25 variables and evaluation

Variables and Evaluation

http://www.lahc.edu/math/frankma.htm

In mathematics we use symbols such as x, y and z to

represent numbers.



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,








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 value “6” for x is called input (value). The answer 12 is

called the output.












called the output. This process of replacing the variables with

input value(s) and find the output is called 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,















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.

Variables and EvaluationWhen evaluating an expression, replace the variables with the

input-values enclosed with ( )’s.

Example A.

a. Evaluate –x if x = –6.



Example A.


When evaluating, insert the input enclosed in a “( )”.



Example A.



Therefore, set x = (–6) we’ve

–x – (–6)



Example A.




–x – (–6) = 6



Example A.




–x – (–6) = 6

b. Evaluate –3x if x = –6.



Example A.




–x – (–6) = 6


–3x –3(–6)



Example A.




–x – (–6) = 6


–3x –3(–6) = 18



Example A.




–x – (–6) = 6


–3x –3(–6) = 18

c. Evaluate –2x2 if x = 6.



Example A.




–x – (–6) = 6


–3x –3(–6) = 18


–2x2 –2(6)2



Example A.




–x – (–6) = 6


–3x –3(–6) = 18


–2x2 –2(6)2 = –2(36)



Example A.




–x – (–6) = 6


–3x –3(–6) = 18


–2x2 –2(6)2 = –2(36) = –72



Example A.




–x – (–6) = 6


–3x –3(–6) = 18


–2x2 –2(6)2 = –2(36) = –72



d. Evaluate –4xyz if x = –3, y = –2, z = –1.

Example A.




–x – (–6) = 6


–3x –3(–6) = 18


–2x2 –2(6)2 = –2(36) = –72




–4xyz

–4(–3)(–2)(–1)

Example A.




–x – (–6) = 6


–3x –3(–6) = 18


–2x2 –2(6)2 = –2(36) = –72




–4xyz

–4(–3)(–2)(–1) = 24

Variables and Evaluatione. Evaluate x – y if x = –3, y = –5.


x – y (–3) – (–5)


x – y (–3) – (–5) = –3 + 5


x – y (–3) – (–5) = –3 + 5 = 2

f. Evaluate 3x2 – y2 if x = 2 and y = –3.


x – y (–3) – (–5) = –3 + 5 = 2


Replace x by (2) and y by (–3) in the expression, we have

3*(2)2 – (–3)2


x – y (–3) – (–5) = –3 + 5 = 2



3*(2)2 – (–3)2

= 3*4 – 9

= 12 – 9


x – y (–3) – (–5) = –3 + 5 = 2



3*(2)2 – (–3)2

= 3*4 – 9

= 12 – 9

= 3


x – y (–3) – (–5) = –3 + 5 = 2

g. Evaluate –x2 + (–8 – y)2 if x = 3 and y = –2.



3*(2)2 – (–3)2

= 3*4 – 9

= 12 – 9

= 3


x – y (–3) – (–5) = –3 + 5 = 2


Replace x by (3), y by (–2) in the expression,

– (3)2 + (–8 – (– 2))2



3*(2)2 – (–3)2

= 3*4 – 9

= 12 – 9

= 3


x – y (–3) – (–5) = –3 + 5 = 2



– (3)2 + (–8 – (– 2))2

= – 9 + (–8 + 2)2



3*(2)2 – (–3)2

= 3*4 – 9

= 12 – 9

= 3


x – y (–3) – (–5) = –3 + 5 = 2



– (3)2 + (–8 – (– 2))2

= – 9 + (–8 + 2)2

= – 9 + (–6)2



3*(2)2 – (–3)2

= 3*4 – 9

= 12 – 9

= 3


x – y (–3) – (–5) = –3 + 5 = 2



– (3)2 + (–8 – (– 2))2

= – 9 + (–8 + 2)2

= – 9 + (–6)2

= – 9 + 36

= 27

h. Evaluate (a – b)(b – c) if a = 3, b = –2, c = –4.



(a – b)(b – c)

((3) – (–2))((–2) – (–4))



(a – b)(b – c)

((3) – (–2))((–2) – (–4))

= (3 + 2)(–2 + 4)



(a – b)(b – c)

((3) – (–2))((–2) – (–4))

= (3 + 2)(–2 + 4)

= (5)(2)



(a – b)(b – c)

((3) – (–2))((–2) – (–4))

= (3 + 2)(–2 + 4)

= (5)(2)

= 10



(a – b)(b – c)

((3) – (–2))((–2) – (–4))

= (3 + 2)(–2 + 4)

= (5)(2)

= 10


i. Evaluate (2b – 3a)2 if a = –4, b = – 3.


(a – b)(b – c)

((3) – (–2))((–2) – (–4))

= (3 + 2)(–2 + 4)

= (5)(2)

= 10



(2(–3) –3(–4))2


(a – b)(b – c)

((3) – (–2))((–2) – (–4))

= (3 + 2)(–2 + 4)

= (5)(2)

= 10



(2(–3) –3(–4))2

= (–6 + 12)2


(a – b)(b – c)

((3) – (–2))((–2) – (–4))

= (3 + 2)(–2 + 4)

= (5)(2)

= 10



(2(–3) –3(–4))2

= (–6 + 12)2

= (6)2 = 36


(a – b)(b – c)

((3) – (–2))((–2) – (–4))

= (3 + 2)(–2 + 4)

= (5)(2)

= 10


j. Evaluate b2 – 4ac if a = –2, b = –3, and c = 5.


(2(–3) –3(–4))2

= (–6 + 12)2

= (6)2 = 36


(a – b)(b – c)

((3) – (–2))((–2) – (–4))

= (3 + 2)(–2 + 4)

= (5)(2)

= 10



(–3)2 – 4(–2)(5)


(2(–3) –3(–4))2

= (–6 + 12)2

= (6)2 = 36


(a – b)(b – c)

((3) – (–2))((–2) – (–4))

= (3 + 2)(–2 + 4)

= (5)(2)

= 10



(–3)2 – 4(–2)(5)

= 9 + 40 = 49


(2(–3) –3(–4))2

= (–6 + 12)2

= (6)2 = 36

Exercise. Evaluate.

A. –2x with the input


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

Variables and EvaluationG. –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

–b2a

I. with the input

35. a = –2, b = – 8 36. a = 2, b = – 12

Variables and Evaluationa – b

c – dK. 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

Education

25 variables and evaluation