In a basketball game, the star player scored a total of 35 points. In basketball, it is possible to make a 3 point basket, a 2 point basket, or a 1 point free throw. He made as many 2 pointers as 3 pointers and free throws combined. He scored 1 point more with 2 pointers than he did with 3 pointers and free throws combined. How many of each type of score did he make?
If there are x 1-pters, y 2-pters and z 3-ptrs, then
x+2y+3z = 35
y = x+z
2y = 3z+x+1
Now crank 'em out
A basketball player only scores two-point and three-point goals during a game. The basketball player scores a total of 25 points with a total of 11 goals during the game.
How many two-point goals does the basketball player score during the game?
Well, it seems like this basketball star isn't just shooting hoops, he's shooting for math problems too! Let's break it down and see if we can score an answer.
Let's say the star player made x number of 3-pointers. That means he scored 3x points from those shots.
Now, he made just as many 2-pointers as 3-pointers and free throws combined. So, the number of 2-pointers he made would be the same as the total number of 3-pointers and free throws, which is 2x.
He also scored 1 point more with 2-pointers than he did with 3-pointers and free throws combined. So, the total points from 3-pointers and free throws would be 3x. Therefore, the total points from 2-pointers would be 3x + 1.
If we add up all his points, we know he scored a total of 35. So, let's add up the points: 3x + 2x + 3x + 1 = 35.
Combining like terms, we have 8x + 1 = 35.
Now, let's subtract 1 from both sides to isolate 8x: 8x = 34.
Lastly, divide both sides by 8: x = 4.25.
Uh-oh, it seems like we've hit a buzzer-beater here. The value of x, which represents the number of 3-pointers, should be a whole number, not a decimal. So, it looks like there might be an error in the problem statement. Let's give it another shot next time!
To solve this problem, we can use a system of equations.
Let's assume the number of 3-pointers the player made is X, the number of 2-pointers is Y, and the number of free throws is Z.
We are given three pieces of information:
1. The star player scored a total of 35 points:
3X + 2Y + 1Z = 35
2. He made as many 2-pointers as 3-pointers and free throws combined:
Y = X + Z
3. He scored 1 point more with 2-pointers than he did with 3-pointers and free throws combined:
2Y = X + Z + 1
Now, we can solve these equations simultaneously.
First, let's simplify equation 2:
Y = X + Z
Substitute this into equation 3:
2(X + Z) = X + Z + 1
2X + 2Z = X + Z + 1
2X - X + 2Z - Z = 1
X + Z = 1 -----> Equation 4
Now we have two equations:
3X + 2Y + 1Z = 35 -----> Equation 1
X + Z = 1 -----> Equation 4
To solve this system of equations, we can use substitution or elimination method.
Let's use substitution method:
Rearrange Equation 4 to solve for X:
X = 1 - Z
Substitute this into Equation 1:
3(1 - Z) + 2Y + 1Z = 35
3 - 3Z + 2Y + Z = 35
-2Z + 2Y = 32
Z - Y = -16 -----> Equation 5
Now we have two equations:
Z - Y = -16 -----> Equation 5
X + Z = 1 -----> Equation 4
To eliminate Y, we can add Equation 5 to Equation 4:
(Z - Y) + (X + Z) = -16 + 1
X + 2Z - Y = -15 -----> Equation 6
Now, we have two equations:
X + 2Z - Y = -15 -----> Equation 6
X + Z = 1 -----> Equation 4
We can add Equation 6 to Equation 4 to eliminate X:
(X + Z) + (X + 2Z - Y) = 1 + (-15)
2X + 3Z - Y = -14 -----> Equation 7
Now, we have two equations:
X + Z = 1 -----> Equation 4
2X + 3Z - Y = -14 -----> Equation 7
To eliminate X, we can multiply Equation 4 by -2 and add it to Equation 7:
(-2)(X + Z) + (2X + 3Z - Y) = (-2)(1) + (-14)
-2X - 2Z + 2X + 3Z - Y = -2 - 14
Z - Y = -16 (same as Equation 5)
Since Equation 5 and Equation 7 are identical, we can conclude that there are infinitely many solutions to the system of equations. The problem states that the star player made as many 2-pointers as 3-pointers and free throws combined, but it does not give us specific values for X, Y, and Z.
Therefore, we cannot determine how many of each type of score the player made with the given information.