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Pre-Algebra!
Vocabulary
Variable –
the letters or symbols used in mathematics to represent unknown quantities
x is commonly used, also a, b, and c.
Coefficient –
the number that is attached to the variable by the multiplication operation
3c = 3 · c, 3 is the coefficient
Algebraic Term –
the product of the coefficient and one or more variables
4b is an algebraic term consisted of the coefficient 4 and the variable b
Writing Algebraic Terms
3 groups of b bananas each = b + b + b or 3b
*We use multiplication to represent
repeated addition*
5 groups of s students each = 5 · s = 5s
1) h + h + h + h = 4 · h = 4h
2
2
2
2
2) g + g + g = 3 · g = 3g
2
3) 3m + 3m + 3m = 3 · (3m) = 9 · m = 9m
Combining Algebraic Terms
Algebraic Terms or “Like Terms” are terms that contain the same variable (with exponent) regardless of the
coefficient.
Like terms
Unlike Terms
3x and 6x are like terms because
4x and 4w are not like terms because
they both contain an x
the variables are different
2
2
6c and 2c are like terms because
2
they both contain a c
2
3
8j and 9j are not like terms because
the variable’s exponents are different
TO COMBINE LIKE TERMS:
1. Determine which terms contain the same variable (with exponent)
2. Add or subtract the coefficients by using the sign in front of the second term
3. Attach the common variable (with exponent)
Example ONE
3x + 6x
1. They both contain the same
variable x
2. Add the coefficients (3 + 6)
3. Attach the variable x on the
end to represent multiplication
= 9x
Example TWO
10a – 3a
1. They both contain the same
variable a
2. Subtract the coefficients (10 –
3) because we take the sign in
front of the second term
3. Attach the variable a on the
end to represent multiplication
= 7a
Example THREE
7b + 3c – 4b
1. We have two variables, b and
c, combine the b terms
2. Subtract the coefficients (7 –
4) because we take the sign in
front of the second like term
3. Attach the variable b on the
end to represent multiplication
and add the last term because
we take the sign in front of the
term.
= 3b + 3c
Algebraic Expressions
Algebraic Expressions are made up of algebraic terms and one or more math operations.
Key Words to Identify Operations
Addition
Subtraction
Sum
Plus
Added to
Multiplication
Difference
Take away
Minus
Division
Product
Times
Of
Quotient
Broke into
Separated by
Putting Algebraic Expressions into Words
3 + n = 3 plus some number n
8x = 8 times some number x
h – 5 = five taken away from some number h
10 = 10 divided by some number t
t
Putting Words into Algebraic Expressions
Some number r plus 10 = r + 10
The product of 9 and some number d = 9d
Some number i subtracted from 20 = 20 – i
the quotient of 25 and some number k =
25
k
Evaluating Algebraic Expressions
Lisa’s long distance plan allows her to talk on the phone for $.02 per minute. How much does each
of the following phones calls cost her?
Her plan is costing $.02 · minutes used or $.02m
Phone Call #1 = 15 minutes
$.02 · 15 = $.30
Phone Call #2 = 34 minutes
$.02 · 34 = $.68
Phone Call #3 = 109 minutes
$.02 · 109 = $2.18
To Evaluate an Expression:
Substitute the given value into the expression and perform the math using the order of operations
5 + n, n = 4
5+4
=9
Example ONE:
5 + n, n = 5
5+5
=10
5 + n, n = 6
5+6
=11
Evaluate 8p + 5 for the given values of p
p=2
8·2+5
16 + 5
= 21
p=3
8·3 + 5
24 + 5
= 29
p=4
8·4 + 5
32 + 5
= 37
2
Example TWO: Evaluate t for the given values of t
t=5
2
5 =5·5=
= 25
t=8
2
8 =8·8=
= 64
t = 10
2
10 = 10 · 10 =
= 100
Solving Algebraic Equations
Algebraic Equations are algebraic statements that say that two expressions are equal.
When we want to solve an algebraic equation we are looking for numerical values for the
variable that make the equation true.
The variable is a missing number. So think of it has a mystery to be solved.
Josie loves collecting stamps. Last time she checked, her collection had 44
stamps. On Thursday her friend came over and gave her more stamps. Her collection is now up to 63
stamps. How many stamps did her friend give her?
The equation…
In words it means…
44 + x =63
44 plus some number equals 63
To solve mysteries we have to back track our moves. Instead of adding (which is what the problem is doing) we are
going to subtract.
Subtract the number on the variable side from both sides.
44 + x = 63
! 44 ! 44
Check your work.
x = 19
0 + x = 19
44
+ 19
63
Example ONE:
Example TWO:
m + 19 = 48
Instead of adding we need to subtract.
Subtract the number on the variable side from both sides.
g – 45 = 33
Instead of subtracting we need to add.
Add the number on the variable side to both sides.
m + 19 = 48
! 19 ! 19
m + 0 = 29 Check your work.
m = 29
29
+ 19
g ! 45 = 33
+ 45 + 45
g + 0 = 78
48
Check your work
g = 78
78
! 45
33
Example THREE:
3 x = 12
Instead of multiplying 3 times x we divide.
Divide both sides by the number on the variable side.
3 x = 12
3 3
x= 4
Example FOUR:
y
= 15
5
Instead of dividing by 5 we need to multiply.
Multiply both sides by 5
5•
y
= 15 • 5
5
y = 75