Solve this application problem using a system of equations: In a basketball game, Alan scored 30 points, consisting only of three-point shots and two-point shots. He made a total of 11 shots. How many shots of each type did he make?
e + w = 11 ... 2 e + 2 w = 22
3 e + 2 w = 30
subtracting equations ... e = 8
To solve this application problem using a system of equations, we need to assign variables to the unknowns in the problem. Let's denote the number of three-point shots as "x" and the number of two-point shots as "y".
Now, let's create a system of equations based on the given information:
Equation 1: The total points scored by Alan is 30. Since a three-point shot earns 3 points and a two-point shot earns 2 points, the equation becomes:
3x + 2y = 30
Equation 2: The total number of shots made by Alan is 11, which means the sum of the three-point and two-point shots is equal to 11:
x + y = 11
Now, we have a system of equations:
3x + 2y = 30
x + y = 11
To solve this system, we can use the method of substitution or elimination. Let's use the method of substitution:
From Equation 2, we can solve for x: x = 11 - y
Now, substitute this value of x into Equation 1:
3(11 - y) + 2y = 30
33 - 3y + 2y = 30
33 - y = 30
-y = 30 - 33
-y = -3
y = 3
Substitute the value of y back into Equation 2 to find x:
x + 3 = 11
x = 11 - 3
x = 8
Therefore, Alan made 8 three-point shots and 3 two-point shots in the basketball game.