A basketball team recently scored a total of 95 points on a combination of 2-point field goals, 3-point field goals, and 1-point foul shots. Altogether, the team made 54 baskets and 16 more 2-pointers than foul shots. How many shots of each kind were made?
How many 1-point foul shots did the team make__?
How many 2-point field goals did the team make___?
How many 3-point field goals did the team make___?
If the 1,2,3 pointers are in amounts of x,y,z, then
x+2y+3z = 95
x+y+z = 54
y = x+16
(x,y,z) = (17,33,4)
Thank you!!
A basketball team recently scored a total of 99 points on a combination of 2-point field goals, 3-point field goals, and 1-point foul shots. Altogether, the team made 54 baskets and 18 more 2-pointers than foul shots. How many shots of each kind were made?
How many 1-point foul shots did the team make?
How many 2-point field goals did the team make?
How many 3-point field goals did the team make?
To determine the number of different kinds of shots made by the basketball team, we need to set up a system of equations based on the given information.
Let's use the variables x, y, and z to represent the number of 1-point foul shots, 2-point field goals, and 3-point field goals made respectively.
From the given information, we have the following equations:
Equation 1: x + y + z = 54 (total number of baskets made by the team)
Equation 2: x = y + 16 (16 more 2-point field goals than foul shots made)
We also know that each 1-point foul shot contributes 1 point, each 2-point field goal contributes 2 points, and each 3-point field goal contributes 3 points to the team's total score of 95 points.
Equation 3: 1x + 2y + 3z = 95 (total points scored by the team)
We can now solve this system of equations to find the values of x, y, and z.
Step 1: Solve Equation 2 for x in terms of y.
x = y + 16
Step 2: Substitute the value of x from Step 1 into Equation 1.
(y + 16) + y + z = 54
Simplifying Equation 1:
2y + z + 16 = 54
2y + z = 38
Step 3: Substitute the values of x and z from Steps 1 and 2 into Equation 3.
1(y + 16) + 2y + 3z = 95
Simplifying Equation 3:
y + 16 + 2y + 3z = 95
3y + 3z = 79
y + z = 26
Step 4: Solve the system of equations from Steps 2 and 3.
2y + z = 38
y + z = 26
By subtracting the second equation from the first equation, we can eliminate z.
(2y + z) - (y + z) = 38 - 26
y = 12
Step 5: Substitute the value of y from Step 4 into Equation 1.
x + y + z = 54
x + 12 + z = 54
x + z = 42
From Step 5, we can determine the value of z.
x + z = 42
x = 42 - z
Now, we can try different values of z that satisfy the equation until we find the one that satisfies the conditions of the problem.
If z = 9, then x would be 42 - 9 = 33.
If z = 8, then x would be 42 - 8 = 34.
However, we are looking for a combination where x and y are both positive values. Since a team cannot score a negative number of foul shots or field goals, we need to keep trying different values of z.
If z = 7, then x would be 42 - 7 = 35. This satisfies the condition of all values being positive.
So, the team made:
- 35 1-point foul shots
- 35 + 16 = 51 2-point field goals
- 7 3-point field goals
Therefore, the team made 35 1-point foul shots, 51 2-point field goals, and 7 3-point field goals.