What Is the Solution to the Linear Equation? –12 + 3b – 1 = –5 – B
Prison cell Telephone Plans
Alignments to Content Standards: eight.EE.C.8
Task
Y'all are a representative for a cell phone company and it is your job to promote different cell phone plans.
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Your boss asks you to visually display three plans and compare them and so you tin can signal out the advantages of each plan to your customers.
- Plan A costs a basic fee of \$29.95 per month and ten cents per text message
- Plan B costs a basic fee of \$ninety.xx per month and has unlimited text messages
- Plan C costs a basic fee of \$49.95 per calendar month and 5 cents per text bulletin
- All plans offering unlimited calling
- Calling on nights and weekends are free
- Long distance calls are included
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A customer wants to know how to decide which programme will save her the nigh money. Determine which plan has the everyman price given the number of text messages a customer is likely to send.
IM Commentary
This task presents a real-world problem requiring the students to write linear equations to model different cell phone plans. Looking at the graphs of the lines in the context of the jail cell telephone plans allows the students to connect the meaning of the intersection points of two lines with the simultaneous solution of two linear equations. The students are required to discover the solution algebraically to complete the chore. Note that the last 3 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 not, relevant to solving the problem at manus.
This task was submitted by James East. Bialasik and Breean Martin for the first Illustrative Mathematics task writing contest 2011/12/12-2011/12/18.
Solution
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All three plans start with a basic monthly fee; in addition, the costs for Plans A and C increment at a steady rate based on the number of text messages sent per month. Therefore, we can find a linear equation for each plan relating $y$, the total monthly cost 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 addition, each text message costs 10 cent or \$0.x. We can write the total price per month equally $$y = 29.95 + 0.10t$$
Plan B has a bones fee of \$xc.20 fifty-fifty if no text messages are sent. In this case the full cost per month, $y$, does not alter for different values of $t$, and then we take $$y = xc.twenty$$
Plan C has a basic fee of \$49.95 even if no text messages are sent. In improver, each text message costs 5 cent or \$0.05. We can write the total price per month equally $$y = 49.95 + 0.05t$$
To visually compare the three plans, we graph the three linear equations. In each case the bones fee is the vertical intercept, since it indicates the toll of a plan fifty-fifty if no text messages are being sent. The graph for the Program B equation is a constant line at $y=90.20$. Plan B has a lower basic fee (\$29.95) than Program A (\$49.95); therefore information technology starts lower on the vertical centrality. Finally, each text message with Plan A costs more 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 nosotros see that the “best†plan volition vary based on the number of text messages a person volition send. For a small number of text messages, Plan A is the cheapest, for a medium number of text letters, Plan C is the cheapest and for a big number of text letters, Plan B is the cheapest.
At an intersection betoken of two lines, the two plans charge the same corporeality for the aforementioned number of text messages. To decide the range of “smallâ€, “medium†and “large†numbers of text messages, we need to discover the $t$-coordinate of the intersection points of the graphs. We can estimate that $t= 400$ is the cutoff bespeak to go from Plan C to Plan A, and $t=800$ is the cutoff point to go from Plan A to Program B.
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To find the exact coordinates of each intersection indicate, nosotros need to solve the corresponding organization of equations. The coordinates of these points correspond to the exact number of text letters for which two plans accuse the same amount. Because we are looking for the number of text messages, $t$, that result in the same cost for two different plans, nosotros can fix the expression that represents the cost of one plan equal to the other and solve for $t$.
Plan A = Program C $$ \brainstorm{align} 0.1t + 29.95 &= .05t + 49.95 \\ .05t &= twenty \\ t &= 400 \quad \text{Text Messages} \end{marshal} $$ Plan C = Plan B $$ \brainstorm{marshal} 0.05t + 49.95 &= xc.20 \\ 0.05t &= 40.25 \\ t &= 805 \quad \text{Text Messages} \end{align} $$ We conclude that Programme A is the cheapest for customers sending 0 to 400 text letters per calendar month, Plan C is cheapest for customers sending between 400 and 805 text messages per calendar month and plan B is cheapest for customers sending more than than 805 text messages per month.
Prison cell Phone Plans
You are a representative for a cell telephone company and it is your chore to promote unlike cell telephone plans.
-
Your boss asks you to visually display 3 plans and compare them so y'all can point out the advantages of each programme to your customers.
- Program A costs a basic fee of \$29.95 per month and 10 cents per text message
- Plan B costs a bones fee of \$xc.twenty per month and has unlimited text messages
- Plan C costs a basic fee of \$49.95 per month and v cents per text bulletin
- All plans offer unlimited calling
- Calling on nights and weekends are free
- Long distance calls are included
-
A client wants to know how to decide which plan will save her the most money. Determine which plan has the lowest cost given the number of text messages a customer is likely to send.
Source: https://tasks.illustrativemathematics.org/content-standards/tasks/469
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