What's the best deal for you? Systems of Equations

Buying an expensive gadget, computer, or phone?  Doing the math can make you feel better about getting the best deal.  What

An Apple I-Phone 4 costs $400 plus a $20 monthly fee.  A Samsung Galaxy Phone costs $350 plus a $25 monthly fee.  Which phone would be the best deal if both phones required a 12 month contract?

Some students sold popcorn for a fundraising project.  On the first week they sold 6 small bags, and 20 large bags of popcorn, and they took in $31.00 the first week. The second week they sold 12 small bags and 28 large bags of popcorn. Sales totaled $47.00 the second week.  What was the price for a small bag, and what was the price for the large bag of popcorn? 

Step 1: Define the variables

s = small bag

L= large bag

Step 2: Put word problem into equations

6s + 20L = 31

12s + 28L = 47

Step 3: Use algebra to solve for one variable 

2 (6s + 20L = 31)

12s + 28L = 47

Subtract the two equations (simplify expressions) First, Eliminate the variable S

12s + 40L = 62

-12s + 28L = 47 

___________________________  Then, Solve for L

12L = 15

L = 1.25

A large bag costs $1.25

Step 3:  Substitute L = 1.25 and solve for S

6s + 20*(1.25)= 31

6s + 25 = 31

6s = 6

s= 1

Small bags costs $1.00 each

Try your hand at some equation problems at www.khanacademy.org

Level 2: Linear Equations

Level 3: Linear Inequalities

  

Visiting a College Campus? Campus Buildings and Designs

College campuses have many attractions.  As we visit a college campus its fun to check out interesting sites.

Reading Room at Cathey Learning Center 

 Cathey Library as seen from Midway Plaissance

We'll find out more about these college buildings and designs on the tour.  Here's a blog post that shows some amazing buildings and designs. A couple of the featured buildings are at the University of Chicago. I wonder what it would be like to read a favorite book inside the immense reading room at the U of Chicago's Cathey Learning Center which stands 39 feet tall?  This building has 2 towers that have an interesting design.  Another library on the campus has robotic arms that take books off shelves for students and guests.  This library is inside a picture perfect glass dome at the U of Chicago's Mansueto Library.  It's neat how historical buildings stand next to modern designs.  What are some buildings and designs that you like?  What Math is involved in creating buildings and designs?  

Functions: Inputs, Outputs, and Beyond!

Try to find patterns below.  How do the numbers compare?  What do you notice from the input to output?
Chicago Parking                 Secure Parking
Input       Output                          Input     Output              

1                7                                      1             5
2                9                                      2             8
3                11                                    3             11
4                13                                  4             14
10              25                                 10            32
20              45                                 20            62
 n              2*x + 5                        n            3*x + 2

Question:  When trying to park in downtown Chicago we are trying to decide between 2 options. Which company offers the better deal if you want to park for a short trip, and which one for a longer trip?

Chicago Parking charges a 5 dollar entry fee and then 2 dollars per hour

Secure Parking offers a 2 dollar entry fee and then a 3 dollars per hour rate.

Equations and Graphs: Compare the equations above.  Let x= the number of hours and
Let y= total cost for parking

Chicago Parking (CP):  y=2x +5.  How? CP costs $2.00 per hour with a starting cost of $5.00.  So, y (total cost) = 2 x (# of hours)  + 5  (starting cost)
Secure Parking (SP): y=3x +2.  How?  SP costs $3.00 per hour, and its start up cost is $2.00. So, y (total cost)= 3 x (# of hours) + 2 (starting cost)

The picture above shows an example graph.  This graph shows the equation y = 2x +3.  We can see the start up cost would be 3 and then it would increase at a rate of $2.00 an hour.
Assignment:
A. Graph the equations using a graphing calculator from this website link:   Holt Mc-Dougal's website
B. What is the "point of intersection" on the graph?  This is where both companies have the same cost.  we
C. Solve the equations using algebra to find the same answer.  Solve the equation below to find x.
      2x + 5 = 3x +2
D. Create your own problem.  Start with 2 equations, or 2 companies that offer slightly different charges.
Share your questions and also some hints for solving the problem.