Answer: A. Let's use x to represent the number of hours Opal works for the photographer, and y to represent the number of hours she coaches soccer. Then we can create the following system of inequalities to represent the situation:
12x + 7y ≥ 150 (Opal needs to earn at least $150 per week)
x + y ≤ 20 (Opal cannot work more than 20 hours per week)
B. There are different ways Opal can meet her goals, but here are two possible solutions:
Solution 1: Opal works for the photographer for 10 hours and coaches soccer for 10 hours. Then her total earnings for the week would be:
12(10) + 7(10) = 120 + 70 = $190
This meets her goal of earning at least $150 per week, and it also satisfies the constraint that she cannot work more than 20 hours per week.
Solution 2: Opal works for the photographer for 15 hours and coaches soccer for 5 hours. Then her total earnings for the week would be:
12(15) + 7(5) = 180 + 35 = $215
This also meets her goal of earning at least $150 per week, and it satisfies the constraint that she cannot work more than 20 hours per week.
C. To check if (10,6) is a solution to the system of inequalities, we need to substitute x = 10 and y = 6 into both inequalities and see if they are true:
12(10) + 7(6) ≥ 150
120 + 42 ≥ 150
162 ≥ 150 (true)
10 + 6 ≤ 20 (true)
Since both inequalities are true, (10,6) is a solution to the system. However, this solution does not meet Opal's goal of earning at least $150 per week, as her total earnings would be:
12(10) + 7(6) = 120 + 42 = $162
So, while (10,6) satisfies the constraints of the system, it is not a valid solution to the problem.
Explanation: