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)
17,33,4
To determine how many shots of each kind were made, we can set up a system of equations based on the information provided.
Let's represent the number of 2-point field goals as "x", the number of 3-point field goals as "y", and the number of 1-point foul shots as "z".
From the given information, we have the following equations:
Equation 1: x + y + z = 54 (Total number of baskets made)
Equation 2: 2x + 3y + z = 95 (Total number of points scored)
Additionally, we know that the team made 16 more 2-pointers than foul shots:
Equation 3: x = z + 16
To solve this system of equations, we can use substitution or elimination method. Let's use substitution method here.
Step 1: Solve Equation 3 for x in terms of z:
x = z + 16
Step 2: Substitute this expression for x in Equation 1:
(z + 16) + y + z = 54
2z + y = 38 (Equation 4)
Step 3: Substitute the expression for x in Equation 2:
2(z + 16) + 3y + z = 95
3z + 3y = 63 (Equation 5)
Step 4: Solve the system of equations (Equation 4 and Equation 5) simultaneously to find the values of z and y.
Multiplying Equation 4 by 3 gives: 6z + 3y = 114
Subtracting Equation 5 from this gives: (6z + 3y) - (3z + 3y) = 114 - 63
3z = 51
z = 17
Substituting the value of z in Equation 4: 2(17) + y = 38
y = 4
Using Equation 1: x + 4 + 17 = 54
x = 33
Therefore, the team made 17 1-point foul shots, 33 2-point field goals, and 4 3-point field goals.
To summarize:
- The team made 17 1-point foul shots.
- The team made 33 2-point field goals.
- The team made 4 3-point field goals.