Investigating an SAT Algebra Problem - Going Beyond the SAT Strategy for Deeper Meaning

[Update as of 6-2-07: Solutions to most parts of the problem are now posted in the Comments.]

The following problem is typical of a somewhat difficult Algebra 2/Precalculus type of SAT problem. Students who have been shown the 'quick and easy way' to solve this definitely have an advantage when taking the test. This has been developed into an activity for Algebra 2 students who might be taking the SAT on June 2nd. However, the activity explores more than just a strategy for solving the problem efficiently to get an answer for the test. Students will be asked to solve the problem the traditional way as well and analyze why this question should not appear on the test. Two separate graphical interpretations are included to deepen student understanding! , one for Algebra 2 students and a more sophisticated one for the Precalculus class. Is it worth spending 25 minutes or more on one problem in class? I'll let you judge...

Consider the system of equations:
1/x + 1/y = 1/4
1/(x+y) = 1/3

(a) (SAT-type of thinking): Show that xy = 12 (without solving for x and y).
Note: I'm giving the 'answer' here so that students will focus on the method; also for part (f).

(b) (Algebra 2): Solve the system (i.e., find all possible solutions for x and y). Write your answer(s) as ordered pairs.

(c) Verify, algebraically, that your solutions satisfy the original equations.

(d) Explain why this exact ques! tion should not appear on the SATs (although this type of question has appeared frequently).

(e) (Precalculus extension): Graph the system:
1/x + 1/y = 1/4
1/(x+y) = 1/3
Notes/Hints:
You must graph each equation separately. Do not replace the first equation by xy = 12.
Hint: Solve each equation for y.

(i) Explain clearly why the graph of the first equation has both vertical and horizontal asymptotes. Label these in your graph.
Note: The customary way is to solve for y first. See if you can also determine the asymptotes without changing the form of the equation!

(ii) Does your graph of the first equation contain the origin? Explain why or why not.

(iii) How many solutions of the system are evident from the graph? Explain the reason for this.

(f) (Algebra 2): Now solve the system
x+y = 3
xy = 12

(i) In what way is this system equivalent to the original system?
(ii) How many solutions of this system are evident from the graph? Explain the reason for this.

solve algebra 2 problems