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!
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a. To write a system of two inequalities that describes this situation, we need to consider the constraints given in the problem.
Let's denote the number of field goals as x and the number of free throws as y.
We know that a field goal is worth 2 points and a free throw is worth 1 point. So, the total points scored in a game can be calculated by 2x + y.
From the given condition, we know that the total points scored in a game should be at least 3. Therefore, the first inequality is:
2x + y ≥ 3
The problem also states that the season-high score is 15 points in one game. So, the second inequality is:
2x + y ≤ 15
Therefore, the system of inequalities is:
2x + y ≥ 3
2x + y ≤ 15
b. To graph the system of inequalities, we need to plot the graph of each inequality on the same coordinate system.
Graph for 2x + y ≥ 3:
First, we assume the equality: 2x + y = 3
To plot this line, we can assign arbitrary values to x or y and find the corresponding values to draw the line.
When x = 0, y = 3
When y = 0, 2x = 3, x = 1.5
Plotting the points (0, 3) and (1.5, 0) and drawing a line through them, we get the graph for 2x + y ≥ 3.
Graph for 2x + y ≤ 15:
Similarly, we assume the equality: 2x + y = 15
When x = 0, y = 15
When y = 0, 2x = 15, x = 7.5
Plotting the points (0, 15) and (7.5, 0) and drawing a line through them, we get the graph for 2x + y ≤ 15.
c. To find one possible solution, we need to find an intersection point of the two inequalities.
Solving the system of inequalities, we find the intersection point to be (6, 3). This means that in one game, you could have made 6 field goals and 3 free throws, totaling 15 points (2x + y = 2 * 6 + 3 * 1 = 12 + 3 = 15). Therefore, one possible solution to the problem is 6 field goals and 3 free throws.
a. Let's define the number of field goals as "x" and the number of free throws as "y". We can write the following inequalities based on the given information:
1. The total score in a game must be at least 3 points: 2x + y ≥ 3
2. The maximum score in any game is 15 points: 2x + y ≤ 15
b. To graph the system, we can plot the inequalities on a coordinate plane. However, since we are only interested in whole numbers of field goals and free throws, we need to graph the solutions as discrete points rather than a continuous line.
c. One possible solution is when you make 4 field goals (x = 4) and 1 free throw (y = 1). This would give you a total score of 9 points (2*4 + 1 = 9), satisfying the minimum requirement of 3 points and being below the maximum score of 15 points.