Any questions on the Section 5.4 homework?

1. Any questions on the Section 5.4 homework?

2. Please CLOSE YOUR LAPTOPS, and turn off and put away your cell phones, and get out your note-taking materials.

3. Reviewing for Quiz 3: Quiz 3 covers: • Section 4.1 on systems of linear equations • Sections 5.1-5.4 on exponents and polynomials

4. Preparing for the Quiz: 1. Review the homework from all six homework assignments using the “Gradebook” function. (All of the quiz problems have been part of one of these homework assignments.) 2. Take the Practice Quiz (under Assignments, Review for Quiz 3). • The practice quiz is longer (18 questions) than the real quiz (12 questions), but has similar problems and will give you a chance to see the difference between doing problems in a homework setting vs. a timed quiz with no “help” or “check answer” functions. • Each time you take the practice quiz you’ll get a different set of questions (not just different numbers in the same questions), so taking it multiple times will make it more likely that you’ll have seem most of the types of problems that will be an the actual quiz. (We’ll give you time to get started on this in class today)

5. Why you should take advantage of the opportunity to take the practice versions of quizzes and tests: Note: Only three students did not take this practice quiz at all; they averaged 55.8% (F)

6. Review of Section 4: Solving Systems of Equations in Two Variables The solution of a system of linear equations in two variables is any ordered pair that solves both of the linear equations. To be a solution of a system of equations, an ordered pair must result in true statements for BOTH equations when the values for x & y are plugged into them. If either one (or both) gives a false statement, the ordered pair is NOT a solution of the system.

7. There are three methods that we have explored to find a solution for a system of equations. • Graphing the lines (Sec 4.1A) • Substitution method (Sec 4.1A) • Addition or elimination method (Sec 4.1B)

8. On a graph, the solution to a system of two linear equations is the intersection (if any) of the two lines. • There are only three possible solution scenarios: • The lines intersect in a single point (so the answer is an ordered pair). • The lines don’t intersect at all, i.e. they are parallel (so the answer is “no solution”.) • The two lines are identical, i.e. coincident, so there are infinitely many solutions (all of the points that fall on that line.)

9. Use of the substitution or addition method to combine two equations might lead you to results like . . . • 5 = 5 (which is always true, thus indicating that there are infinitely many solutions, since the two equations represent the same line), or • 0 = 6 (which is never true, thus indicating that there are no solutions, since the two equations represent parallel lines).

10. There are three types of answers you will encounter in the systems of linear equations problems, corresponding to the three different ways two lines can intersect: 1. Intersection in a single point(Answer is an ordered pair.) (The two lines have different slopes) 2. No common intersection (parallel lines) (Answer: NS) (The lines have the same slope, different y-intercepts; all variables drop out by elimination, leaving a false statement such as “0 = 3”) 3. The two equations represent the same line, so the Intersection is all the points on the line Answer: {(x,y)| y = mx + b} (Lines have same slope AND y-intercept; all variables drop out by elimination, leaving a true statement such as “0 = 0” )

11. What would be the best first step in solving each of these systems? • 2x + 3y = 10 and 2x – 3y = -2 • 2x + 3y = 10 and 5x + 6y = 7 • 4x + 7y = 8 and 3x + 19y = -12 • 4x + 7y = 8 and x = 2y - 1

12. REMINDER: Always check the proposed solution in the original equations. REMINDER: ALWAYS CHECK ANSWERS IN SYSTEMS OF EQUATIONS PROBLEMS!!

13. Example: X WRONG!!! How to check: plug in ¾ for x and -1 for y in BOTH of the original equations to see if you get true statements for BOTH. Result:This is the WRONG answer. A small negative sign error gave y = -1 instead of the correct y = +1. The correct answer is (1/2, 1). CHECK THIS NOW!!!

14. Brief Review of Sections 5.1-5.4: All of the rules of exponents are on the pink formula sheet you will have available for use during the quiz. Make sure that you know when the rules apply, e.g., when to add exponents, when to multiply them, when to subtract them.

15. Power of a Quotient Quotient Rule for exponents Negative exponent Summary of exponent rules: If m and n are integers and a and b are real numbers, then: Product Rule for exponentsam• an = am+n Power Rule for exponents(am)n = amn Power of a Product(ab)n = an• bn Zero exponenta0 = 1, a 0

16. Something to watch out for that we have noticed lots of students having trouble with on these problems: When working with numbers as bases, treat them just like you treat variables, i.e. don’t combine exponents for terms with different bases, and add/subtract/multiply the exponents only, not the base numbers. Examples: 23 = 8 (2∙2∙2) 23 is NOTthe same as 2∙3 !!

17. Example:

18. Don’t forget to review problems using scientific notation.

19. Do any of you who have already started the practice quiz have any questions you’d like to have explained?

20. Go ahead and start the practice quiz, and we’ll come around to help if you have questions. Remember, the open lab next door will be staffed till 7:30 PM at night and starting at 8:00 AM every Monday through Thursday. Just drop in – no appointments needed.