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Algebra: B

Learn how to solve linear systems, rate-time-distance problems, and problems involving mean, median, and mode. Explore Venn diagrams, recursion, functions, and properties of arithmetic sequences. Find all the resources and solutions on our website!

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Algebra: B

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  1. Algebra: B

  2. MAKE SURE BEFORE WE CONTINUE • If you have not already contacted me in some way, shape, or form, no worries: we now have a website! • Go to wmsmc.weebly.com. All of this stuff will be on the website so that you can review if you want to. • I’m going to try to make a forum on this site for problem discussion. This way you all can help each other solve problems!

  3. Make sure you know: if not ASK QUESTIONNNNNSSSSS!!!!!! • Solving linear systems in any number of variable • Solving rate-time-distance problems and proportions • Solving simple problems involving mean, median, and mode • To be honest, that’s all you need for today!

  4. Making systems easier: Part I • If you are given an equation Ax+By=C and A, B, and C have a common factor greater than 1, then it is usually a good idea to divide out this common factor. • Ex: 6x+2y=18, common factor 2, divide through by 2 to get 3x+y=9. • Sample problem on board

  5. Dividing out may not always be the best idea: moral of the story: THINK!! • Sample problem on board

  6. Quickly determine one solution • There are some cases in which one of the variables is obviously 0. • Sample problem on board

  7. Dealing with symmetry • Systems of equations can easily be solved when there is symmetry involved • Ex. 2x+3y=26, 3x+2y=24 • Normally, for these, the elimination/substitution can get messy, but exploiting symmetry makes it a lot cleaner

  8. Any questions?

  9. Diophantine linear systems • Diophantine=integer solutions ONLY • The Diophantine equation 2x+3y=5 would have only one solution: (1,1)

  10. Sample problem • Find all integer solutions to the equation 12x+20y=132

  11. Rate-time-distance • Really, you only get good at these problems by doing many of them (they’re computationally intensive) • By the time you’ve narrowed it down to an equation or a system of equations, you can finish the problem off in style by exploiting the numbers

  12. Intro to Venn Diagrams • Formally: a problem involving the union, intersection, or both, of some sets • Union is EVERYTHING, intersection is the stuff between the circles • Example: 15 students attend derp high school. If 12 students are taking a troll class and 13 are taking a blowoff class, then how many students are taking both?

  13. Principle of Inclusion-Exclusion • Called PIE • Explaining it on board now • If you didn’t understand it, let me know. For further reading, visit the page http://scimath.unl.edu/MIM/files/MATExamFiles/Derr_FinalMEXP_C5.pdf.

  14. What is recursion? • Sequence that builds upon itself • Great example=Fibonacci numbers • Sample problem: write the function f(x)=2x recursively • Important: learn to recognize recursive patterns quickly!

  15. Functions • An operation  is defined as a  b = 6a − b. What is the value of 4  22? • Functions can be manipulated to give clean results. Example: on board

  16. Eh, all of that is useful and all… • Let’s get to the meat of MATHCOUNTS. Are you ready…?

  17. Arithmetic Sequences • Defined as a sequence with a constant nonzero difference between consecutive terms • 2,5,8,11,etc., is an arithmetic sequence • Let’s name a few more arithmetic sequences!

  18. Properties of Arithmetic Sequences • Constant, nonzero difference • If there are an odd number of terms in an arithmetic sequence, then the middle term is the average (let’s prove this) • If the first term is a and the common difference is d, then the nth term is a+(n-1)d • Graphically, arithmetic sequences are linear equations. What is the slope?

  19. Sample Problem (taken directly from 2012 MATHCOUNTS) • Seven consecutive positive integers have a sum of 91. What is the largest of these integers? • I have 2 solutions in mind. One of them is a smart way, one of them is a bleh way

  20. Let’s solve an interesting problem! • Last week, we proved that the sum of the first n positive integers was n(n+1)/2 • What is the sum of the first n odd integers? How about first n even integers? • Discuss among your groups

  21. In conclusion… • You can find what we did today under the Sept. 14 section of the website on the home page. • The handout completely reviews everything that we did today. IT IS VERY HELPFUL TO DO IT! • Now, let’s do something fun.

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