1. AIM Algebra In Motion Fall 2008
On the MENU below please find links to three of the Power Point presentations we have created as part of our AIM Project. Thanks for looking!
Kathy Monaghan & Pat Peterson
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1.1 Introduction to Algebra
3.3a The Slope of a Line
7.2 Solving Systems by Substitution
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Thanks for Looking!
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3. Getting Started with Algebra
What is
Algebra ?
Algebra
is a branch of mathematics in which symbols, usually letters, are used to represent quantities that can be replaced by a number or an expression.
( ) 4 X 3 2
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4. Getting Started with Algebra
Who invented
Algebra ?
Algebra
is a reasoning
skill and language that developed and evolved along with civilization No one person invented
Algebra
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5. Getting Started with Algebra
Where did the word
Algebra
originate ?
The word is from
Kitab al-Jabr wa-l-Muqabala
which was a book written in approximately 820 A.D. by a Persian mathematician.
Algebra
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6. a letter used to represent various numbers.
Variables
variable
A is
a letter used to represent various numbers. X
“x” is frequently used as the variable, but many other letters can be used.
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7. leg inseam measurement L
Variables
For example, jean sizes are often given by waist and leg inseam measurements.
waist measurement W
leg inseam measurement L
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8. Here are three examples:
Define each Variable
We must always define what quantity or measurement the letter represents.
Here are three examples: X
= unknown number W
= waist measurement
= leg inseam measurement L
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9. = the speed of light in Einstein’s equation E = mc2 c
Constants
A is
constant
a letter used to represent a number that doesn’t change its value in the problem. For example:
= “pi” = …. p
= the speed of light in Einstein’s equation E = mc2 c
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10. Algebraic ExpressionsAn
algebraic expression
is a mathematical phrase using variables, constants, numerals, & operation signs.
algebraic expression An
will NOT have any of the following symbols: = > < > <
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11. Algebraic Expressions: Examples
x is the variable.
+ is the operation
5 is a numeral and a constant.
is the
algebraic expression
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12. Algebraic Expressions: Examples
p is the variable.
. is the indicated operation
3 is a numeral and a constant.
is the
algebraic expression
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13. Algebraic Expressions: Examples
z is the variable.
- is the operation
9 is a numeral and a constant.
is the
algebraic expression
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14. Algebraic Expressions: Examples
y is the variable.
÷ is the operation
5 is a numeral and a constant.
is the
algebraic expression
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15. Algebraic Expressions: Examples
x and y are variables.
 and + are operations
2 and 5 are numerals and constants.
is the
algebraic expression
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16. Algebraic Expressions: Examples
x and y are variables.
+ ÷  are operations
5 and 2 are numerals and constants.
is the
algebraic expression
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17. Substitution
When a variable is replaced with a numerical value, that is called substitution.
Sometimes, in higher mathematics, a variable is replaced with an expression. That is also called substitution.
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18. Evaluate an Algebraic Expression
When a numerical value is substituted into an algebraic expression and then simplified, that is called evaluating the expression.
Evaluating means you will compute a numerical value.
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19. Evaluate an Expression: Example 1a
when
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20. Evaluate an Expression: Example 1b
when
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21. Evaluate an Expression: Example 1c
when
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22. Evaluate an Expression: Example 1d
when
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23. Evaluate an Expression: Example 2
a) when
b) when
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24. Evaluate an Expression: Example 3
when
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25. Evaluate an Expression: Example 4a
when
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26. Evaluate an Expression: Example 4b
when
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27. Evaluate an Expression: Example 5
When and
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28. Evaluate an Expression: Example 6a
When and
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29. Evaluate an Expression: Example 6b
When and
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30. Evaluate an Expression: Example 6c
When and
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31. Evaluate an Expression: Example 6d
When and
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32. Application: Area of a Rectangle
The AREA of a Rectangle
Area = length x width
A=l.w
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33. Translating: English into Algebra
In order to solve problems, English phrases must be translated into the language of algebra.
The following slides list keywords which can help us translate.
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34. English & Algebra ADDITION
The following words translate as ADDITION:
Plus
Sum
Add
Added to
Total
More than
Increased by
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35. The following phrases would translate to :
X + 7
The following phrases would translate to :
A number plus seven
The sum of a number and seven
Add a number and seven
Seven added to a number
The total of seven and a number
Seven more than a number
A number increased by seven
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36. English & Algebra SUBTRACTION
The following words translate as SUBTRACTION:
Minus
Difference
Subtract
Subtracted From
Take away
Less Than
Decreased by
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37. The following phrases would translate to :
X - 7
The following phrases would translate to :
A number minus seven
The difference of a number and seven
Subtract a number and seven
Seven subtracted from a number
Seven take away a number
Seven less than a number
A number decreased by seven
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38. English & Algebra MULTIPLICATION
The following words translate as MULTIPLICATION:
Multiplied by
Multiply
Product
Times Of
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39. The following phrases would translate to :7x
The following phrases would translate to :
A number multiplied by seven
Multiply seven and a number
The product of a number and seven
The product of seven and a number
Seven times a number
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40. English & Algebra DIVISION
The following words translate as DIVISION:
Divided by
Divide
Quotient
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41. The following phrases would translate to :
x/7
The following phrases would translate to :
A number divided by seven
Divide a number by seven
The quotient of a number and seven
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42. The following phrases would translate to :
7/x
The following phrases would translate to :
Seven divided by a number
Divide a seven by a number
The quotient of seven and a number
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43. “Half of a number” would beor or
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44. “Thirty percent of a number” is:or
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45. “Twice” or “Double” “Twice a number” is: “Double a number” is: SAMPLE
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46. Translate a Phrase “Seven more than twice a number” Seven more than
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47. Translate a Phrase “Seven less than twice a number” Seven less than
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48. Translate a Phrase (watch for the comma)
“the quotient of seven and a number increased by two”
“the quotient of seven and a number, increased by two”
comma
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49. Salary Increase? Suppose you will get a salary increase of 3%.
Let s represent your old salary.
The increase is 3% of your current salary, so 0.03s is the increase.
Your new salary will be the sum of the old salary and the increase.
So, s s is your new salary.
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50. Discount?
Suppose the bookstore has all merchandise on sale for 15% off.
Let p represent the regular price.
The discount is 15% of the regular price, so 0.15p is the discount.
The sale price will be the difference of the regular price and the discount
So, p p is the sale price.
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51. Click here to go back to the menu.
That’s All for Now!
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52. Slope
Slope
of a
Straight Line
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53. Definition: Slope The slope of the line containing points
P1(x1, y1) and P2(x2, y2) is given by
The denominator can’t be zero.
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54. A Slope Triangle RISE y2 - y1
x2 - x1
RUN
P2(x2, y2)
RISE
y2 - y1
P1(x1, y1)
RUN
x2 - x1
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55. m
Calculus
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56. The line is rising (going upwards)
Compute the Slope
The line is rising (going upwards)
The slope is positive.
(4, -1)
(-2, -3)
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57. Another Slope
(0, 5)
(-4, -3)
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58. The line is falling (going downwards).
Compute this Slope
The line is falling (going downwards).
The slope is negative.
(-5, 2)
(4, -1)
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59. HORIZONTAL Compute the Slope The line is horizontal.
The slope is zero, 0.
HORIZONTAL
(-5, -1)
(4, -1)
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60. VERTICAL Compute the Slope The line is vertical.
The slope is undefined.
vertical
california
VERTICAL
(-4, 0)
(-4, -5)
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61. SLOPE BASICS POSITIVE SLOPE The line rises from left to right.
ZERO SLOPE
The line is HORIZONTAL (zero rise)
NEGATIVE SLOPE
The line falls from left to right.
UNDEFINED SLOPE
The line is VERTICAL (zero run)
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62. Determine the slope of the line containing points P1 and P2.
Practice Problem 1a
Determine the slope of the line
containing points P1 and P2.
RISE = 2
RUN = 3
The line rises from left to right.
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63. Determine the slope of the line containing points P1 and P2.
Practice Problem 1b
Determine the slope of the line
containing points P1 and P2.
The slope is undefined.
Vertical Line
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64. Determine the slope of the line containing points P1 and P2.
Practice Problem 1c
Determine the slope of the line
containing points P1 and P2.
RUN = 2
RISE = -1
The line falls from left to right.
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65. Determine the slope of the line containing points P1 and P2.
Practice Problem 1d
Determine the slope of the line
containing points P1 and P2.
RISE = 0
RUN = 6
The line is Horizontal.
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67. Solving Systems of Linear Equations Substitution Method
Systems of Equations
Solving Systems of
Linear Equations
Substitution Method
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68. In order to solve a system of equations by substitution:
Systems of Equations
In order to solve a system of equations by substitution:
Solve one equation for x or for y (pick the easiest one).
Substitute for that letter in the OTHER equation.
Solve this equation for the variable.
Substitute the answer back into the original equation.
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69. No Graph?
WHAT LUCK!
Just Substitute for x!
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70. Let’s Check the Answer
Replace x with 2
& y with -4
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71. Another Substitution 4 U !
Substitute for x
in the second equation!
Solve this equation for y
and then determine x.
is the solution set.
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72. Be Sure to Check the Answer!
Replace x with 16
& y with 5
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73. Solve this equation for x to get:
Would I Try to Fool You?
Substitute for y
in the second equation!
Solve this equation for x to get:
YIKES! That is FALSE!
NO SOLUTION!
is the solution set.
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74. Substitute for x in the second equation! is the solution set.
Guess what?
Substitute for x
in the second equation!
is the solution set.
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75. Check To Be Sure!
Replace x with 0
& y with 0
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76. Solve the second equation for x.
OK…last one for now!
Solve the second equation for x.
Substitute for x in the the first equation.
Simplify.
continued on next page...
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77. Always TRUE means they are the same line.
Problem continued!
TRUE!
Always TRUE means they are the same line.
Write the answer using the slope intercept form of the equation.
{(x, mx + b)}
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