In a basketball tournament, the Titans defeated the Panthers by a score of 65 to 53. The Titans won by scoring a combination of one-point free throws, two-point and three-point field goals. The number of free throws was one less than the number of three-point field goals and 7 less than the number of two-point goals. Find the combination of scores that the Titans scored to win the tournament.
f = foul shots , d = two pointers , t = three pointers
f + 2d + 3t = 65
t = f + 1
d = f + 7
substituting ... f + 2f + 14 + 3f + 3 = 65
...solve for f , then substitute back to find t and d
Well, well, well, looks like it's game time for the Titans and the Panthers! Let's break down the score like a magician breaking down a trick.
So, we know the Titans won with a score of 65 points, defeating the Panthers who only managed 53 points. Impressive, Titans!
Now, let's assume the Titans scored x one-point free throws, y two-point field goals, and z three-point field goals. And here's where the real fun begins!
According to the information provided, we know that:
- The number of free throws was one less than the number of three-point field goals, so x = (z - 1).
- The number of free throws was 7 less than the number of two-point field goals, so x = (y - 7).
Now let's put on our mathematician hats and solve this puzzle:
Since x = (z - 1) and x = (y - 7), we can substitute these expressions into the equation 65 = (x * 1) + (y * 2) + (z * 3):
65 = [(z - 1) * 1] + [(y - 7) * 2] + (z * 3).
Now, let's simplify this equation step by step:
65 = z - 1 + 2y - 14 + 3z
= 4z + 2y - 15.
Next, let's gather all the like terms:
4z + 2y = 65 + 15
= 80.
Using this equation, we can find different combinations of z and y that satisfy the conditions.
For example, if we let z = 10 and y = 35, then 4z + 2y = 80. Awesome three-pointers and two-pointers!
Therefore, the Titans scored 10 three-point field goals, 35 two-point field goals, and 9 one-point free throws. What a winning combination!
Keep shootin' hoops, Titans!
Let's assign variables to represent the number of one-point free throws, two-point field goals, and three-point field goals scored by the Titans.
Let x represent the number of one-point free throws.
Let y represent the number of two-point field goals.
Let z represent the number of three-point field goals.
From the given information, we can make the following equations:
Equation 1: x = z - 1 (The number of free throws was one less than the number of three-point field goals)
Equation 2: x + y + z = 65 (The total number of points scored by the Titans)
Equation 3: y = z + 7 (The number of free throws was 7 less than the number of two-point goals)
We have a system of three equations with three variables. Let's solve this system to find the values of x, y, and z.
From Equation 1, we can express z in terms of x: z = x + 1.
Now, let's substitute z = x + 1 in Equation 3:
y = (x + 1) + 7
y = x + 8
Substitute z = x + 1 and y = x + 8 in Equation 2:
x + (x + 8) + (x + 1) = 65
3x + 9 = 65
3x = 56
x = 56/3
x ≈ 18.67
Since x represents the number of one-point free throws, which cannot be a decimal value, we need to find the nearest whole number. Considering x is less than 19, x = 18 is the most reasonable approximation.
Substitute x = 18 in Equation 1:
18 = z - 1
z = 19
Substitute x = 18 in Equation 3:
y = 18 + 8
y = 26
Therefore, the combination of scores that the Titans scored to win the tournament is:
- One-point free throws: 18
- Two-point field goals: 26
- Three-point field goals: 19
To find the combination of scores that the Titans scored to win the tournament, we can set up a system of equations based on the given information.
Let's assign variables to each unknown value:
Let F = number of free throws
Let T = number of three-point field goals
Let P = number of two-point field goals
Based on the information given, we know that:
1) The Titans defeated the Panthers by a score of 65 to 53, so the total score of the Titans is 65. Therefore, the equation is: F + 3T + 2P = 65.
2) The number of free throws was one less than the number of three-point field goals. So, the equation is: F = T - 1.
3) The number of free throws was 7 less than the number of two-point goals. So, the equation is: F = P - 7.
We now have a system of three equations:
F + 3T + 2P = 65
F = T - 1
F = P - 7
To solve this system of equations, we can substitute the value of F from equation 2) into equations 1) and 3).
From equation 2), F = T - 1, we substitute into equation 1):
(T - 1) + 3T + 2P = 65
4T + 2P = 66 (Equation 4)
From equation 2), F = T - 1, we substitute into equation 3):
(T - 1) = P - 7
T - P = 6 (Equation 5)
We now have a system of two equations with two variables:
4T + 2P = 66 (Equation 4)
T - P = 6 (Equation 5)
Now we can solve the system of equations 4) and 5) simultaneously.
Multiplying equation 5) by 2, we get:
2(T - P) = 2(6)
2T - 2P = 12 (Equation 6)
Adding equation 6) and equation 4), we can eliminate the P term:
4T + 2T - 2P + 2P = 66 + 12
6T = 78
T = 13
Substituting T = 13 into equation 5):
13 - P = 6
P = 13 - 6
P = 7
Substituting T = 13 and P = 7 into equation 2):
F = T - 1
F = 13 - 1
F = 12
So, the combination of scores that the Titans scored to win the tournament is:
Free throws: 12
Three-point field goals: 13
Two-point field goals: 7