A baseball player's batting average is the number of hits the player has divided by the number of at-bats. At the beginning of a game, a player has a batting average of .360. During the game, the player gets 3 hits during 5 at-bats, and his batting average changes to .375.
a. Write a system of linear equations that represents the situation.
b. How many at-bats has the player had so far this season?
c. Another player on the team has a batting average of .240 at the beginning of the same game. During the game, he gets 3 hits during 5 at-bats, and his batting average changes to .300. Has this player had more at-bats so far this season than the other player? Explain.
Help explain this. Can’t figure out
a. To represent the situation with a system of linear equations, we need to define the variables and the relationships between them:
Let's say the number of hits the player has before the game is x and the number of at-bats is y.
From the given information, we know that the player's batting average at the beginning of the game is .360. This means that x divided by y is equal to .360:
x/y = .360
During the game, the player gets 3 hits during 5 at-bats, so we can update the values:
x + 3 hits = new total hits
y + 5 at-bats = new total at-bats
The player’s batting average changes to .375, so we can write another equation using the updated values:
(new total hits) / (new total at-bats) = .375
b. To find out how many at-bats the player has had so far this season, we need to solve the system of linear equations. Let's use the substitution method.
From equation 1 (x/y = .360), we can rewrite x in terms of y:
x = .360y
Now we can substitute this expression for x in equation 2:
(.360y + 3) / (y + 5) = .375
We can then solve this equation to find the value of y, which represents the number of at-bats:
(.360y + 3) = .375(y + 5)
.360y + 3 = .375y + 1.875
.375y - .360y = 1.875 - 3
.015y = -1.125
y = -1.125 / .015
y = 75
Therefore, the player has had 75 at-bats so far this season.
c. To determine if the other player has had more at-bats so far this season, we need to compare their total at-bats.
The other player's batting average at the beginning of the game is .240. We can set up a similar equation as before:
z / w = .240
During the game, the player gets 3 hits during 5 at-bats, so we can update the values:
z + 3 hits = new total hits
w + 5 at-bats = new total at-bats
The player’s batting average changes to .300, so we can write another equation using the updated values:
(new total hits) / (new total at-bats) = .300
Now we can compare the values of y and w to determine if the other player has had more at-bats. If w is greater than y, then the other player has had more at-bats so far this season.