Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. Dalton, an office manager, needs to find a courier to deliver a package... Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. Dalton, an office manager, needs to find a courier to deliver a package. The first courier charges a fee of $11 plus $5 per kilogram. The second charges $12 plus $4 per kilogram. Dalton determines that, given his package's weight, the two courier services are equivalent in terms of cost. What is the weight? How much will it cost? At a package weight of ___ kilograms, the two couriers both cost $___.
Understand the Problem
The question is asking to set up a system of equations based on the costs of two different courier services depending on a package's weight. We need to find the weight at which their costs are equivalent and then calculate that cost.
Answer
At a package weight of 1 kilograms, the two couriers both cost $16.
Answer for screen readers
At a package weight of 1 kilograms, the two couriers both cost $16.
Steps to Solve
- Define the cost equations
For the first courier:
The cost, $C_1$, can be defined as:
$$ C_1 = 11 + 5w $$
where ( w ) is the weight of the package in kilograms.
For the second courier:
The cost, $C_2$, can be defined as:
$$ C_2 = 12 + 4w $$
- Set the equations equal to each other
Since the costs are equivalent, we set the two cost equations equal:
$$ 11 + 5w = 12 + 4w $$
- Solve for the weight ( w )
Start by isolating ( w ). Subtract ( 4w ) from both sides:
$$ 11 + 5w - 4w = 12 $$
This simplifies to:
$$ 11 + w = 12 $$
Now, subtract ( 11 ) from both sides:
$$ w = 12 - 11 $$
Simplifying gives:
$$ w = 1 $$
- Calculate the cost at this weight
Now substitute ( w = 1 ) into either cost equation to find the cost. Using the first courier's cost equation:
$$ C_1 = 11 + 5(1) $$
This simplifies to:
$$ C_1 = 11 + 5 = 16 $$
At a package weight of 1 kilograms, the two couriers both cost $16.
More Information
Dalton will pay the same amount whether he uses the first or the second courier at a weight of 1 kilogram. This scenario illustrates how to set up and solve a system of equations based on linear cost relationships.
Tips
- Forgetting to set the cost equations equal to each other.
- Miscalculating when isolating ( w ).
AI-generated content may contain errors. Please verify critical information
