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Algebra Notes

Algebra Notes. Writing Algebraic Expressions. Let Statement: math sentence used to define a variable to represent the unknown quantities. Laura has twice as much homework as Ann. The Bills won five more games than they lost. Seven more than three times a number is 25.

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Algebra Notes

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  1. Algebra Notes

  2. Writing Algebraic Expressions

  3. Let Statement:math sentence used to define a variable to represent the unknown quantities.

  4. Laura has twice as much homework as Ann. The Bills won five more games than they lost. Seven more than three times a number is 25. The length of a rectangle is 3 cm more than the width. Let Ann = a Let Laura = 2a Let games lost = g Let Bills won = 5 + g Let Yankees = y Let Tigers = 3y Let width = w Let length = 3 + w

  5. Mike is three years older than Jim. Eight more than twice a number is 32 Seven more than three times a number is 25. Twice a number increased by four is 16. Let Jim = j Let Mike = 3+ j Let number = n 2n + 8 = 32 Let number = n 3n + 7 Let number = n 2n + 4 = 16

  6. Six less than three times a number is 21. Fifteen less than twice a number is 25. Sixty-six is eleven more than five times a number. Let number = n 3n – 6 = 21 Let number = n 2n – 15 = 25 Let number = n 66 = 5n + 66

  7. Writing Algebraic Expressions

  8. Setting up & Solving word problems

  9. Write your let statement Write your equation Solve Check Write an answer sentence

  10. A cell phone company charges $39 a month plus $.15 per text message sent. If Jan sends 35 text messages this month, how much does she owe before taxes are added? The Bills won five more games than they lost. Let text message = t Jam owes $44.25 39 + 0.15t t = 44.25 Let text message = s 12 + 2s s = 4 4 snacks

  11. A rental car company ABC charges $25 per day plus $.15 per mile. Rental car company XYZ charges $18 per day plus $.25 per mile. If you plan to drive 50 miles, who is the cheaper rental company? Joe attends a carnival. The admission is $48. Tickets for rides cost $4 each. Joe needs one ticket for each ride. Write an equation Joe can use to determine the number of ride tickets, r, he can buy if he has $200 before he pays the admission fee. Let miles = m ABC: 25 + 0.15m $32.50 XYZ: 18 + 0.25 $30.50 XYZ is cheaper Let number or rides = r 48 + 4r = 200 r = 38 38 rides

  12. substitutions

  13. Evaluate if s = 4 • 4s → • 4 + s → • 5 – s → • 12 ÷ s → 4(4) 16 4 + 4 8 5 - 4 1 12 ÷ 4 3

  14. Evaluate if s = -6 • 7s → • 3 + s → • 7 – s → • 18 ÷ s → 7(-6) -42 3 + (-6) -3 7 – (-6) 13 18 ÷ (-6) -3

  15. Evaluate if n = 3 and r = 5 • n² + 7r • 9n - r² • 2nr + 6n 3² + 7(5) 9 +12 21 9(3) - 5² 27 – 25 2 2(3)(5) + 6(3) 30 + 18 48

  16. Evaluate if p = 12 and q = -8 • p + q +6 • p – q + 3 • p – q + q² 12 + (-8) + 6 -4 + 6 2 12 – (-8) + 3 20 + 3 23 12 – (-8) + (-8)² 20 + 64 84

  17. Evaluate if a = -2 and b = 6 • 3a² + 5b² • 4a³ + 3b • 7a² - (b²/3) 3(-2)² + 5(6)² 3(4) + 5(36) 192 4(-2)² + 3(6) 4(-8) + 18 -14 7(-2)² - (6²/3) 7(4) – (36/3) 28 – 12 16

  18. Like Terms

  19. Terms of an Expression • Termsare parts of a math expression separated by addition or subtraction signs. 3x + 5y – 8 has 3 terms.

  20. Like Terms • Like Terms: have thesame variablesto the same powers 8x²+2x²+5a +a 8x²and 2x² are like terms 5a and a are like terms

  21. LIKE terms: Yes or No? 3x + 7x Yes - Like 5x + 5y No - Unlike 4c + c Yes - Like 4d + 4 No - Unlike

  22. LIKE terms: Yes or No? 3ab – 6b No - Unlike 2a – 5a Yes - Like x andx² No - Unlike Yes - Like 6 and 10

  23. Identify the LIKE terms 3m – 2m + 8 – 3m + 6 5x + b – 3x + 4 + 2x – 1 – 3b -6y + 4yz + 6x² + 2yz – 4y + 2x² - 5

  24. Coefficients • A Coefficient: a numberwritten in front of the variable. Example: 6x The coefficient is 6. Example: x The coefficient is 1.

  25. Simplify • Simplify: means to combine like terms. • Combine LIKE terms by adding their coefficients.

  26. Write an expression: + 3c + 4c =7c

  27. Write an expression: - 8a - 1a = 7a

  28. Write an expression: + 5c + 4d

  29. Write an expression: - 5a – 4b This expression cannot be simplified. Why not?

  30. Simplify the following expressions

  31. 2x + 4x • 2a + 5a + 6 • 3xy – xy +2x • -4c + 8c – 6c • 3a + 7a • 3½y + 5y -4y • cd + 4cd – 2a • ½e – 2e + ¾ e • 6xy – 2xy • 5d – 6d – 3d • 4s – 4s • 5x + 4x + 4x + 11x 6x 10a 4xy 4½y 7a + 6 -4d 2xy + 2x 5cd – 2a 0 -¾e 24x -2c

  32. Challenge questions

  33. 1. –5x – 3x 2. 8x – 2x 3. –7x – (–3x) 4. 6x – (–4x) 5. –10x –14x 6. –9x – (–x) 7. 3x – 8x 8. x – (–5x) 9. a² + b² + 2a² + 5b² 10. 7h² + 3 – 2h² + 4 -8x 6x -4x 10x -8x -24x -5x 6x 5h² + 7 3a + 6b²

  34. 11. 3x + 3y + x + y + z 12. 5b +5b + 6b² - 10 – 3b 13. Find the perimeter of the rectangle: A 4x + 3y B 8x + 6y C 12xy D 4x²+ 3y² 4x + 4y +z 6a² + 7b - 10

  35. Adding & subtracting Polynomials

  36. Adding • Combine like term • Add the coefficients to simplify Example: Add   2x² + 6x + 5   and  3x² - 2x – 1 • Start with: 2x² + 6x + 5     +3x² - 2x – 1 • Place like terms together: _______+ ________+ ________ • Add the like terms: _________+ __________+ _________ • Final answer: 2x² - 3x² 6x – 2x 5 – 1 5x² 4x 4 5x² + 4x + 4

  37. Subtracting Change the subtraction sign to addition and reverse the sign of each term that follows Then add as usual Example: Subtract   5y² + 2xy - 5   and  3x² - 2x – 1 Start with: 5y² + 2xy - 5     -2y² - 3xy + 3 Place like terms together: _______+ ________+ ________ Add the like terms: _________+ __________+ _________ Final answer: - - + + 5y² - 2y² 2xy – 3xy -5 + 3 3y² -xy -2 3y² - xy - 2

  38. Try the following

  39. 1. (2x + 3y) + (4x + 9y) 2.(3a + 5b + 7c) - (5a – 2b + 9c) • 3. (3x – 5) + (x – 7) + (7x + 12) • 4.(3a + 5b + 7c) + (8a – 2b – 9c) • 5. –4x³ + 6x² – 8x – 10 and 7x³ – 4x² + 9x + 3 • 6. Subtract (5m – 6n + 12) from (2m + 3n – 5). • (2m + 3n – 5) - (5m – 6n + 12) -2a + 7b – 2c 6x + 12y 11x 11a + 3b – 2c 3x³ + 2x² - x - 7 -3m + 9n -17

  40. 7.Subtract 8a + 5b – 6c from 10a + 8b + 7c (10a + 8b + 7c) - (8a + 5b – 6c) 8. (4x + 8y + 9z – 7a + 5b) – (4b + 5x + 7y + 3z + 2a) 9. (– 3x2 + 4x – 11) – (–6x2 – 8x + 10) . 10. (7e² + 3e +2) + (9 – 6e + 4e²) + (9e + 2 – 6e²) 2a – 3b + 13c -x – y + 6z – 9a + b 3x² + 12x - 21 5e² + 6e + 13

  41. challenge

  42. Some of the measures of the polygons are given. P represents the measure of the perimeter. Find the measure of the other side or sides. x² - 15x + 3 2x + y 4x - 3 14x² - 4x + 7

  43. The distributive property

  44. The Distributive Property • Distributive Property: the process of distributing the number on the outside of the parentheses to each term in the inside. a(b + c) = ab + ac Example: 5(x + 7) = 5x + 35 5•x + 5•7

  45. Practice #1 3(m - 4) 3 • m - 3 • 4 3m – 12 Practice #2 -2(y + 3) -2 • y + (-2) • 3 -2y + (-6) -2y - 6

  46. Simplify the following: 3(x + 6) = 3x + 18 4(4 – y) = 16 – 4y 7(2 + z) = 14 + 7z 5(2a + 3) = 10a + 15

  47. Simplify the following: 6(3y - 5) = 18y – 30 3 +4(x + 6) = 4x + 27 2x + 3(5x - 3) + 5 = 17x – 4

  48. Distributive practice

  49. 2(4 + 9x) 2. 7(x + -1) 3. 12(a + b + c) • 7(a + c + b) 5. -10(3 + 2 + 7x) 6. -1(3w + 3x + -2z) • -1(x + 2) 8. 3(-2 + 2x2y3 + 3y2) 9. 5(5 + 5x) • y(1 + x) 11. 12x(3x + 3) 12. 9(9x + 9y) 8 + 18x 7x - 7 12a + 12b + 12c -3w – 3x + 2z 7a + 7c + 7b -70x - 50 -x – 2 25x + 25 -6 + 6x²y³ + 9y² y + yx 36x² + 36x 81x + 8y

  50. Factoring

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