In a basketball game you score 2 points for a field goal and 1 point for a free throw. Suppose that you have scored at least 5 points a game and have a season high of 12 points a game. How many field goals and free throws could you have made in one game?
5 field goals (5*2=10) 2 free throws (1*2=2). 10+2=12. That's just one combination (possibility)
3 field goals and 1 free throw
Let's solve this step-by-step:
Step 1: Let's assume the number of field goals as "x" and the number of free throws as "y".
Step 2: According to the given information, scoring a field goal gives 2 points, and scoring a free throw gives 1 point.
Step 3: As per the problem, you have scored at least 5 points in a game. This translates to the equation: 2x + y ≥ 5.
Step 4: Your season-high score is 12 points in a game. This translates to the equation: 2x + y = 12.
Step 5: Now, we have a system of equations:
2x + y ≥ 5
2x + y = 12
Step 6: To solve this system, we can subtract the first equation from the second equation to eliminate "y":
(2x + y) - (2x + y) = (12) - (5)
0 = 7
Step 7: We obtained 0 = 7, which is not a valid equation. This indicates that there is no solution that satisfies these conditions.
Therefore, it is not possible to determine the number of field goals and free throws you made in one game based on the given information.
To answer this question, we need to find the number of field goals and free throws that would result in a point total between 5 and 12. Let's create a system of equations to solve this problem.
Let's define two variables:
- F: The number of field goals made in one game.
- T: The number of free throws made in one game.
According to the given information, each field goal is worth 2 points and each free throw is worth 1 point.
1. The total point score can be calculated using the equation: 2F + T = total points.
2. We also know that the total points scored is between 5 and 12, so we can write the inequality as: 5 ≤ 2F + T ≤ 12.
To determine the possible values of F and T, we will solve this system of equations and inequalities.
1. Start with the lower bound: 5 ≤ 2F + T.
- Substitute T = 5 - 2F into the upper bound inequality.
- Simplify the inequality: 5 ≤ 10 - 4F.
- Rearrange the inequality: 4F ≤ 5.
- Divide both sides by 4: F ≤ 5/4.
Since the number of field goals should be a whole number, the maximum value for F is 1. Therefore, F can be 0 or 1.
2. Now let's consider the upper bound: 2F + T ≤ 12.
- Substitute F = 1 (the maximum value we found above) into the inequality.
- Simplify the inequality: T ≤ 10.
To summarize the possibilities:
1. If F = 0, we have 0 field goals and T can be any value between 5 and 10.
2. If F = 1, we have 1 field goal and T can be any value between 5 and 10.
Therefore, in a single game, you could have made either 0 or 1 field goal, and the number of free throws could have ranged from 5 to 10.