Bella finds a stadium that charges $8/hour to use the jogging track. She also finds a bike rental shop that charges $10/hour for a bike rental. She pays a total of $44 to train for the day. Jamar does a little bit better. He can use his high school's track for $4/hour and his uncle will rent him a bike for $5/hour. He pays a total of $22 to train for the day. If Bella and Jamar use the track they found for x hours and rent the bike they discover for y hours, how many hours does each one spend at the track?
please help i need to know the answer and if it has a solution no solutions or infinite solutions plz help its my last question
8x+ 10y =44
4x + 5y =22
Probably the best way to find the solution is to multiply the second equation by -2 because we are using the elimination (addition) method.
8x+ 10y =44
-8x -10y =-44
When you add these everything cancels and you get 0=0
Any time you get a true statement, the answer is any solution.
Any value of x will work.
Actually, if you graphed these lines, they would graph on top of each other.
If the answer was no solution. You would have come up with something like 0=5 which is false. In that case, your two lines would be parallel to each other and would never cross (intersect).
Huh? You say
If Bella and Jamar use the track they found for x hours and rent the bike they discover for y hours, how many hours does each one spend at the track?
Clearly, they spend x hours at the track.
Is that each? Together? And where does the cost come into play?
Very poorly worded question.
Let's break down the information given in the problem:
For Bella:
Jogging track cost: $8/hour
Bike rental cost: $10/hour
Total cost: $44
For Jamar:
Jogging track cost: $4/hour
Bike rental cost: $5/hour
Total cost: $22
We need to find the number of hours Bella and Jamar spend at the track, given by x.
Let's set up the equations based on the given information:
For Bella:
8x + 10y = 44 ----(1)
For Jamar:
4x + 5y = 22 ----(2)
To solve this system of equations, we can use the method of substitution or elimination. Let's use the method of substitution:
From equation (1), we can express y in terms of x:
y = (44 - 8x) / 10
Substituting this value of y into equation (2):
4x + 5((44 - 8x) / 10) = 22
Multiplying through by 10 to eliminate the fraction:
40x + 5(44 - 8x) = 220
40x + 220 - 40x = 220
220 = 220
The equation simplifies to 220 = 220, which means the equation is true for all values of x. This indicates that there are infinite solutions.
Therefore, there is no unique solution to the problem. The number of hours Bella and Jamar spend at the track can vary.
To solve this problem, we need to set up a system of equations based on the given information.
Let's assume Bella spends x hours using the jogging track and y hours renting a bike. According to the given information, the cost equation for Bella would be:
8x + 10y = 44 ---(Equation 1)
In the same way, let's assume Jamar spends x hours using the track and y hours renting a bike. The cost equation for Jamar would be:
4x + 5y = 22 ---(Equation 2)
Now, we can solve this system of equations to find the values of x and y.
To eliminate one variable, multiply Equation 1 by 2 and Equation 2 by -1:
16x + 20y = 88 ---(Equation 3)
-4x - 5y = -22 ---(Equation 4)
Adding Equation 3 and Equation 4 together:
(16x + 20y) + (-4x - 5y) = 88 + (-22)
12x + 15y = 66
Now, we can solve this equation for either of the variables. Let's solve it for x:
12x = 66 - 15y
x = (66 - 15y) / 12
Since we want to find the number of hours, x and y should be non-negative integers. We can try different values for y and see which ones yield integer values for x.
Let's start by checking y = 0:
x = (66 - 15(0)) / 12
x = 66 / 12
x = 5.5
Since x should be an integer, y = 0 is not a valid solution.
Let's try y = 1:
x = (66 - 15(1)) / 12
x = 51 / 12
x ≈ 4.25
Again, x is not an integer, which means y = 1 is not a valid solution.
Let's try y = 2:
x = (66 - 15(2)) / 12
x = 36 / 12
x = 3
Finally, we have found an integer solution. If Bella spends 3 hours on the track and 2 hours renting a bike, she will pay a total of $44.
Now that we have determined Bella's values, we can substitute these values into either Equation 1 or Equation 2 to find Jamar's values.
Let's use Equation 1:
8x + 10y = 44
8(3) + 10y = 44
24 + 10y = 44
10y = 44 - 24
10y = 20
y = 2
Therefore, Jamar spends 3 hours at the track, same as Bella, and also spends 2 hours renting a bike.
In summary, both Bella and Jamar spend 3 hours at the track.