Joe currently has 18 rocks in her collection and gains for each week. Louis currently has 30 rocks in his collection and gains three each week. Set up a system of equations to show how many rocks each has in their collection after how many weeks with Joel and Lewis have the same number of rocks in their collection
Let's say that after x weeks, Joe will have y rocks in her collection.
Similarly, after x weeks, Louis will have z rocks in his collection.
Given that Joe currently has 18 rocks in her collection, and she gains 2 rocks every week, the equation representing Joe's collection after x weeks would be y = 2x + 18.
Also, given that Louis currently has 30 rocks in his collection, and he gains 3 rocks every week, the equation representing Louis's collection after x weeks would be z = 3x + 30.
In order for Joe and Louis to have the same number of rocks in their collections after x weeks, the equation would be: y = z.
By substituting the values from the given equations, we have:
2x + 18 = 3x + 30.
Let's denote the number of weeks as "w". To set up a system of equations, we can use the number of rocks each person has after "w" weeks.
Joe's equation:
Number of rocks for Joe = 18 + w
Louis's equation:
Number of rocks for Louis = 30 + 3w
To find the number of weeks at which Joe and Louis have the same number of rocks in their collection, we can set the two equations equal to each other:
18 + w = 30 + 3w
Thus, the system of equations is:
18 + w = 30 + 3w
Let's assign variables to represent the unknowns in this problem.
Let's say the number of weeks is represented by "w" (a positive whole number).
Now, let's establish the equations to represent the number of rocks each person has in their collection after some number of weeks.
Let's use "J" to represent the number of rocks in Joe's collection, and "L" to represent the number of rocks in Louis' collection.
After a certain number of weeks:
Joe's collection: J = 18 + w
Louis' collection: L = 30 + 3w
Now, if we want to find the number of weeks it takes for Joe and Louis to have the same number of rocks in their collection, we can set their respective equations equal to each other.
So, we have the equation:
18 + w = 30 + 3w
Simplifying this equation, we get:
18 = 30 + 2w
Subtracting 30 from both sides:
-12 = 2w
Dividing both sides by 2:
w = -6
It seems that the number of weeks, "w," is negative, which does not make sense in this context. Therefore, it appears that Joe and Louis will never have the same number of rocks in their collection based on the given information.