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# Graphs of Linear Equations

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# Finding the Cheapest Cell Phone Plan - Answer Key

## Real World Applications – Algebra I

### Topic

How do you know which cell phone plan is the cheapest?

### Student Exploration

The student will investigate, compare and determine which cell phone plan is the least expensive over time.

1. Search for two different cell phone companies (i.e. AT&T, Verizon, Sprint, T-Mobile, Boost, Metro PCS).
2. The most organized way to list the cost of each cell phone plan is to make a table. Make a table for each cell phone company. The input $(x)$ values should be the # of months you have the phone, and the output $(y)$ values should be the total cost. “Month 0” should be the initial cost of purchasing the cell phone itself.
3. Create a graph for each table of points. What are realistic values of $x$ and $y$ on this graph? Why?
4. Create an equation for each cell phone plan represented in slope-intercept form.
5. In each equation, what is the slope? What does this represent in the cell phone plan? What is the $y-$intercept in each equation? What does this represent in the cell phone plan? Justify your thinking.
6. Which cell phone plan is the cheapest, and for how long?
7. When are both cell phone plans the same cost? Use the Substitution Method to solve this system of linear equations you created. What do “$x$” and “$y$” each represent in this solution?

Any phone and phone plan can be compared. For the sake of this exercise, we’re going to compare the cell phone plans for MetroPCS and Verizon Wireless with a Blackberry cell phone. After conducting some research on the internet, I found that the Blackberry phone costs $149 and the plan costs$60 per month for MetroPCS. For Verizon, the phone costs $50, and the plan costs$100/month.

We can determine the two equations for the two cell phone plans. The equation that represents the total cost of a Blackberry and plan for Metro is $y = 60x + 149$, where $x$ represents the number of months that have passed after having the phone, and $y$ represents the total cumulative cost of having the phone. The equation for Verizon is $y = 100x + 50$.

The graph above represents the two lines for the two different cell phone plans. We can see that the intersection point is where the two cell phones will be the same cost and at what time. By looking at the graph, it looks like that after 2 and a half months, they will both cost about $300. We can solve a system of linear equations to find the exact time and month that these two plans are the same. We can use substitution, and substitute one of the $y$ values for its counterpart in the other equation. $Y = 60x + 149$ and $y = 100x + 50$ We can substitute $y$ in the second equation with $60x + 149$, since $y$ and $60x + 149$ are the same. $60x + 149 = 100x + 50$ Now we should subtract $60x$ from both sides and also subtract 50 from both sides. $99 = 40x$ Now we want to divide both sides by 40 to solve for $x$. $2.475 = x$ Now we want to find out the corresponding $y$ value. We can do this by substituting 2.475 in for $x$ in either equation and solve for $y$. $Y &= 60x + 149\\Y &= 60(2.475) + 149\\Y &= 148.5 + 149\\Y &= 297.50$ We now know that in 2.475 months, both plans’ total cost would be$297.50.