In basketball you score 2 points for a field goal and 1 point for a free throw. Suppose that you have scored at least 3 points in every game this season, and have a season high score of 15 points in one game. How many field goals and free throws could you have made in any one game?
a. Write a sytem of two inequalities that describes this situation.
b. graph the system to show all possible solutions.
c. Write 1 possible solution to the problem.
Please help!
a. To write a system of two inequalities that describes this situation, we can define the variables x and y, where x represents the number of field goals made and y represents the number of free throws made in a single game.
The first inequality can be written as: 2x + y ≥ 3, since the player has scored at least 3 points in every game.
The second inequality can be written as: 2x + y ≤ 15, since the player's season high score is 15 points in one game.
b. To graph the system of inequalities, plot the points that satisfy both inequalities. The shaded region represents all possible solutions.
c. One possible solution could be:
- The player made 5 field goals (x = 5) and 5 free throws (y = 5) in a single game.
This solution satisfies both inequalities:
2x + y = 2(5) + 5 = 15, which is the maximum score achieved in one game.
2x + y = 2(5) + 5 = 15 is greater than or equal to the minimum score required in every game.
a. To describe this situation with a system of inequalities, we need to set up inequalities for the minimum and maximum number of points you can score in a game.
Let's denote the number of field goals as F and the number of free throws as T.
The minimum number of points you can score in a game is 3, so we can set up the inequality:
2F + 1T ≥ 3
The maximum number of points you can score in a game is 15, so we can set up the inequality:
2F + 1T ≤ 15
b. To graph the system of inequalities, we can first solve each inequality for T.
From the first inequality: T ≥ 3 - 2F
From the second inequality: T ≤ 15 - 2F
Now we can graph these two inequalities on the T-F plane.
The first inequality, T ≥ 3 - 2F, is a solid line because it includes the points on the line itself. Start with plotting the points (0, 3) and (1.5, 0) and draw a straight line passing through them.
The second inequality, T ≤ 15 - 2F, is also a solid line. Plot the points (0, 15) and (7.5, 0), and draw a straight line passing through them.
Shade the region that satisfies both inequalities. This region represents all the possible solutions for the number of field goals and free throws you could have made in any one game.
c. One possible solution could be:
Field goals (F) = 4
Free throws (T) = 7
This gives us a total of 2 * 4 (8 points) from field goals and 1 * 7 (7 points) from free throws, resulting in a total of 15 points, which matches the highest score you mentioned in the problem.