph both equations and determine the point where their sales are equal.

• (1, 7)

• (4, 14)

• (3, 11)

• (2, 9)

3. A school is planning a field trip and has a budget of $300. The cost per student for transportation is represented by the equation y = 15x, and the cost per student for admission is y = 20x + 50. Graph these equations and find the number of students that can attend the trip within budget.

• 10

• 5

• 7

• 12

4. A car rental company charges a flat fee of $30 plus $0.20 per mile driven. The equation for the total cost is y = 0.2x + 30. Another company charges $25 plus $0.25 per mile. Graph both equations and find the point where the costs are the same.

• The costs are the same at 100 miles for $50.

• The costs are the same at 50 miles for $40.

• The costs are the same at 120 miles for $60.

• The costs are the same at 75 miles for $45.

5. A local gym offers two membership plans. Plan A costs $40 per month, and Plan B has a one-time fee of $100 plus $20 per month. Write the equations for both plans, graph them, and identify the point where the total costs are equal.

• The total costs are equal after 7 months.

• The total costs are equal after 3 months.

• The total costs are equal after 5 months.

• The total costs are equal after 10 months.

6. A concert venue has a seating capacity of 500. The ticket sales for general admission are represented by the equation y = 10x, and for VIP tickets, it is y = 20x - 50. Graph these equations and find the intersection point that indicates the number of tickets sold.

• (3, 30)

• (10, 100)

• (7, 70)

• (5, 50)

7. A bakery sells muffins for $2 each and cookies for $1.50 each. If the total sales can be represented by the equation y = 2x + 1.5z, where x is the number of muffins and z is the number of cookies, graph the equations for different values of x and z to find the combinations that yield $30 in sales.

• (1, 18)

• (5, 15)

• (10, 12)

• (0, 20), (3, 18), (6, 16), (9, 14), (12, 12), (15, 10)

8. A company produces two products, A and B. The profit from product A is represented by the equation y = 5x, and from product B by y = 3x + 10. Graph both equations and determine the production level where profits are equal.

• 5

• 3

• 7

• 10

9. A delivery service charges a base fee of $15 plus $0.50 per mile. Another service charges $10 plus $0.75 per mile. Write the equations, graph them, and find the distance at which both services charge the same amount.

• 20 miles

• 15 miles

• 25 miles

• 30 miles

10. A student is saving money for a new laptop. She saves $20 a week from her allowance and $10 a week from her job. Write the equations for her savings from both sources, graph them, and find the point where her total savings reach $500.

• 15 weeks

• 20 weeks

• 17 weeks

• 12 weeks