Joelle currently has 18 rocks in her collection and gains 4 each week. Lewis currently has 30 rocks in his collection and gains 3 each week. Set up a system of equations to show how many rocks each has in their collection. (6 points)
a) Joelle has y = _ x + _
b) Lewis has y = _ x + _
c) After how many weeks will Joelle and Lewis have the same number of rocks in their collections?
_ weeks
d) How many rocks will Joelle and Lewis have when the amount of rocks in their collection is equal? They will each have _ rocks in their collection.
a) Joelle has y = 18 + 4x
- The initial number of rocks in Joelle's collection is 18.
- Joelle gains 4 rocks each week.
b) Lewis has y = 30 + 3x
- The initial number of rocks in Lewis' collection is 30.
- Lewis gains 3 rocks each week.
c) To find the number of weeks when Joelle and Lewis have the same number of rocks in their collections, we can set their equations equal to each other and solve for x:
18 + 4x = 30 + 3x
d) To find the number of rocks Joelle and Lewis will have when their collections are equal, we can substitute the value of x obtained in part c) into one of the equations and solve for y.
a) Joelle has y = 18 + 4x
(The variable x represents the number of weeks, and the equation shows that Joelle starts with 18 rocks and gains 4 rocks per week.)
b) Lewis has y = 30 + 3x
(The variable x represents the number of weeks, and the equation shows that Lewis starts with 30 rocks and gains 3 rocks per week.)
c) To find the number of weeks it will take for Joelle and Lewis to have the same number of rocks in their collections, we need to set their equations equal to each other and solve for x:
18 + 4x = 30 + 3x
Solving this equation, we find:
x = 12 weeks
Therefore, after 12 weeks, Joelle and Lewis will have the same number of rocks in their collections.
d) To find the number of rocks in their collections when they have an equal amount, we can substitute the value of x=12 into one of the equations:
Joelle: y = 18 + 4(12) = 18 + 48 = 66 rocks
Lewis: y = 30 + 3(12) = 30 + 36 = 66 rocks
Therefore, when they have an equal amount of rocks, Joelle and Lewis will each have 66 rocks in their collections.
To set up a system of equations to represent the number of rocks in Joelle and Lewis's collections, we need to define some variables.
Let's say x represents the number of weeks that have passed.
For Joelle's collection:
- She currently has 18 rocks, so her starting point is y = 18.
- She gains 4 rocks each week, so the rate of change is +4 (or 4 additional rocks each week). Therefore, the equation for Joelle's collection is y = 4x + 18.
For Lewis's collection:
- He currently has 30 rocks, so his starting point is y = 30.
- He gains 3 rocks each week, so the rate of change is +3 (or 3 additional rocks each week). Therefore, the equation for Lewis's collection is y = 3x + 30.
Now let's answer the remaining questions using this system of equations:
c) To find the number of weeks when Joelle and Lewis have the same number of rocks, we need to find the value of x where their respective equations intersect. We can do this by setting the equations equal to each other and solving for x:
4x + 18 = 3x + 30
Simplifying the equation, we get:
x = 12
Therefore, after 12 weeks, Joelle and Lewis will have the same number of rocks in their collections.
d) To find the number of rocks Joelle and Lewis will have when their collections are equal, we substitute x = 12 into either equation. Let's use Joelle's equation:
y = 4x + 18
y = 4(12) + 18
y = 48 + 18
y = 66
Therefore, when the amount of rocks in their collections is equal, Joelle and Lewis will each have 66 rocks in their collections.