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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___?
To solve this problem, we need to set up a system of equations. Let's define some variables:
Let x be the number of 2-point field goals.
Let y be the number of 3-point field goals.
Let z be the number of 1-point foul shots.
We know that the team made a total of 54 baskets. Since each basket is either a 2-point field goal, a 3-point field goal, or a 1-point foul shot, we can write the equation:
x + y + z = 54
We also know that the team scored a total of 95 points. Since each 2-point field goal is worth 2 points, each 3-point field goal is worth 3 points, and each 1-point foul shot is worth 1 point, we can write another equation:
2x + 3y + z = 95
Finally, we know that the team made 16 more 2-pointers than foul shots:
x = z + 16
Now we have a system of three equations. We can solve this system by substitution or elimination.
First, let's solve the third equation for z:
z = x - 16
Substitute this expression for z into the first equation:
x + y + (x - 16) = 54
Combine like terms:
2x + y = 70
Next, substitute both expressions for z and x into the second equation:
2(x) + 3y + (x - 16) = 95
Combine like terms:
3x + 3y = 111
Now we have a system of two equations:
2x + y = 70
3x + 3y = 111
We can solve this system by elimination.
Multiply the first equation by 3:
6x + 3y = 210
Now subtract this equation from the second equation:
(3x + 3y) - (6x + 3y) = 111 - 210
Simplify:
-3x = -99
Divide both sides by -3:
x = 33
Substitute this value for x into the first equation:
2(33) + y = 70
Simplify:
66 + y = 70
Subtract 66 from both sides:
y = 4
Finally, substitute the values of x and y into the third equation:
z = (33) - 16
Simplify:
z = 17
Therefore, the team made 17 1-point foul shots, 33 2-point field goals, and 4 3-point field goals.