25 variables and evaluation
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Transcript of 25 variables and evaluation
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
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
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
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
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
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
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
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
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
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.
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,
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.
Variables and EvaluationWhen evaluating an expression, replace the variables with the
input-values enclosed with ( )’s.
Example A.
a. Evaluate –x if x = –6.
Variables and EvaluationWhen evaluating an expression, replace the variables with the
input-values enclosed with ( )’s.
Example A.
a. Evaluate –x if x = –6.
When evaluating, insert the input enclosed in a “( )”.
Variables and EvaluationWhen evaluating an expression, replace the variables with the
input-values enclosed with ( )’s.
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 EvaluationWhen evaluating an expression, replace the variables with the
input-values enclosed with ( )’s.
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 EvaluationWhen evaluating an expression, replace the variables with the
input-values enclosed with ( )’s.
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 EvaluationWhen evaluating an expression, replace the variables with the
input-values enclosed with ( )’s.
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 EvaluationWhen evaluating an expression, replace the variables with the
input-values enclosed with ( )’s.
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 EvaluationWhen evaluating an expression, replace the variables with the
input-values enclosed with ( )’s.
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 EvaluationWhen evaluating an expression, replace the variables with the
input-values enclosed with ( )’s.
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 EvaluationWhen evaluating an expression, replace the variables with the
input-values enclosed with ( )’s.
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 EvaluationWhen evaluating an expression, replace the variables with the
input-values enclosed with ( )’s.
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 EvaluationWhen evaluating an expression, replace the variables with the
input-values enclosed with ( )’s.
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 EvaluationWhen evaluating an expression, replace the variables with the
input-values enclosed with ( )’s.
d. Evaluate –4xyz if x = –3, y = –2, z = –1.
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 EvaluationWhen 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)
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 EvaluationWhen 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
f. Evaluate 3x2 – y2 if x = 2 and y = –3.
Variables and Evaluatione. Evaluate x – y if x = –3, y = –5.
x – y (–3) – (–5) = –3 + 5 = 2
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 Evaluatione. Evaluate x – y if x = –3, y = –5.
x – y (–3) – (–5) = –3 + 5 = 2
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 Evaluatione. Evaluate x – y if x = –3, y = –5.
x – y (–3) – (–5) = –3 + 5 = 2
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 Evaluatione. 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.
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 Evaluatione. 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
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 Evaluatione. 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
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 Evaluatione. 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
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 Evaluatione. 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
h. Evaluate (a – b)(b – c) if a = 3, b = –2, c = –4.
(a – b)(b – c)
((3) – (–2))((–2) – (–4))
Variables and Evaluation
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
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
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
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.
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
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
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
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
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
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
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
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