In a basketball game, Garrett and Logan scored a total of 19 points. Logan and Ray scored a total of 14 points. Logan scored as many points as Garrett and Ray together. How many points did each player score?
Logan______ Garrett______ Ray_____
To solve this problem, let's assign variables to each player's number of points. Let L represent Logan's points, G represent Garrett's points, and R represent Ray's points.
We are given that Garrett and Logan scored a total of 19 points, so we can write the equation: G + L = 19.
We are also given that Logan and Ray scored a total of 14 points, so we can write the equation: L + R = 14.
Finally, we are given that Logan scored as many points as Garrett and Ray together, which means L = G + R.
Now, we have a system of three equations:
G + L = 19 (Equation 1)
L + R = 14 (Equation 2)
L = G + R (Equation 3)
We need to solve this system of equations to find the values of L, G, and R.
One way to solve this is by substitution:
From Equation 3, we can rewrite it as G = L - R.
Substitute this expression for G in Equation 1:
(L - R) + L = 19.
Simplify the equation: 2L - R = 19 (Equation 4).
Now, we have a system of two equations:
2L - R = 19 (Equation 4)
L + R = 14 (Equation 2)
We can solve this system using various methods such as substitution, elimination, or graphing.
Let's solve it using the substitution method:
R = 14 - L (Equation 2)
Substitute R in Equation 4:
2L - (14 - L) = 19.
Simplify: 2L - 14 + L = 19.
Combine like terms: 3L - 14 = 19.
Add 14 to both sides: 3L = 33.
Divide by 3: L = 11.
Now that we have found the value of L, we can substitute it back into Equation 2 to find R:
11 + R = 14.
Subtract 11 from both sides: R = 3.
Finally, substitute L = 11 and R = 3 into Equation 3 to find G:
G = L - R = 11 - 3 = 8.
Therefore, Logan scored 11 points, Garrett scored 8 points, and Ray scored 3 points.
Logan: 11
Garrett: 8
Ray: 3