The Bracket Method: there is a simple way to learn algebra

Transcription

1 No More Double Signs Page 1

2 No more double signs A +, or - sign can mean many things; create a list of words for both signs: + : positive, plus, add, - : negative, subtract, minus, Circle all double signs and replace with one sign, then punch the problem in your calculator exactly as you see it! TI-83 The colored buttons on your calculator are math operations (add and subtract). Use these buttons only! Do not touch the negative sign if you want to add or subtract! 1: : Add Subtract What happens when the first number is -? You must tell the calculator the starting point is negative! The only time you use the negative sign when adding or subtracting is to tell the calculator the first number is negative. TI-83 The negative (-) button is only used to tell the calculator the starting point of an addition or subtraction problem is negative. If the first number is negative, use this button. 3: : : : : : Negative The Big Ugly (TBU): No More Double Signs Page 2

3 No More Double Signs Everything in math is either positive or negative; the sign of the number tells you both: (1) if it is positive or negative; and (2) to add or subtract when combining. No more or A number is either + or! Write the following real life scenarios as a mathematical expression with one sign in front of each number 1: balance of your checking account is $134 2: write a check for $103 3: the temperature drops by 16 degrees, then falls another 6 degrees. 4: you sell $350 at a garage sale 5: you lose 5 points on an exam for not writing your name, but you got the extra credit right for 2 extra points 6: 60 students left the concert early. 23 students came in late. 7: the fence has to be 7 feet longer A little tougher remember one sign for each number! 9: Starting temperature is 67 degrees; the temperature rises 15 degrees at noon, then goes up 8 more degrees by dinner time; the temperature then drops 19 degrees at sundown, and decreases another 5 during the night. b: What is the final temperature? 10: The business account had a starting balance of $1340. You write a check for $245; then you take $160 out at an ATM; then you make a deposit for $378; finally, you write another check for $29. b) What is the final balance? Speak Geek Expression: numbers and/or variables put together in a mathematical sentence with + or signs (no equal sign). No More Double Signs Page 3

4 No more double signs Find the Mistakes: 1: : : Speak Geek Commutative Property: Do you see the Co in the word Commutative? Remember Co - change order. It says you can change the order of an addition (or subtraction) problem, without changing the order. Prove the Commutative property by going through the following examples. a & b are the same problem, just a different order. Put both in the calculator; see if you get the same answer. 4a: a: a: b: b: b: Now that you know the Commutative Property works: find the mistakes; why don t these problems work? 7a: a: a: b: b: b: Fraction Time: Get rid of double signs; Use your calculator to find the answer; put the answer as a fraction. 10: : TBU: TI-83 A fraction is a division problem in disguise. To enter a fraction, simply type the top number,, and the bottom number. To be safe, put all fractions inside (); do not put the operation sign inside the () example: (3 5) (5 7) 1: [Math] Don t worry about simplifying or reducing your answers, the calculator can do that: Press [Math], [1], [Enter]. You should be able to change any answer from decimal to fraction with the press of 3 buttons: Math, one, enter. 2: [1] 3: [Enter] No More Double Signs Page 4

5 How to Bracket Page 5

6 How to Bracket Bracketing Terms: the first step to the bracket method is to be able to bracket the terms in an expression What is the rule for bracketing: Bracket the following expressions 1: 5x 2 6(3x) + 5y(8x) : -6x(4y) 7x(2y-7t) + 6z 8(5) + 19(-t) 3: -2(5x) + 6(2x-7+8y) + 9(3)(-7) 4: 6x2(4x) 2(5y)(-6)(-9y) 6t 7t : y (5)(-8x)(6y) - 8t(-6x)(3y4) 4(-7x) (-x) Don t panic, stick with your rules! 6: x 8 7x (2!)(6x 5y) 7: 5x 7y 6x 2 + sin 6x cos(3x 9) TBU: -6(5x) + 6x(6y)(-6) (4x +5-7y) 8(4)(-6) + 5x 6(-x)(-y) + 4 6(-8) 2 + 9x(-3x)(-6x) How well have you trained your eyes? Without bracketing, write the number terms in each expression? 8: 5x(-40) 7(-3) (3x-4) (3-6y) 9: 4(3x)(4y)(-5z) 2(3) (53y) 2(-5) 10: -6(3x)(-6y) 6x 6(4x-7) - 6(7) 11: -5x(7y)(-6) 2x 2 + 5(-3x-2y 2 ) x(5y) + (-t) Speak Geek Term: a part of a mathematical expression or equation separated by + or signs. How to Bracket Page 6

7 How to Mash Page 7

8 Multiplication = Mash potatoes In algebra, the easiest operation is multiplication. Just put it all the signs, numbers, letters in one pile (like your uncle s mash potato plate at thanksgiving: potato s, butter, peas, carrots, etc.), Multiply each term 1: 4(3x)(6y) 3: 7x(7y)(5t)(2) 4: 6(2x)(5)(2t)(10) 5: x(y)(t)(z) 6: 2(x)(5)(2y) Don t forget about signs? (In multiplication and division, every 2 negative signs is equal to a positive!) Even # of signs = Odd # of signs = Multiply each term: (1) signs; (2) numbers; (3) letters 7: -3(4x)(7y)(-5t) 8: 4t(6y)(-5)(-8z) 9: -6(-4x)(-7y)(5t)(10z) 10: 4(-3x)(12y)(-4)(-9z) 11: -(2)(3x)(-7y)(-10)(-8t) 12: -x(y)(-z)(t) 13): -6.8x(2.3y)(-7.1z) 14: 2.5(-6.8x)(6.25y)(-6.5t) 15: -(3.4x)(8.4y)(-9.2t)(-4) TBU: 3 5 7x y 5t t ( 8z) Math Geeks Only: 16: Is 5(4)(-6)(-10) the same as -6(5)(-10)(4)? 17: What property allows you to change the order and keep the same answer? (ahem change order) How to Mash Page 8

9 Bracket and Mash Each term (bracket) is a potential multiplication (mash) group. After you bracket, mash each term. Bracket and then mash each term separately (signs, numbers, letters) 1: 4(3x) - 4(-5x)(2y) - 3(4y) 2: -5x(4y)(2) 5(-2x) + 7(3t)(-5x) 3: 3(-5) + 3x(-7y) - 3(-x)(y)(-t) 4: -(4)(-5) 5(-2x)(-7) + 4(-6x)(y) 5: -(-4x) + (-6)(-5t)(7z) x(-y)(-z) 6: 7(3) 4(-8x) + 4x(2y) 6(-8t) 7: x 4t(6)(3) 6 + 2x(-6) 8: -6(-4)(-5)(7) (2x)(3y)(-6y)3 9: -(-1)(-1)(-5) + 3x(-3z) + 5(-y) 10: 3.2(-6.5) + 4x(-5.7)(-2.1) 3.6z(-4.5t) 11: -12(7x)(-15y) 5(-19)(6t) (-12x)(-25) 12: -t (-x)(-t)(-y) (v)(-z)(-t) x(z) TBU: 3 5 x 7y x 4 7 t + 5.6t 5x 2 5 y 7.2t y 6 5 t x y How to Mash Page 9

10 Part I: No more double signs you know what to do! Mixed Review 1: : : : Part II: Bracket the following terms don t do the math, just see how many terms there are! 5: 5(2x) 7(-3) + x(y)(-z) 6: sin32 14(x) + 4cos(3x) 2(3)(4) 7: 3 4 4x x x (y) 8: 4 (7-2x) (x)(2y) + 17(4) 2x3 + 16(-x 2 )(x) 3 Part III: Mash the following terms 9: 2(-3x)(-6y) 10: -(-3x)(4y)(-7z) 11: -2(-x)(-7)(-y) 12: 7(-6y)(5x)(-t) 13: 3 5y 2 t ( 3 x) 14: -(-x)(-y)(-t)(-z) Part IV: Bracket terms; mash each term. 15: 2(-4) + 4x(-6y) - 3(-x)(5)(-t) 16: -(3)(-5) 2(-3x)(-5) + 3(-6x)(y) 17: -(-4x) + (-3)(-2t)(7z) x(-2)(-z) 18: 5(4) (-8z) + 4x(2y) 3(-7t) How to Mash Page 10

11 How to Distribute (Pizza Delivery) Page 11

12 How to Distribute (Pizza Delivery) Page 12

13 Distribute: the second type of multiplication Write M (mash) or D (distribute) above each term then do it! 1: 6(3x)(-5) 2: 7(5x + 7) 3: 7(x + 9) 4: -5(6 x + 7y) 5: -(6x)(+7) 6: -(7 5x) 7: 5(4x)(7y)(6z) 8: 3(5x-8y+9) Distribute each problem (draw those arrows if you are not sure!) 9: 6( 4x + 8) 10: -5(7 + 3t) 11: -7(6 8x) 12: 2y( -7 4x) 13: - ( 5x 8) 14: 5(t x) 15: -5(2x y + 5) 16: (6 5x + 7y) TBU1: -2xy(6t 9 + 4z u) TBU2: 3 4 (5x 1 3 y z) How to Distribute (Pizza Delivery) Page 13

14 Part I: No more double signs you know what to do! Mixed Review 1: : : : Part II: Bracket the following terms don t do the math, just see how many terms there are! 5: 3(-2x) 5(-2) + x(3y)(-4) 6: 32(7!) (2x) + tan(3x) 2!(3)(4) 7: x + x 2 4 (y) 3 8: (y -2x) (-x 2 )(-18x) 3 - (5x)(3y) + 17(4) 2x 3 (5x) Part III: Bracket terms; mash each term 9: 2(5x) - 4(-x)(2y) - 8(5y) 10: 6(-2x) - 5x(4y)(2) + (4t)(-7x) 11: -(-7) - 2(-x)(y)(-7) + 3x(-7y) 12: -(3)(-2)(7)(v) 5(-2x)(-7t)(-w) + 4(-6x)(y)+(2) Part IV:. Mash or Distribute the following terms 13: 6(3x)(-5) 14: 7(5x + 7) 15: -(4x 2 5x 8) 16: 2 3 (1 x 1 ) 17: -(-4x)(-7)(-6y) 18: -5(6 x + 7y) 2 3 How to Distribute (Pizza Delivery) Page 14

15 PEMD: The Bracket Way Page 15

16 Bracket and Identify Find Terms (Bracket), then write M (mash) or D (distribute) or S (solo) above the bracket; do not do the math! 1: 4(2x) 3(2 +6y) 2: -2(5y) + 4(2)(5) 3(2x) 3: -3(4-5x) 2(4y) + 2(3t)(-z) 4: 5(3x-6y) 4(-5)(-3) 4(t) 5: -(4x)(-5) (5y 7) 6: -3(2x) 4(5 3y) 4w + 2(3xy 5t) Put it all together now: bracket, decide and label each term (M, D or S); do the math one term at a time! 7: 4y + 7(3x 8) 8: 5x + 6y(3x) 7y(-3) 9: 7 3y(-4)(-7) + 4(5x) 10: 5x + 3(5y 6t) : -(-3x)(-8) (5y 10) 12: -3t(2z) (4x - 8) + (-8y) 13: -(x)(-y)(-t) + cos(90) 3x(2t) 14: 3 4 5x 2 3 x 4 7 y + 3x(5 6 t) 15: 3 2x 3y 7x 2 5 y ( t) 16: 3x y 17z 4x 1 3 6y x( 9t) TBU: 3x 5 7y y 6t + 3t 7y z 4x 5 8 y t x 1 2 z ( 3 4 ) PEMD: The Bracket Way Page 16

17 Algebra Multiplication how good are you? Part I: Bracket terms; don t do the math; just label each term M, D or S. 1: 5x(-5) (-3)(-5) 7y(3x 10) 2: 4!(-7y) -.387(2x) 4cosx 3: 7 4(3x 12) + 8 7y 4: x 2 (2y 3 8) 2 (4)sin x Part II: Mash or Distribute each term 5: -(3x 5) 6: -(4x)(-5) 7: 2(4x)(+7)(-8) 8: -2(7 x) Part III: Find the mistakes! 9: 5(3x) 7y (4t 8) 10: 6 + 4(4x 8y) 15x 7y -4t x 8y 11: 5x 7x(3t) + 4 (2)(-x) 12: -(3x 7y) 3 + (3t 8z) 2x + 3t + 8x -3x -7y -9y +24z Part IV: Put it all together now: bracket, decide and label each term with (M or D or S), and do the math one term at a time. 13: 6(3x) + 5(3y 17) (-6)(-3v) 14: -(-6)(-5x) 7 + 3(4y 8z) 15: -5(2y) 9x(-7y)(-2) 8(-3) 16: 6(4 7x) 2y (7t 9z) 3(-2x)(-z)(-t) PEMD: The Bracket Way Page 17

18 Basic Exponents Page 18

19 What About Exponents? Speak Geek Exponent: math notation used to show repeat multiplication (mashing) of numbers or variables (ie. xxx=x 3 ) Expanded Notation: writing terms without the use of exponents (ie. xxxyy instead of x 3 y 2 ) Alphabetical Order: it doesn t really matter, but mathematicians do prefer you write variables in alphabetical order Mash and write the answers in expanded notation (the long way) 1: 2x(3x)(-5x) 2: -4x(2y)(-3x) 3: -x(4x)(2t)(-5x) 4: -2x(3t)(4x)(-10t)(-t) Part b: Now, in the box below, rewrite your answer using exponents (exponential notation) Distribute and write the answers in expanded notation 5: 3x(2x + 5y) 6: -2y(6x 5xy) 7: -xy(2x + 4y 2xy) 8) x(x y + 2xy) Part b: Now below your expanded answer, write the answer using exponents Put it all together: find terms, mash or distribute; use exponents if necessary 9: 2x(-3x) - (2y)(3y)(-4y) + 3x(2x)(-5x) 10: -3(2x)(-x) + 3(-5y)(-6y) (-2y)(3x) 11: 3x(2 4x) + 3(2y)(-3y) x(xy) 12: 2xy(3y 4x) 5y(y)(-y) 13: -(2x)(-5x)(-y) + 5y(3x)(-2y) 14: 3(4x)(5x) 5y(2y)(-10y) + 6(2)(-10) TBU: 2 3 x 4x 5 4 3x x 7 8 x 8 9 y + 2y 2 5 y + 4xy 4 3 (6xy)(8t) Basic Exponents Page 19

20 What About Exponents? Write the following expanded answers in exponential form 1: 7xx 2 x 3 2: -42y 3 xy 2 x 5 3: -x 4 x 2 txt 3 y 2 4: -2xy 3 t 4 xt 3 t 2 y Math Geeks Only: What property allows you to re-arrange 5x 3 yxy 2 into 5xx 3 yy 2? (hint: change order?!) Mash and write the answers in expanded notation (the long way) 1: 2x(3x)(-5x 2 ) 2: -4x 3 (2y)(-3x 2 ) 3: -x(4x 3 )(2t)(-5x) 4: -2x(3t 2 )(4x 2 )(-10t)(-t 3 ) Part b: Now, below your expanded answer, write the answer using exponents Distribute and write the answers in expanded notation 5: 3x(2x 3 + 5y) 6: -3y 2 (6x 5xy) 7: -xy 2 (2x + 4y 2x 3 y) 8: x(x 2 y + 2x 3 y) Part b: Now below your expanded answer, write the answer using exponents Put it all together: bracket, mash or distribute; use exponents if necessary 9: 2x(-3x) - (2y)(3y)(-4y) + 3x(2x)(-5x) 10: -3(2x)(-x) + 3x(-5y)(-6y) x(-2y)(3x) 11: 3x(2x 4y) + 3(2y)(-3y) x(2xy) 12: 2x(3 4x) 5y(2y)(-y) 13: -(2x)(-5x)(-y) + 5(3y)(-2y) 3(2x)(-5) 14: 3(4x)(5x) 5y(2y)(-10y) + 6(2)(-10) Basic Exponents Page 20

21 The Challenge The problems below are considered tough! If you can handle these, you officially have algebra skills! 1: 5x(3x) 4(x 7) (-8)(-y) 2: 4 5(4x 8y) (x)(3x)(-x) 3: 5y(3x 4y) 4xy(5x) + 6y(-y)(2y) 4: 5x 3 (-3x 2 ) 2x(5x 2 7y 2 ) + y 2 (4y 3 ) 5: -x(-x 2 )(-x 3 ) + x 2 (y 2 ) (x 2 y 2 ) 6: -2xy(4x 2 ) + 2xy(3x 4y) 4x 2 (5x 3 ) 7: 2x 3 (3x) 4y 2 (3y 4x) - 5x 3 8: -(-3x 2 )(x 4 ) 5(4x 7y) 6 + (3x)(-2x 2 )(3y) 9: 3x 2 y 3 (5x 2 7y 3 ) + x 3 y(5xy 2 7xy) 10: 2x(-2y) 4x 2 (x 2-5x) xy(-x 3 )(-y 2 ) 11: 2 5 x 15x2 1 3 y2 9y 21y 3 12: 2 3 x 4x 3 5 x2 1 3 y2 ( y3 ) TBU: ( 3x) 5x 3x 2 5x x 6x2 7y x2 y xy2 3 4 x(5 8 x3 9y) Basic Exponents Page 21

22 Clean-up: Combine Like Terms Page 22

23 Clean-up: Combine Like Terms Page 23

24 Combine = Clean-up Learn how to identify like terms: use underlines, double underlines, and circle constants (regular numbers). 1: 5t 7y + 4-8t 12t + 8x 2: 3x + 6x 2-7xy - 18x x 17xy 3: 4xy 7x xy 2 + 8xy x + 3x 2 y 4: 3x 5y 3x 2 6x 3 + x 7x x 3 5: : 5t 7t t t 3 5 6t 2 + 9t 3 6 7: 6x 3x 2 10xy +8x 2 y 6xy 2 + 6x 2 8x 3 + x 2 7xy + 15x 2 y -3x 2 y xy x +2x 3 Clean these up...combine the following expressions 8: 7x 12x + 8x 20x + 11x 9: 3y + 11y + 8y + 20y 40y 10: 3t 2 + 9t 2-4t 2 21t 2 + 6t 2 11: -8x 5 + 9x 5 3x 5 x 5 +8x 5 12: -101x + 56x 68x -71x -211x + 57x 13: 15y + 9y -12y + y 5y Cleaning up (combining) is all about adding or subtracting; the sign tells you what to do. 14: 5x - 6x + 8x + 9y + 18y 7y 15: -9y + 7x 13x + 18y + 9x -21y 5x 16: 8x 5x 2 11x 2 + 7x - 17x 2 + 8x 17: 12x 15y y 17 9y + 10x - 5 Speak Geek Coefficient: a number in front of a variable Constant: a number by itself (no variables): a regular number (all constants/regular numbers are like terms) Like Terms: terms where the non-coefficient parts (the stuff after the front number) are the same Clean-up: Combine Like Terms Page 24

25 Combine = Clean-up Clean-up the mess: Identify a term, find all of them, and combine (add or subtract depending on the sign) 1: Answer: 2: Answer: 3: Answer: Clean-up: Combine Like Terms Page 25

26 Combine = Clean-up Clean-up the mess: Identify a term, find all of them, and combine (add or subtract depending on the sign) 1: Answer: 2: Answer: Now that you have seen the worst, try some basic problems: do the same thing! 3: -5x + 17y xy - 3y 11x + 8xy : 4x 2 5x 3 + 9xy 10x xy 2 2x 2 y + 25xy 14x 2 y xy 2 5: Clean-up: Combine Like Terms Page 26

27 Combining Like Terms (Clean-up) Math Geeks Only: If xx = x 2, then why is x+x not x 2? Find the Mistakes 1: 4x 5x x 2 11x + 3x 2-7 2: 6x + 18y y y + 8x - 9-4x - 11x x + 35y + 13 Identify and combine (clean-up) 3: 3x 5y +6x y 84x -17 4: 5x 2 7x 8x 2-18x + 34x 2 17x 2 5: 5x + x x 4x x + 5x : 16x + 14 xy 5y 8xy 7y y 2 - xy 7: : 6x 3 4x x x x 2 + 8x : x + xy y + 4xy 4y + x + xy 3x 6y 10: 3x 2 4xy 2 + 7y 2 7x 2 y 9y xy TBU1: 3 4 x 1 2 x x 2 3 x TBU2: y 1 4 y2 1 6 y + y 1 3 y y2 Clean-up: Combine Like Terms Page 27

28 Simplify: The Bracket Way Page 28

29 Simplify = Bracket Find the Mistakes 1: 5x 3(5 8x) + 3(-4) 2: 7 + 3(6x 2) - 5x 5x x (6x 2) x x x - 25 Simplify: (1) Bracket terms, (2) Mash, Distribute (or Solo) each term, (3) Clean-up 3: 17 5(3x 8) 4: 5x + 2(-4) 7x(-5) : 20(-2x) 7(-8) + 2x 3(-7) 6: 5(2x - 8) 4(9x - 5) 7: -3(4) (3x 6) (-4x) 8: 5x 7(-2x)(-5) 2(x 4) 9: 3(4x 2 5x -8) 2x 2 4(3x) - (-2) 10: -(-4)(-3y) + 3x -2(-4x -7y) + 5(-4y) TBU: 7(3x)(-7) 4x 6(-3)(-1) - (4x 8y + 7) + 14y x(-5) 8(-3y + 8x) (- 7)(-6) x y Simplify: The Bracket Way Page 29

30 The Challenge (Part II) If you can simplify these you can simplify anything! Remember: (1) find terms, (2) Mash or Distribute each term, (3) Clean-up (combine) 1: 5x(2x) 5(3x 2 7x - 8) + (-7) 2: -2(-3x 3 ) 2x(-5x) 6x 2 + 5x(-2x)(-7x) 3: 2(x 3 18x 2 + 9x 34) (x 3 8x 2 14x - 21) 4: 5(2x 6y) (4x)(-3y) 7(-2x) + x(3y) (-x) 5: x(x) 2(3x 2 7x 9) 3(2x) : -(-6x 3 ) + 18x 2 2x 2 (-5x) x(x 5) 7: 5(-2y) 7y(4y 8) 9 +2(7y) (-4y 2 ) 8: 2xy(3x 8y) 3x(-4y 2 ) x(9x)(-3y) + 7y(-2xy) 9: x(x 3 9x 2 + 4x 21) 2x(x 2 4x) 2(-11) 10: 3 8 x 4x 5x x2 3x ( 7) TBU: -4x 2 (3x 2 ) 7x 3 (-5) x(x 3 6x 2 10) x(-4x)(-7x) + 18x 2 (-x) + 5x 3 (-7x) 6x(18 7x x2 + 5x 3 ) Simplify: The Bracket Way Page 30

31 Solving: Junk & Divide Page 31

32 Solving: Junk & Divide Page 32

33 Solving: Junk & Divide Page 33

34 Solving Equations: Junk & Divide Speak Geek Equation: when one mathematical expression is set equal to another. Solving Basic Rules: 1. Draw the 2. Work on the 3. Whatever you do to one side of the wall, Draw the wall; put an * above the junk (do not solve!) 1: 7x 27 = 76 2: 35 = x 3: 16 8x = 123 4: -352 = 21x + 7 5: 4x - 7π = 237 6: 71 = 2 3 7x 7: 9x 4 = 78 8: 6 + 7x = Now go back (1-8) and show how you would get rid of the junk (do not dive yet, that is next) Solve: Put it all together now Junk & Divide 9: 17 9x = 71 10: -5x 35 = : 89 = 4x : -57 = x 13: 12 7x = : 13x + 78 = : 171 = 5 5.2x 16: 17 x = 58 17: 5x = 16 Solving: Junk & Divide Page 34

35 Solve: Junk & Divide 1: -17x = : x = : 54.5 = x 4: 1235 = 123x 86 5: 1 x = 0 6: 67 12x = -782 These are so easy, they are tough (sometimes you only do junk, and sometimes you skip right to divide)!? Speak Geek One Step Equations: equations which can be solved in one mathematical step. 7: 5x = 876 8: x = 91 9: -17 = -4x 10: -346 = 89 + x 11: - x = : x 13 = -21 Don t panic, they re only fractions (remember to use parenthesis) junk and divide! 13: 5x 2 3 = 28 14: x = 34 15: 174 = x 16: 6 = 3 8 x : 5 9 x 27 3 = 34 18: 46 7 = 1 4 x Solving: Junk & Divide Page 35

36 Simplify and Solve Bracket, Junk & Divide 1: 7 + 2(3x 8) = 123 2: 5x 2(3x) 18(-4) = 231 3: 4(-2) 6(3x 8) + 8(-3x) = : -5x 4(x 8) = : 1056 = 4(-2x) 18(3) 7 6: -78 = 7(2) 3(4x 8) + 2(-5x) 7: -(3x 8) x (-7x) = : 78 = 17 2(x - 8) + 5(-7x) (-5)(-8) TBU: (5x)(-17)(-4) (12x 2 6) 5x 4(5 x) (-2)(-8)(-3x) + 2(3x)(-7) + 4(3x 2 7x +9) = Solving: Junk & Divide Page 36

37 X s on Both Sides make one disappear, junk & divide 1: 5x 19 = x 2: x = x 3: 24 8x = x 4: 34 18x = x 5: 8x + 19 = -6x 23 6: -16x + 31 = -7x : x = 25.4x : 5.26x = : x = 40 8x These are strange think about which x you want to disappear!? 10: x = 6x 11: -78 7x = -12x 12: 15x = x 13: -19x = -7x : x = x 15: x = 19x Hmmm?16: 6x 8 = x Hmmm?17: 21 7x = -7x + 21 Solving: Junk & Divide Page 37

38 The Challenge (Part III) If you can solve these you can solve anything! 1: 6 + 2(3x 8) = 7x -9 2: -3(x 8) = -4(5x 7) 3: -4(-3) (6x 8) + 7 = 5x 18 4: 7x 8(-5x) + 7 = 17 5x 5: 5(2x) 17(-2) = 2(7x 8) 6: -(-5) 4(2x 8) = 3x 7(-2) + 8x 7: 4x 2(-3x) + 2(5) = 7 3(2x 8) 9x 8: 4(3x + 21) = 5x 7x + 15x 9: 5(-2)(3) (4x 9) 4(-7) = 3(-2) -3(4)(-5x) 9 10: 2(-4) x 2(3) = 8 + 8x x 11: 3 4 2x 8 = 2 3 6x 14( 3) 12: x 3 8 = 1 2 4x ( 2 21 ) Solving: Junk & Divide Page 38

39 Solving for Letters Speak Geek Implicit Formulas: solving a formula for another variable When solving for formulas, nothing changes (except the answers are ugly looking). Identify what you are trying to solve for; get rid of the junk; divide. 1: solve for x: 5x t = 4g 2: solve for t: 4g 7t = cy 3: solve for v: 3tv = 35g 4: solve for g: 16t 2 5g = F 5: Solve for r: 6πrh = 156 6: solve for h: fm gh = 38t 7: solve for v: 6t 2 7t 5vh = 324 8: solve for g: 7y 6t 8g = 56 9: solve for r: 7d 3πrh = -72g One of the most common things you will do in algebra is to solve for y (get y alone!). 10: y + 4x = 7 11: 5x - y = 18 12: 5x + 8y = 21 13: 2x 7y = : 7y 4x 8 = 18 15: 15 = 5x 3y Solving: Junk & Divide Page 39

40 Adding to your repertoire Junk & Divide can be used to solve most any type of linear algebraic equation. However, once you get good at this method, there are certain methods you should learn to make solving faster or easier! 1) Cross Multiplication: used anytime you have fractions set equal to each other. 2) Dot Method, or IHF (I hate fractions), or GCF (this is the official geek name: it stands for greatest common factor): used when you have a lot of fractions in an equation, and you just want to get rid of them! 3) Graph the equation and find the intersect: every equation can be solved this way. 4) Two Sticks, Two Equations: used to solve absolute value problems. 5) The Inverse of Square is Square Root: Used to solve x 2 or x problems Solving: Junk & Divide Page 40