Juan is buying flowers for his mother. He has $18 to spend and sees that roses are $3 each and carnations are $1.50 each. He wants to buy 3 times as many carnations as roses and plans to spend all of his money on flowers. Write a system of equations to model this situation. Is there a viable solution that meets Juans conditions? Explain.
what i dont understand I dont even know how to do this
3r + 1.5c = 18
r = 2c
3(2c) + 1.5c = 18
Solve for c, then insert in first equation to find r.
PsyDAG, wouldn't it be r=3c since it's he wants to buy 3 times as many carnations as roses?
You have to look at the problem carefully. It says "Is there a VIABLE solution?" This question is asking whether or not there is a solution possible. 3r + 1.5c = 18 doesn't work out. I have no idea how PsyDAG is able to solve it, considering the answer would be a decimal number. So in conclusion I don't think there is a viable answer.
Sure, here's a system of equations to model this situation:
Let x be the number of roses bought.
Let y be the number of carnations bought.
Equation 1: 3x + 1.5y = 18 (total cost of flowers should be $18)
Equation 2: y = 3x (he wants to buy 3 times as many carnations as roses)
To check if there is a viable solution that meets Juan's conditions, we can solve the system of equations.
Substitute Equation 2 into Equation 1:
3x + 1.5(3x) = 18
3x + 4.5x = 18
7.5x = 18
x = 2.4
Since the number of roses has to be a whole number, there is no viable solution that meets Juan's conditions. Therefore, he would need to reconsider his plan in order to spend all of his money. Perhaps he can adjust the quantities or prices of the flowers he desires.
To model this situation, we can use the following system of equations:
Let's assume that Juan buys x roses and y carnations.
Equation 1: 3y = x
This equation represents Juan's condition of wanting to buy 3 times as many carnations as roses.
Equation 2: 3x + 1.5y = 18
This equation represents the total cost of the flowers, where the cost of each rose is $3 and each carnation is $1.50. The total cost should be equal to the $18 that Juan has to spend.
Now, let's determine if there is a viable solution that meets Juan's conditions.
Substituting Equation 1 into Equation 2, we get:
3(3y) + 1.5y = 18
9y + 1.5y = 18
10.5y = 18
y ≈ 1.71
Since the number of flowers cannot be a fraction, we need to round up or down. Let's round y to the nearest whole number:
If we round y down to 1, then x would be 3 (using Equation 1).
However, if we round y up to 2, then x would be 6 according to Equation 1.
Let's calculate the total cost in each case:
Case 1 (rounded down):
3x + 1.5y = 3(6) + 1.5(1) = 18. This meets Juan's condition.
Case 2 (rounded up):
3x + 1.5y = 3(3) + 1.5(2) = 13.5. This does not meet Juan's condition.
Therefore, the only viable solution that meets Juan's conditions is when he buys 6 roses and 2 carnations.