What Is the Solution to the System of Equations? Y = 2x – 3.5 X – 2y = –14

Cell Telephone Plans

Alignments to Content Standards: 8.EE.C.8

Job

You are a representative for a prison cell phone visitor and information technology is your chore to promote different cell phone plans.

1. Your boss asks you to visually display three plans and compare them so yous can indicate out the advantages of each plan to your customers.

   • Plan A costs a bones fee of \$29.95 per month and x cents per text message
   • Program B costs a bones fee of \$90.20 per calendar month and has unlimited text letters
   • Program C costs a basic fee of \$49.95 per calendar month and 5 cents per text message
   • All plans offer unlimited calling
   • Calling on nights and weekends are costless
   • Long distance calls are included

2. A customer wants to know how to make up one's mind which programme volition save her the nigh money. Determine which programme has the everyman toll given the number of text messages a customer is probable to send.

IM Commentary

This task presents a real-world problem requiring the students to write linear equations to model different prison cell phone plans. Looking at the graphs of the lines in the context of the cell telephone plans allows the students to connect the pregnant of the intersection points of two lines with the simultaneous solution of two linear equations. The students are required to find the solution algebraically to complete the task. Note that the last three pieces of information describing the plans are superfluous; it is important for students to be able to sort through information and decide what is, and is non, relevant to solving the problem at manus.

This task was submitted past James East. Bialasik and Breean Martin for the first Illustrative Mathematics chore writing contest 2011/12/12-2011/12/18.

Solution

1. All 3 plans start with a basic monthly fee; in addition, the costs for Plans A and C increase at a steady charge per unit based on the number of text messages sent per month. Therefore, we can notice a linear equation for each plan relating $y$, the total monthly price in dollars, to $t$, the number of text messages sent.

   Plan A has a basic fee of \$29.95 even if no text messages are sent. In improver, each text message costs 10 cent or \$0.10. We tin write the total toll per calendar month as $$y = 29.95 + 0.10t$$

   Plan B has a bones fee of \$xc.20 even if no text messages are sent. In this instance the total cost per calendar month, $y$, does non change for different values of $t$, so we accept $$y = 90.20$$

   Plan C has a bones fee of \$49.95 fifty-fifty if no text messages are sent. In improver, each text message costs v cent or \$0.05. We can write the total cost per month as $$y = 49.95 + 0.05t$$

   To visually compare the three plans, we graph the three linear equations. In each case the basic fee is the vertical intercept, since information technology indicates the cost of a programme even if no text messages are being sent. The graph for the Plan B equation is a constant line at $y=90.twenty$. Plan B has a lower basic fee (\$29.95) than Plan A (\$49.95); therefore it starts lower on the vertical centrality. Finally, each text bulletin with Plan A costs more than than with Plan B, therefore, the slope of the line for Plan A is larger than the slope of the line for Plan B.

   

   From the graphical representation we run across that the “best” programme volition vary based on the number of text messages a person will send. For a pocket-size number of text messages, Plan A is the cheapest, for a medium number of text messages, Programme C is the cheapest and for a large number of text messages, Programme B is the cheapest.

   At an intersection bespeak of two lines, the two plans accuse the same amount for the same number of text messages. To determine the range of “small”, “medium” and “large” numbers of text letters, we demand to find the $t$-coordinate of the intersection points of the graphs. We can estimate that $t= 400$ is the cutoff betoken to get from Plan C to Program A, and $t=800$ is the cutoff point to go from Plan A to Programme B.

2. To observe the verbal coordinates of each intersection betoken, we need to solve the corresponding system of equations. The coordinates of these points correspond to the exact number of text messages for which two plans charge the same amount. Because nosotros are looking for the number of text messages, $t$, that issue in the aforementioned price for 2 different plans, we can set the expression that represents the price of one program equal to the other and solve for $t$.

   Plan A = Plan C
   $$ \begin{align} 0.1t + 29.95 &= .05t + 49.95 \\ .05t &= 20 \\ t &= 400 \quad \text{Text Letters} \finish{align} $$
   Plan C = Plan B
   $$ \begin{align} 0.05t + 49.95 &= 90.20 \\ 0.05t &= 40.25 \\ t &= 805 \quad \text{Text Messages} \end{align} $$

   We conclude that Programme A is the cheapest for customers sending 0 to 400 text messages per month, Plan C is cheapest for customers sending between 400 and 805 text messages per calendar month and programme B is cheapest for customers sending more than 805 text messages per month.

Cell Phone Plans

You lot are a representative for a cell phone company and it is your task to promote different cell phone plans.

1. Your dominate asks you to visually brandish three plans and compare them so y'all can point out the advantages of each plan to your customers.

   • Program A costs a bones fee of \$29.95 per month and 10 cents per text message
   • Plan B costs a basic fee of \$ninety.20 per calendar month and has unlimited text letters
   • Program C costs a bones fee of \$49.95 per month and five cents per text message
   • All plans offer unlimited calling
   • Calling on nights and weekends are free
   • Long altitude calls are included

2. A customer wants to know how to decide which plan volition salvage her the near money. Determine which plan has the lowest toll given the number of text messages a customer is likely to transport.

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Source: https://tasks.illustrativemathematics.org/content-standards/tasks/469

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