Question

1) A family is looking to enroll their child in a pre-school. They have narrowed their search down to two choices and are now looking at the prices to help decide. Little Learners PreSchool: Pay a one-time initial $50 enrollment fee, then $100 each week after they enroll. Growing Place Preschool: Pay a one-time initial $300 enrollment fee then $75 each week after they enroll. a) Let y represent the total cost of the pre-school and x represent the number of weeks. Write a system of equations in Slope Intercept Form (y = mx + b) that represents the total cost for both schools. Equation for Little Learners: $50 + ($100 * 10) = $1050 Equation for Growing Place: $300 + ($75 * 10) = $1050 b) Graph the system of equations that you wrote in both parts of (a) above. c) What is the solution to the system of equations? __________ Identify the solution of the system on your graph above and clearly indicate this by writing “Solution” by it. d) Explain what the significance of this point is in terms of the pre-school situation. e) Using the Substitution Method, solve the system you wrote in (a) to verify algebraically that the solution you found in (c) is correct. f) The family thinks Little Learners is the best choice since their enrollment fee is less. Are they correct? Is there a scenario where Little Learners is not the best choice? Explain and justify your reasoning. 2) Solve the following system using the elimination method. 2 − 9 = −8 5 + = −20

Answer 1

a) The equations for the total cost of the two preschool options are: Little Learners - y = 50 + 100x, Growing Place - y = 300 + 75x. b) The graph of the system of equations shows the total cost for both schools. c) The solution to the system is (10, 1050), indicating the number of weeks and total cost when both schools have the same price. d) The solution point represents the number of weeks when both schools have the same total cost. e) The solution can be verified algebraically using the substitution method. f) Little Learners is the best choice based on the lower enrollment fee, but factors like length of enrollment may affect this decision.

Step-by-step explanation:

a) Let y represent the total cost of the pre-school and x represent the number of weeks.
Equation for Little Learners: y = 50 + 100x
Equation for Growing Place: y = 300 + 75x

b) Graph the system of equations representing the total cost for both schools.
Little Learners: y = 50 + 100x
Growing Place: y = 300 + 75x

c) The solution to the system of equations is (10, 1050). This point is the intersection point of the two lines on the graph and indicates the number of weeks (x) and the total cost (y) for which both schools have the same price.

d) The significance of this point is that it represents the number of weeks at which both schools have the same total cost. It helps the family make an informed decision by showing that after 10 weeks, both schools will cost $1050.

e) Using the substitution method, we can verify that the solution (10, 1050) is correct for both equations:
Equation for Little Learners: 1050 = 50 + 100 * 10
Equation for Growing Place: 1050 = 300 + 75 * 10

f) The family's claim that Little Learners is the best choice based on the lower enrollment fee is correct. However, there may be scenarios where Little Learners is not the best choice, such as if the family plans to enroll their child for a longer period of time where the lower weekly cost of Growing Place could result in a lower total cost. It depends on the specific needs and circumstances of the family.