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Consistent and Inconsistent Linear Systems

Systems with parallel or intersecting lines

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AT&T & Verizon - Same Cost?
Teacher Contributed

Real World Applications – Algebra I


Comparing Iphone Plans – Same Cost?

Student Exploration

In this concept, you learned how to differentiate between a consistent and inconsistent system of linear equations. In this demonstration, you will see how an inconsistent system of equations is represented between the most popular cell phone and cell phone carriers of 2012 – AT&T Wireless and Verizon Wireless. Both service providers carry very similar phones, such as the Iphone 4S. As of July 2012, their competitive prices are actually quite similar – so similar that the relationship between the two wireless providers could represent an inconsistent system of linear equations.

Let’s take a look at AT&T Wireless first. An Iphone 4S costs $199. The unlimited talk plan costs $70/month, “unlimited” data (up to 3 gigabytes) costs $30/month, and the one-time activation fee costs $36.

Now let’s look at Verizon Wireless. An Iphone 4S also costs $199. The unlimited talk and text plan costs $60/month, which also includes up to 2 gigabytes of data. There is also a monthly fee of $40 for mandatory access to the network. The one-time activation fee costs $35.

If we were to make an equation for each cell phone provider, we have to identify the two unknowns. Let \begin{align*}x\end{align*} represent the number of months you would use this plan, and let \begin{align*}y\end{align*} represent the total cost. The equation for AT&T Wireless would be \begin{align*}y = (70+30)x + 36\end{align*}. The equation for Verizon Wireless would be \begin{align*}y = (60+40)x + 35\end{align*}. If we simplified both of these equations, we would have:

AT&T: \begin{align*}y = 100x + 36\end{align*}

Verizon: \begin{align*}y = 100x + 35\end{align*}

Does this system of equations represent a consistent or inconsistent system?

This system represents an inconsistent system of equations because the slope is the same. In this case, you would pay $100 each month, no matter which cell phone provider you have. But, if you look at what each provider gives you, there are differences. With this $100 plan, AT&T wouldn’t give you any texting (unlimited texting costs an additional $20/month), whereas Verizon has this included with the talk plan. But, AT&T also gives an additional 1gig of data than Verizon. Which plan would you choose?

Extension Investigation

Most people nowadays want more. Let’s say you chose AT&T and want the additional $20 of unlimited texting. What would the equation look like? If you compare this equation with Verizon’s equation, would this represent a consistent or inconsistent system of equations? Why? If you were to try to solve this system of equations, is there a solution? If so, what would this solution mean?

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