Question

Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

Hector and Joel are comparing the international calling plans on their cell phones. On his plan, Hector pays $3 just to place a call and $2 for each minute. When Joel makes an international call, he pays $5 to place the call and $1 for each minute. A call of a certain duration would cost exactly the same under both plans. What is the cost? What is the duration?


Under each plan, a call would cost what if it were how many minutes in duration.

Answer 1

To set up a system of equations for Hector's and Joel's calling plans, let \( x \) represent the number of minutes of the call.

**Hector's cost** for a call can be expressed by the equation:
\[
H = 3 + 2x
\]
where \( H \) is the total cost of the call for Hector.

**Joel's cost** for a call can be expressed by the equation:
\[
J = 5 + x
\]
where \( J \) is the total cost of the call for Joel.

Now, we want to find the duration for which their costs are equal, so we set the two equations equal to each other:
\[
3 + 2x = 5 + x
\]

Next, we can solve this equation step-by-step:

1. Subtract \( x \) from both sides:
\[
3 + 2x - x = 5
\]
which simplifies to:
\[
3 + x = 5
\]

2. Subtract 3 from both sides:
\[
x = 5 - 3
\]
which gives us:
\[
x = 2
\]

Now that we've found \( x \), we can plug this value back into either of the cost equations to find the total cost of the call.

Let's use Hector's equation:
\[
H = 3 + 2(2) = 3 + 4 = 7
\]

Now we can summarize our results:
- The cost of a call for both plans, when lasting 2 minutes, is **$7**.
- The duration of the call that costs the same under both plans is **2 minutes**.

Filling in the blanks:
Under each plan, a call would cost **$7** if it were **2** minutes in duration.