To create the system, write an equation for each kennel. In order to solve the system, write an equation which the cost of Alpha Kennel is equal to the cost of Beta Kennel.
System for One Dog:C_A=30n+75 C_B=34.95n Solution for One Dog: 15.2 days, see solution. System for Two Dogs:C_(A2)=54n+135 C_(B2)=69.90n Solution for Two Dogs: 8.5 days, see solution.
Let C_A and C_B be the costs of the Alpha Kennel and Beta Kennel, respectively, for n days.
Alpha Kennel
Beta Kennel
Verbal Expression
Algebraic Expression
Verbal Expression
Algebraic Expression
Initial fee
$75
Initial fee
$0
Daily rate
$30
Daily rate
$34.95
Charge for n days
$30* n
Charge for n days
$34.95* n
Add the initial fee and daily rate
$30* n+ $75
Add the initial fee and daily rate
$34.95* n+ $0
Form the equation
C_A= $30* n+ $75
Form the equation
C_B= $34.95* n
Thus, we have the system of linear equations.
C_A=30n+75 & (I) C_B=34.95n & (II)
Now, in order to solve the system we will create an equation which C_A is equal to C_B.
C_A&=C_B
30n+75&=34.95n
Let's solve it for n.
The solution to the linear system is the point where boarding a dog at either kennel costs the same amount. If the family was going away for 15 days, they would want to use Beta Kennel. If they were going away for longer, they would want to use Alpha Kennel.
Solution for Two Dogs
This time, we will write the system of linear equations for two dogs. Let C_(A2) and C_(B2) be the costs of the Alpha Kennel and Beta Kennel, respectively, for two dogs.
Alpha Kennel
Beta Kennel
Verbal Expression
Algebraic Expression
Verbal Expression
Algebraic Expression
Initial fee for two dogs
$150
Initial fee for two dogs
$0
Charge of two dogs for n days
$60* n
Charge of two dogs for n days
$69.90* n
Add the initial fee and daily rate
$60* n+ $150
Add the initial fee and daily rate
$69.90* n+ $0
Charge after discount
$60* n+ $150-0.1( $60* n+ $150)
Charge after discount
$69.90* n+ $0
Form the equation
C_(A2)= $54* n+ $135
Form the equation
C_(B2)= $69.90* n
Therefore, we can write the new system as the following.
C_(A2)=54n+135 & (I) C_(B2)=69.90n & (II)
Let's solve the system in the same way as we did in the previous part.
The solution to the new linear system is the point where boarding two dogs at either kennel costs the same amount. If the family was going away for 8 days, they would want to use Beta Kennel. If they were going away for a more extended period of time, they would want to use Alpha Kennel.
