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A phone company offers two long-distance plans. The first charges a flat value of $\$ 5.00$ per month plus $\$ 0.99$ for each minute used, and the second charges $\$10.00$ a month plus $\$0.79$ for each minute used.



If $x$ represents the number of minutes used each month, which system of equations represents the total amount of money, $y$, you would spend for each long-distance plan?



a.)

$x=0.99 y+5$

$x=0.79 y+10$



b.)

$y=0.99 x+5$

$y=0.79 x+10$



c.)

$x=5 y+0.99$

$x=10 y+0.79$



d.)

$\begin{array}{l}

y=5 x+0.99 \\

y=10 x+0.79

\end{array}$

Verified Answer :

  • Plan 1 cost: \(y = 0.99x + 5\).
  • Plan 2 cost: \(y = 0.79x + 10\).
  • System of equations: \(\begin{cases} y = 0.99x + 5 \ y = 0.79x + 10 \end{cases}\).
  • Final answer: \(\boxed{\text{b}}\).

Explanation

  1. Analyzing the Cost Structures

To determine the system of equations, we must analyze the cost structure for each phone plan. The total monthly cost, denoted as \(y\), is composed of a fixed monthly fee and a variable charge based on the number of minutes used, denoted as \(x\).

For Plan 1, there is a flat monthly fee of \(\$5.00\) and a variable cost of \(\$0.99\) per minute. Since the variable cost is per minute, we multiply \(\$0.99\) by the number of minutes \(x\). The total cost \(y\) is the sum of these two:
\[y = 0.99x + 5.00\]

For Plan 2, there is a flat monthly fee of \(\$10.00\) and a variable cost of \(\$0.79\) per minute. Similarly, we multiply \(\$0.79\) by the number of minutes \(x\) and add the fixed fee:
\[y = 0.79x + 10.00\]

  1. Constructing the System of Equations

Combining the two individual cost equations into a system, we have:
\[ \begin{cases} y = 0.99x + 5 \ y = 0.79x + 10 \end{cases} \]
Comparing this result to the given options, we can see that it matches option b. This system correctly relates the total cost \(y\) to the usage duration \(x\) for each plan.

  1. Final Conclusion

The system of equations representing the total amount of money \(y\) for each long-distance plan is given by:
\[\boxed{\begin{cases} y = 0.99x + 5 \ y = 0.79x + 10 \end{cases}}\]
This corresponds to choice b.

Examples

Understanding systems of equations is fundamental in comparing different pricing models, such as choosing between utility providers, internet plans, or even comparing savings accounts with different interest structures. By setting up these linear equations, you can easily determine the "break-even point"—the exact number of minutes where both plans cost the same—which is crucial for making informed financial decisions. For instance, if you solve the system \(\begin{cases} y = 0.99x + 5 \ y = 0.79x + 10 \end{cases}\) by setting them equal, \(0.99x + 5 = 0.79x + 10\), you find \(0.20x = 5\), meaning at \(x = 25\) minutes, both plans cost exactly \(\$29.75\).

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