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 53 baskets and 13 more 2-pointers than foul shots. How many shots of each kind were made?
We know that a minimum of 14 2-pointers have to be made to have at least 1-point shot. So we can calculate the total points with the 14 2-pointers (hint: 14x2).
We know that 53 baskets were made and out of that at least 14 were 2 pointers and 1 was one pointer. That leaves us with 38 baskets and the remaining number of points. I think that should get you started!
f = foul shots , d = two pointers , t = three pointers
d = f + 13
f + 2 d + 3 t = 95 ... substituting ... f + 2 f + 26 + 3 t = 95 ... 3 f + 3 t = 69
f + d + t = 53 ... substituting ... f + f + 13 + t = 53 ... 2 f + t = 40
solve the system
To solve this problem, we can set up a system of equations. Let's denote 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".
Based on the given information, we can form the following equations:
1) x + y + z = 53 (The total number of baskets made is equal to 53)
2) 2x + 3y + z = 95 (The total number of points scored is equal to 95)
3) x = z + 13 (The number of 2-point field goals is 13 more than the number of foul shots)
Now, let's solve the system of equations:
From equation 3, we can substitute x in equation 1 with (z + 13):
(z + 13) + y + z = 53
2z + y + 13 = 53
2z + y = 40
Rearranging equation 2 to solve for x gives us:
2x = 95 - 3y - z
x = (95 - 3y - z)/2
Substituting the value of x from equation 3 into the rearranged equation 2:
(95 - 3y - z)/2 = z + 13
Multiplying both sides by 2 to eliminate the fraction:
95 - 3y - z = 2z + 26
Rearranging the equation:
3y + 3z = 69
Now, we have a system of equations:
2z + y = 40
3y + 3z = 69
Dividing the second equation by 3, we get:
y + z = 23
Solving the above equation simultaneously with the first equation:
y + z = 23 ---(4)
2z + y = 40 ---(5)
Subtracting equation (4) from equation (5):
(2z + y) - (y + z) = 40 - 23
2z + y - y - z = 17
z = 17
Substituting the value of z back into equation (4):
y + 17 = 23
y = 6
Now, substituting the values of y and z into equation (3):
x = z + 13
x = 17 + 13
x = 30
Therefore, the number of shots made of each kind are as follows:
2-point field goals: 30
3-point field goals: 6
1-point foul shots: 17