1 variables, evaluation
- 2. In mathematics we use symbols such as x, y and z to represent numbers. Variables and Evaluation
- 3. In mathematics we use symbols such as x, y and z to represent numbers. These symbols are called variables because their values change depending on the situation . Variables and Evaluation
- 4. In mathematics we use symbols such as x, y and z to represent numbers. These symbols are called variables because their values change depending on the situation . We use variables and mathematics operations to make expressions which are calculation procedures. Variables and Evaluation
- 5. In mathematics we use symbols such as x, y and z to represent numbers. These symbols are called variables because their values change depending on the situation . We use variables and mathematics operations to make expressions which are calculation procedures. For example, if an apple cost $2 and x represents the number of apples, Variables and Evaluation
- 6. In mathematics we use symbols such as x, y and z to represent numbers. These symbols are called variables because their values change depending on the situation . We use variables and mathematics operations to make expressions which are calculation procedures. For example, if an apple cost $2 and x represents the number of apples, then “2x” is the expression for the cost for x apples. Variables and Evaluation
- 7. In mathematics we use symbols such as x, y and z to represent numbers. These symbols are called variables because their values change depending on the situation . We use variables and mathematics operations to make expressions which are calculation procedures. For example, if an apple cost $2 and x represents the number of apples, then “2x” is the expression for the cost for x apples. Suppose we have 6 apples, set x = 6 in the expression 2x, Variables and Evaluation
- 8. In mathematics we use symbols such as x, y and z to represent numbers. These symbols are called variables because their values change depending on the situation . We use variables and mathematics operations to make expressions which are calculation procedures. For example, if an apple cost $2 and x represents the number of apples, then “2x” is the expression for the cost for x apples. Suppose we have 6 apples, set x = 6 in the expression 2x, we obtain 2(6) = 12 for the total cost. Variables and Evaluation
- 9. In mathematics we use symbols such as x, y and z to represent numbers. These symbols are called variables because their values change depending on the situation . We use variables and mathematics operations to make expressions which are calculation procedures. For example, if an apple cost $2 and x represents the number of apples, then “2x” is the expression for the cost for x apples. Suppose we have 6 apples, set x = 6 in the expression 2x, we obtain 2(6) = 12 for the total cost. The value “6” for x is called input (value). Variables and Evaluation
- 10. In mathematics we use symbols such as x, y and z to represent numbers. These symbols are called variables because their values change depending on the situation . We use variables and mathematics operations to make expressions which are calculation procedures. For example, if an apple cost $2 and x represents the number of apples, then “2x” is the expression for the cost for x apples. Suppose we have 6 apples, set x = 6 in the expression 2x, we obtain 2(6) = 12 for the total cost. The value “6” for x is called input (value). The answer 12 is called the output. Variables and Evaluation
- 11. In mathematics we use symbols such as x, y and z to represent numbers. These symbols are called variables because their values change depending on the situation . We use variables and mathematics operations to make expressions which are calculation procedures. For example, if an apple cost $2 and x represents the number of apples, then “2x” is the expression for the cost for x apples. Suppose we have 6 apples, set x = 6 in the expression 2x, we obtain 2(6) = 12 for the total cost. The value “6” for x is called input (value). The answer 12 is called the output. This process of replacing the variables with input value(s) and find the output is called evaluation. Variables and Evaluation
- 12. In mathematics we use symbols such as x, y and z to represent numbers. These symbols are called variables because their values change depending on the situation . We use variables and mathematics operations to make expressions which are calculation procedures. For example, if an apple cost $2 and x represents the number of apples, then “2x” is the expression for the cost for x apples. Suppose we have 6 apples, set x = 6 in the expression 2x, we obtain 2(6) = 12 for the total cost. The value “6” for x is called input (value). The answer 12 is called the output. This process of replacing the variables with input value(s) and find the output is called evaluation. Variables and Evaluation Each variable can represent one specific measurement only.
- 13. In mathematics we use symbols such as x, y and z to represent numbers. These symbols are called variables because their values change depending on the situation . We use variables and mathematics operations to make expressions which are calculation procedures. For example, if an apple cost $2 and x represents the number of apples, then “2x” is the expression for the cost for x apples. Suppose we have 6 apples, set x = 6 in the expression 2x, we obtain 2(6) = 12 for the total cost. The value “6” for x is called input (value). The answer 12 is called the output. This process of replacing the variables with input value(s) and find the output is called evaluation. Variables and Evaluation Each variable can represent one specific measurement only. Suppose we need an expression for the total cost of apples and pears and x represents the number of apples,
- 14. In mathematics we use symbols such as x, y and z to represent numbers. These symbols are called variables because their values change depending on the situation . We use variables and mathematics operations to make expressions which are calculation procedures. For example, if an apple cost $2 and x represents the number of apples, then “2x” is the expression for the cost for x apples. Suppose we have 6 apples, set x = 6 in the expression 2x, we obtain 2(6) = 12 for the total cost. The value “6” for x is called input (value). The answer 12 is called the output. This process of replacing the variables with input value(s) and find the output is called evaluation. Variables and Evaluation Each variable can represent one specific measurement only. Suppose we need an expression for the total cost of apples and pears and x represents the number of apples, we must use a different letter, say y, to represent the number of pears since they are two distinct measurements.
- 15. Variables and Evaluation When evaluating an expression, replace the variables with the input-values enclosed with ( )’s.
- 16. Example A. a. Evaluate –x if x = –6. Variables and Evaluation When evaluating an expression, replace the variables with the input-values enclosed with ( )’s.
- 17. Example A. a. Evaluate –x if x = –6. When evaluating, insert the input enclosed in a “( )”. Variables and Evaluation When evaluating an expression, replace the variables with the input-values enclosed with ( )’s.
- 18. Example A. a. Evaluate –x if x = –6. When evaluating, insert the input enclosed in a “( )”. Therefore, set x = (–6) we’ve –x – (–6) Variables and Evaluation When evaluating an expression, replace the variables with the input-values enclosed with ( )’s.
- 19. Example A. a. Evaluate –x if x = –6. When evaluating, insert the input enclosed in a “( )”. Therefore, set x = (–6) we’ve –x – (–6) = 6 Variables and Evaluation When evaluating an expression, replace the variables with the input-values enclosed with ( )’s.
- 20. Example A. a. Evaluate –x if x = –6. When evaluating, insert the input enclosed in a “( )”. Therefore, set x = (–6) we’ve –x – (–6) = 6 b. Evaluate –3x if x = –6. Variables and Evaluation When evaluating an expression, replace the variables with the input-values enclosed with ( )’s.
- 21. Example A. a. Evaluate –x if x = –6. When evaluating, insert the input enclosed in a “( )”. Therefore, set x = (–6) we’ve –x – (–6) = 6 b. Evaluate –3x if x = –6. –3x –3(–6) Variables and Evaluation When evaluating an expression, replace the variables with the input-values enclosed with ( )’s.
- 22. Example A. a. Evaluate –x if x = –6. When evaluating, insert the input enclosed in a “( )”. Therefore, set x = (–6) we’ve –x – (–6) = 6 b. Evaluate –3x if x = –6. –3x –3(–6) = 18 Variables and Evaluation When evaluating an expression, replace the variables with the input-values enclosed with ( )’s.
- 23. Example A. a. Evaluate –x if x = –6. When evaluating, insert the input enclosed in a “( )”. Therefore, set x = (–6) we’ve –x – (–6) = 6 b. Evaluate –3x if x = –6. –3x –3(–6) = 18 c. Evaluate –2x2 if x = 6. Variables and Evaluation When evaluating an expression, replace the variables with the input-values enclosed with ( )’s.
- 24. Example A. a. Evaluate –x if x = –6. When evaluating, insert the input enclosed in a “( )”. Therefore, set x = (–6) we’ve –x – (–6) = 6 b. Evaluate –3x if x = –6. –3x –3(–6) = 18 c. Evaluate –2x2 if x = 6. –2x2 –2(6)2 Variables and Evaluation When evaluating an expression, replace the variables with the input-values enclosed with ( )’s.
- 25. Example A. a. Evaluate –x if x = –6. When evaluating, insert the input enclosed in a “( )”. Therefore, set x = (–6) we’ve –x – (–6) = 6 b. Evaluate –3x if x = –6. –3x –3(–6) = 18 c. Evaluate –2x2 if x = 6. –2x2 –2(6)2 = –2(36) Variables and Evaluation When evaluating an expression, replace the variables with the input-values enclosed with ( )’s.
- 26. Example A. a. Evaluate –x if x = –6. When evaluating, insert the input enclosed in a “( )”. Therefore, set x = (–6) we’ve –x – (–6) = 6 b. Evaluate –3x if x = –6. –3x –3(–6) = 18 c. Evaluate –2x2 if x = 6. –2x2 –2(6)2 = –2(36) = –72 Variables and Evaluation When evaluating an expression, replace the variables with the input-values enclosed with ( )’s.
- 27. Example A. a. Evaluate –x if x = –6. When evaluating, insert the input enclosed in a “( )”. Therefore, set x = (–6) we’ve –x – (–6) = 6 b. Evaluate –3x if x = –6. –3x –3(–6) = 18 c. Evaluate –2x2 if x = 6. –2x2 –2(6)2 = –2(36) = –72 Variables and Evaluation When evaluating an expression, replace the variables with the input-values enclosed with ( )’s. d. Evaluate –4xyz if x = –3, y = –2, z = –1.
- 28. Example A. a. Evaluate –x if x = –6. When evaluating, insert the input enclosed in a “( )”. Therefore, set x = (–6) we’ve –x – (–6) = 6 b. Evaluate –3x if x = –6. –3x –3(–6) = 18 c. Evaluate –2x2 if x = 6. –2x2 –2(6)2 = –2(36) = –72 Variables and Evaluation When evaluating an expression, replace the variables with the input-values enclosed with ( )’s. d. Evaluate –4xyz if x = –3, y = –2, z = –1. –4xyz –4(–3)(–2)(–1)
- 29. Example A. a. Evaluate –x if x = –6. When evaluating, insert the input enclosed in a “( )”. Therefore, set x = (–6) we’ve –x – (–6) = 6 b. Evaluate –3x if x = –6. –3x –3(–6) = 18 c. Evaluate –2x2 if x = 6. –2x2 –2(6)2 = –2(36) = –72 Variables and Evaluation When evaluating an expression, replace the variables with the input-values enclosed with ( )’s. d. Evaluate –4xyz if x = –3, y = –2, z = –1. –4xyz –4(–3)(–2)(–1) = 24
- 30. Variables and Evaluation e. Evaluate x – y if x = –3, y = –5.
- 31. Variables and Evaluation e. Evaluate x – y if x = –3, y = –5. x – y (–3) – (–5)
- 32. Variables and Evaluation e. Evaluate x – y if x = –3, y = –5. x – y (–3) – (–5) = –3 + 5
- 33. Variables and Evaluation e. Evaluate x – y if x = –3, y = –5. x – y (–3) – (–5) = –3 + 5 = 2
- 34. f. Evaluate 3x2 – y2 if x = 2 and y = –3. Variables and Evaluation e. Evaluate x – y if x = –3, y = –5. x – y (–3) – (–5) = –3 + 5 = 2
- 35. f. Evaluate 3x2 – y2 if x = 2 and y = –3. Replace x by (2) and y by (–3) in the expression, we have 3*(2)2 – (–3)2 Variables and Evaluation e. Evaluate x – y if x = –3, y = –5. x – y (–3) – (–5) = –3 + 5 = 2
- 36. f. Evaluate 3x2 – y2 if x = 2 and y = –3. Replace x by (2) and y by (–3) in the expression, we have 3*(2)2 – (–3)2 = 3*4 – 9 = 12 – 9 Variables and Evaluation e. Evaluate x – y if x = –3, y = –5. x – y (–3) – (–5) = –3 + 5 = 2
- 37. f. Evaluate 3x2 – y2 if x = 2 and y = –3. Replace x by (2) and y by (–3) in the expression, we have 3*(2)2 – (–3)2 = 3*4 – 9 = 12 – 9 = 3 Variables and Evaluation e. Evaluate x – y if x = –3, y = –5. x – y (–3) – (–5) = –3 + 5 = 2 g. Evaluate –x2 + (–8 – y)2 if x = 3 and y = –2.
- 38. f. Evaluate 3x2 – y2 if x = 2 and y = –3. Replace x by (2) and y by (–3) in the expression, we have 3*(2)2 – (–3)2 = 3*4 – 9 = 12 – 9 = 3 Variables and Evaluation e. Evaluate x – y if x = –3, y = –5. x – y (–3) – (–5) = –3 + 5 = 2 g. Evaluate –x2 + (–8 – y)2 if x = 3 and y = –2. Replace x by (3), y by (–2) in the expression, – (3)2 + (–8 – (– 2))2
- 39. f. Evaluate 3x2 – y2 if x = 2 and y = –3. Replace x by (2) and y by (–3) in the expression, we have 3*(2)2 – (–3)2 = 3*4 – 9 = 12 – 9 = 3 Variables and Evaluation e. Evaluate x – y if x = –3, y = –5. x – y (–3) – (–5) = –3 + 5 = 2 g. Evaluate –x2 + (–8 – y)2 if x = 3 and y = –2. Replace x by (3), y by (–2) in the expression, – (3)2 + (–8 – (– 2))2 = – 9 + (–8 + 2)2
- 40. f. Evaluate 3x2 – y2 if x = 2 and y = –3. Replace x by (2) and y by (–3) in the expression, we have 3*(2)2 – (–3)2 = 3*4 – 9 = 12 – 9 = 3 Variables and Evaluation e. Evaluate x – y if x = –3, y = –5. x – y (–3) – (–5) = –3 + 5 = 2 g. Evaluate –x2 + (–8 – y)2 if x = 3 and y = –2. Replace x by (3), y by (–2) in the expression, – (3)2 + (–8 – (– 2))2 = – 9 + (–8 + 2)2 = – 9 + (–6)2
- 41. f. Evaluate 3x2 – y2 if x = 2 and y = –3. Replace x by (2) and y by (–3) in the expression, we have 3*(2)2 – (–3)2 = 3*4 – 9 = 12 – 9 = 3 Variables and Evaluation e. Evaluate x – y if x = –3, y = –5. x – y (–3) – (–5) = –3 + 5 = 2 g. Evaluate –x2 + (–8 – y)2 if x = 3 and y = –2. Replace x by (3), y by (–2) in the expression, – (3)2 + (–8 – (– 2))2 = – 9 + (–8 + 2)2 = – 9 + (–6)2 = – 9 + 36 = 27
- 42. h. Evaluate (a – b)(b – c) if a = 3, b = –2, c = –4. Variables and Evaluation
- 43. h. Evaluate (a – b)(b – c) if a = 3, b = –2, c = –4. (a – b)(b – c) ((3) – (–2))((–2) – (–4)) Variables and Evaluation
- 44. h. Evaluate (a – b)(b – c) if a = 3, b = –2, c = –4. (a – b)(b – c) ((3) – (–2))((–2) – (–4)) = (3 + 2)(–2 + 4) Variables and Evaluation
- 45. h. Evaluate (a – b)(b – c) if a = 3, b = –2, c = –4. (a – b)(b – c) ((3) – (–2))((–2) – (–4)) = (3 + 2)(–2 + 4) = (5)(2) Variables and Evaluation
- 46. h. Evaluate (a – b)(b – c) if a = 3, b = –2, c = –4. (a – b)(b – c) ((3) – (–2))((–2) – (–4)) = (3 + 2)(–2 + 4) = (5)(2) = 10 Variables and Evaluation
- 47. h. Evaluate (a – b)(b – c) if a = 3, b = –2, c = –4. (a – b)(b – c) ((3) – (–2))((–2) – (–4)) = (3 + 2)(–2 + 4) = (5)(2) = 10 Variables and Evaluation i. Evaluate (2b – 3a)2 if a = –4, b = – 3.
- 48. h. Evaluate (a – b)(b – c) if a = 3, b = –2, c = –4. (a – b)(b – c) ((3) – (–2))((–2) – (–4)) = (3 + 2)(–2 + 4) = (5)(2) = 10 Variables and Evaluation i. Evaluate (2b – 3a)2 if a = –4, b = – 3. (2(–3) –3(–4))2
- 49. h. Evaluate (a – b)(b – c) if a = 3, b = –2, c = –4. (a – b)(b – c) ((3) – (–2))((–2) – (–4)) = (3 + 2)(–2 + 4) = (5)(2) = 10 Variables and Evaluation i. Evaluate (2b – 3a)2 if a = –4, b = – 3. (2(–3) –3(–4))2 = (–6 + 12)2
- 50. h. Evaluate (a – b)(b – c) if a = 3, b = –2, c = –4. (a – b)(b – c) ((3) – (–2))((–2) – (–4)) = (3 + 2)(–2 + 4) = (5)(2) = 10 Variables and Evaluation i. Evaluate (2b – 3a)2 if a = –4, b = – 3. (2(–3) –3(–4))2 = (–6 + 12)2 = (6)2 = 36
- 51. h. Evaluate (a – b)(b – c) if a = 3, b = –2, c = –4. (a – b)(b – c) ((3) – (–2))((–2) – (–4)) = (3 + 2)(–2 + 4) = (5)(2) = 10 Variables and Evaluation j. Evaluate b2 – 4ac if a = –2, b = –3, and c = 5. i. Evaluate (2b – 3a)2 if a = –4, b = – 3. (2(–3) –3(–4))2 = (–6 + 12)2 = (6)2 = 36
- 52. h. Evaluate (a – b)(b – c) if a = 3, b = –2, c = –4. (a – b)(b – c) ((3) – (–2))((–2) – (–4)) = (3 + 2)(–2 + 4) = (5)(2) = 10 Variables and Evaluation j. Evaluate b2 – 4ac if a = –2, b = –3, and c = 5. (–3)2 – 4(–2)(5) i. Evaluate (2b – 3a)2 if a = –4, b = – 3. (2(–3) –3(–4))2 = (–6 + 12)2 = (6)2 = 36
- 53. h. Evaluate (a – b)(b – c) if a = 3, b = –2, c = –4. (a – b)(b – c) ((3) – (–2))((–2) – (–4)) = (3 + 2)(–2 + 4) = (5)(2) = 10 Variables and Evaluation j. Evaluate b2 – 4ac if a = –2, b = –3, and c = 5. (–3)2 – 4(–2)(5) = 9 + 40 = 49 i. Evaluate (2b – 3a)2 if a = –4, b = – 3. (2(–3) –3(–4))2 = (–6 + 12)2 = (6)2 = 36
- 54. Exercise. Evaluate. A. –2x with the input Variables and Evaluation 1. x = 3 2. x = –3 3. x = –5 4. x = –1/2 B. –y – 2x with the input 5. x = 3, y = 2 6. x = –2, y = 3 7. x = –1, y = –4 8. x = ½, y = –6 C. (–x)2 with the input 9. x = 3 10. x = –3 11. x = –5 12. x = –1/2 D. –x2 with the input 13. x = –2 14. x = –3 15. x = –9 16. x = –1/3 E. –2x3 with the input 17. x = 3 18. x = –2 19. x = –1 20. x = –½ F. 3x2 – 2x – 1 with the input 21. x = – 4 22. x = –2 23. x = –1 24. x = ½
- 55. Variables and Evaluation G. –2y2 + 3x2 with the input 25. x = 3, y = 2 26. x = –2, y = – 3 27. x = –1, y = –4 28. x = –1, y = –1/2 J. b2 – 4ac with the input 37. a = –2, b = 3, c = –5 38. a = 4, b = –2, c = – 2 39. a = –1, b = – 2, c = –3 40. a = 5, b = –4, c = 4 H. x3 – 2x2 + 2x – 1 with the input 29. x = 1 30. x = –1 31. x = 2 32. x = ½ 33. a = –1, b = – 2 34. a = 2, b = –4 –b 2a I. with the input 35. a = –2, b = – 8 36. a = 2, b = – 12
- 56. Variables and Evaluation a – b c – d K. with the input 43. a = –2, b = 3, c = –5, d = 0 44. a = –1, b = –2, c = –2, d = 14 41. a = 1, b = –2, c = 2, d = – 2 42. a = –4, b = –2, c = –1, d = –4 (a – b)(b – c) (c – d)(d – a) L. with the input 47. a = –2, b = 3, c = –5, d = 0 48. a = –1, b = –2, c = –2, d = 14 45. a = 1, b = –2, c = 2, d = 2 46. a = –4, b = –2, c = –1, d = –4 M. b2 – a2 – c2 if 49. a = –2, b = 3, c = –5 . 50. a = 4, b = –2, c = – 2 N. b2 – 4ac if 51. a = –2, b = 3, c = –5 . 52. a = 4, b = –2, c = – 2