Bails – Math 094 Notes
Section 2.1 Introduction to Variables
Variable: letter that represents part of a rule that varies.
Constant: part of a rule that does not change.
Coefficient: the number in front of the variable.
Expression: combination of variables and constants terms separated by operations.
For the following expressions, identify the variable term, coefficient, and constant term.
1.
3x  7
variable term =
coefficient =
constant term =
2.
4y  11
variable term =
coefficient =
constant term =
3.
x  6
variable term =
coefficient =
constant term =
4.
y
variable term =
coefficient =
constant term =
5.
30
variable term =
coefficient =
constant term =
Review of Exponents:
Recall 32 = 3  3 = 9
so
3x2 = 3x  x and
Rewrite each expression without exponents.
6.
7.
2x 3
8.
x4y 3
9.
x2y3 = x  x y  y  y
xy 3
9x 2 yz3
Steps for evaluating an expression:
1.
Write the original expression
2.
Copy the original again, but replace the _______________ with a set of parenthesis
3.
Place given value of variable inside parenthesis
4.
Simplify
Evaluate each expression when a  3 , b  1, and c  2
3abc
10.
11.
a3
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Chapter 2 Understanding Variables and Solving Equations
Evaluate each expression when a  3 , b  1, and c  2
12.
2a2 b2
13.
ab  bc
14.
2a  c
b
15.
3c 2  5b
3  a
16.
The expression for determining the cost per ounce is c  z where c is the total cost
and z is the number of ounces. Evaluate the expression when the total cost of caviar
is $48 for 16 oz.
17.
The expression for determining the perimeter of a rectangle is 2L + 2W, where L is
the length and W is the width. Evaluate the expression when
a.
The length is 15 centimeters (cm) and the width is 11 centimeters.
b.
The length is 20 feet (ft) and the width is 15 feet.
22
Bails – Math 094 Notes
Section 2.2 Simplifying Expressions
Similar (Like) Terms: have the same variable(s) and exponents. The number in front of the
variable (coefficient) can differ.
Like Terms Examples:
3x, 4x
2xy, -5xy
4y2, 10y2
Unlike Terms Examples:
3x, 3y
3x2, 4x
4x, 5xy
Steps for simplifying an expression:
1.
Rewrite expression without parenthesis
If necessary, make sure to distribute negative through parentheses
2.
Identify like ______________________
3.
Add/subtract the ______________________ (number in front of variable) & copy the variable
**Variables always stay the same in addition and subtraction
4.
Write each answer with the variables in alphabetical order and any constant term last.
Simplify (combine like terms) if possible.
3x  8x
1.
2.
xxx
3.
3y  7y  y
4.
2x 2  5 x 2
5.
8 xy  9xy  xy
6.
2x  3  4 x  1
7.
3x  5  9x  2
8.
4y  5 x  3y  7 x
9.
3 x 2  4 x  x 2  11x
10.
6x 3  7x 2  10x
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Chapter 2 Understanding Variables and Solving Equations
Review of Associative Property of Multiplication:
(a  b)  c = a  (b  c)
Simplify by using the associative property of multiplication.
11.
8  3x  =
12.


9 3x 2 =
Review of Distributive Property: a  b  c   ab  ac
Note: If there is no number (coefficient) in front of the variable, the number is 1. Therefore, x = 1x
Use the distributive property to simplify each expression.
13.
2  6 x  1 =
14.
3  x  2 =
15.
4  4 x  1 =
16.
5  7 x  3  =
18.
3  x  2   15
Simplify each expression.
2  6x  5  2
17.
19.
4  6  9x  2
20.
6  10  x  1
21.
16 x  6  9 x  2  18
22.
3  5  2x  7   15 x
23.
2  3 x   7  4  8 x  1
24.
5 x  3  2x  9   4  3 x  2   11
24
Bails – Math 094 Notes
Section 2.3 Solving Equations Using Addition
Addition Property of Equality
If
A=B
Then
A+C=B+C
A, B, C are algebraic expressions
Adding/subtracting the same number to both sides of an equation keeps the equation balanced.
Note: A solution for an equation is a number that makes the equation a true statement.
Solve each equation and check the answer.
1.
x  7  11
2.
x  7  11
3.
x  7  11
4.
x  7  11
5.
33  27  x
6.
52  x  61
7.
x  17  21  28
8.
16  x  30  14
25
Chapter 2 Understanding Variables and Solving Equations
For each equation, simplify each side (if possible) then solve and check the answer.
9.
x  8  4  11  9
10.
6x  4  5x  3
11.
7 x  8  6 x  3  1
12.
3  6  9  9x  5  8x
13.
6 x  7 x  9  19  8
14.
 x  3  7 x  9 x  4  7  9
26
Bails – Math 094 Notes
Section 2.4 Solving Equations Using Division
Division Property of Equality
If A = B, then
A B

C C
as long as C  0
Dividing by the same nonzero number on both sides of an equation keeps the equation balanced.
Solve each equation and check the answer.
1.
6 x  42
2.
5 x  40
3.
8 x  48
4.
121  11x
5.
27 x  0
6.
14   x
For each equation, simplify each side (if possible) then solve and check the answer.
7.
2 x  3 x  17  23
8.
8 x  15 x  71  15
27
Chapter 2 Understanding Variables and Solving Equations
For each equation, simplify each side (if possible) then solve and check the answer.
9.
8  2x   32
10.
60  2  15x 
11.
14 x  10 x  36  44
12.
7 x  5 x  7  4  59
13.
82  96  5  4 x   8  2x   3 x
14.
36  36  41  50  x  3  2x   4  4 x 
28
Bails – Math 094 Notes
Section 2.5 Solving Equations with Several Steps
Steps for solving equations with variables on both sides:
1.
Simplify each side of equation by removing parentheses and combining like terms
2.
Use _____________________ property to move all variables (letters) to one side.
3.
Use _____________________ property to move all constant terms (numbers) to the
opposite side of the equation.
4.
Use division property to get a _____________________ (the number in front of the
variable) of 1.
5.
Check the solution by going back to the original equation.
Solve each equation and show all work.
1.
5 x  10  55
2.
7 x  4  31
3.
3 x  6  36
4.
42  3  x  7 
5.
3 x  2 x  40
6.
11x  4 x  8  15
29
Chapter 2 Understanding Variables and Solving Equations
Solve each equation and show all work.
7.
5 x  3  2x  3
8.
2x  7  4x  1
9.
5 x  10  19  8 x
10.
15 x  1  4 x  20
11.
8  2x  1  6  3 x  5 
12.
7  x  8   2  x  13 
13.
12  x  2  5  2x  1
14.
x  28  10  4  x  6   27
30