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Transcript 
Linear Equations in One Variable
Basic Ideas and Definitions
An equation is a sentence that
expresses the equality of two
algebraic expressions.
Basic Ideas and Definitions
An equation is a sentence that
expresses the equality of two
algebraic expressions.
Consider the equation
2x+1=7
Basic Ideas and Definitions
An equation is a sentence that
expresses the equality of two
algebraic expressions.
Consider the equation
2x+1=7
We say 2(3) +1=7 is true.
Basic Ideas and Definitions
An equation is a sentence that
expresses the equality of two
algebraic expressions.
Consider the equation
2x+1=7
We say 2(3) +1=7 is true.
We say that 3 satisfies the equation.
Basic Ideas and Definitions
Algebraic Expressions
8x + 9
y-4
3
x y
z
8
Basic Ideas and Definitions
A Linear Equation in one variable
involves only real numbers and one
variable raised to the first power
x + 1 = -2
x-3=5
2k + 5 = 10
Basic Ideas and Definitions
A Linear Equation in one variable
can be written in the form
Ax + B = C
Basic Ideas and Definitions
Any number that satisfies the
equation is called a solution or root
to the equation.
Solution Set
The set of all
solutions to
an equation is
called the
solution set to
the equation.
Example 1
Determine whether each equation
is satisfied by the number following
the equation
3x  7  8, - 5
Example 1
Determine whether each equation
is satisfied by the number following
the equation
3x  7  8, - 5
3 5  7  8
Example 1
Determine whether each equation
is satisfied by the number following
the equation
3x  7  8, - 5
3 5  7  8
15  7  8
True
Example 1
Determine whether each equation
is satisfied by the number following
the equation
2x 1  2x  3, 4
Example 1
Determine whether each equation
is satisfied by the number following
the equation
2x 1  2x  3, 4
24 1  24  3
Example 1
Determine whether each equation
is satisfied by the number following
the equation
2x 1  2x  3, 4
24 1  24  3
23  8  3
False
Solving Equations
To solve an equation means to find its
solution set.
The most basic method for solving
equations involves the properties of
equality.
Properties of Equality
Addition-Subtraction Property of
Equality
The same real number may be
added to or subtracted from each
side of an equation without
changing the solution set.
Properties of Equality
Multiplication-Division
Property of Equality
Each side of an equation may be
multiplied by or divided by he
same real number without
changing the solution set.
Example 1
Solve the equation
4x - 2x - 5 = 4 + 6x + 3
Example 1
Solve the equation
4x - 2x - 5 = 4 + 6x + 3
2x - 5
Example 1
Solve the equation
4x - 2x - 5 = 4 + 6x + 3
2x - 5  6x  7
Example 1
Solve the equation
4x - 2x - 5 = 4 + 6x + 3
2x - 5  6x  7
 2x
 2x
Example 1
Solve the equation
4x - 2x - 5 = 4 + 6x + 3
2x - 5  6x  7
 2x
 2x
5 
Example 1
Solve the equation
4x - 2x - 5 = 4 + 6x + 3
2x - 5  6x  7
 2x
 2x
 5  4x
Example 1
Solve the equation
4x - 2x - 5 = 4 + 6x + 3
2x - 5  6x  7
 2x
 2x
 5  4x  7
Example 1
Solve the equation
4x - 2x - 5 = 4 + 6x + 3
2x - 5  6x  7
 2x
 2x
 5  4x  7
7
7
Example 1
Solve the equation
4x - 2x - 5 = 4 + 6x + 3
2x - 5
 2x
5
7
 12
 6x  7
 2x
 4x  7
7

Example 1
Solve the equation
4x - 2x - 5 = 4 + 6x + 3
2x - 5
 2x
5
7
 12
 6x  7
 2x
 4x  7
7
 4x
Example 1
Solve the equation
4x - 2x - 5 = 4 + 6x + 3
2x - 5
 2x
5
7
 12
 142
 6x  7
 2x
 4x  7
7
 4x

4x
4
Example 1
Solve the equation
4x - 2x - 5 = 4 + 6x + 3
2x - 5
 2x
5
7
 12
 142
3
 6x  7
 2x
 4x  7
7
 4x

4x
4
Example 1
Solve the equation
4x - 2x - 5 = 4 + 6x + 3
2x - 5
 2x
5
7
 12
 142
 6x  7
 2x
 4x  7
7
 4x

3  x
4x
4
Example 2
Solve the Equation
2(k - 5) + 3k = k + 6
Example 2
Solve the Equation
2(k - 5) + 3k = k + 6
2k
Example 2
Solve the Equation
2(k - 5) + 3k = k + 6
2k - 10
Example 2
Solve the Equation
2(k - 5) + 3k = k + 6
2k - 10  3k  k  6
Example 2
Solve the Equation
2(k - 5) + 3k = k + 6
2k - 10  3k  k  6
5k
Example 2
Solve the Equation
2(k - 5) + 3k = k + 6
2k - 10  3k  k  6
5k - 10  k  6
Example 2
Solve the Equation
2(k - 5) + 3k = k + 6
2k - 10  3k  k  6
5k - 10  k  6
k
k
Example 2
Solve the Equation
2(k - 5) + 3k = k + 6
2k - 10  3k  k  6
5k - 10  k  6
k
k
4k
Example 2
Solve the Equation
2(k - 5) + 3k = k + 6
2k - 10  3k  k  6
5k - 10  k  6
k
k
4k - 10 
Example 2
Solve the Equation
2(k - 5) + 3k = k + 6
2k - 10  3k  k  6
5k - 10  k  6
k
k
4k - 10 
6
Example 2
Solve the Equation
2(k - 5) + 3k = k + 6
2k - 10  3k  k  6
5k - 10  k  6
k
k
4k - 10 
6
 10
 10
Example 2
Solve the Equation
2(k - 5) + 3k = k + 6
2k - 10  3k  k  6
5k - 10  k  6
k
k
4k - 10 
6
 10
 10
4k
Example 2
Solve the Equation
2(k - 5) + 3k = k + 6
2k - 10  3k  k  6
5k - 10  k  6
k
k
4k - 10 
6
 10
 10
 16
4k
Example 2
Solve the Equation
2(k - 5) + 3k = k + 6
2k - 10  3k  k  6
5k - 10  k  6
k
k
4k - 10 
6
 10
 10
 16
4k
16
4k

4
4
Example 2
Solve the Equation
2(k - 5) + 3k = k + 6
2k - 10  3k  k  6
5k - 10  k  6
k
k
4k - 10 
6
 10
 10
 16
4k
16
4k

4
4
k 
Example 2
Solve the Equation
2(k - 5) + 3k = k + 6
2k - 10  3k  k  6
5k - 10  k  6
k
k
4k - 10 
6
 10
 10
 16
4k
16
4k

4
4
4
k 
Example 3 Solve the Equation
x  7 2x  8

 4
6
2
Example 3 Solve the Equation
x  7 2x  8

 4
6
2
x7

6
2x  8

2
4
Example 3 Solve the Equation
x  7 2x  8

 4
6
2
x7
2x  8
6 6  6 2  6  4
Example 3 Solve the Equation
x  7 2x  8

 4
6
2
x7
2x  8
6 6  6 2  6  4
1 x  7
Example 3 Solve the Equation
x  7 2x  8

 4
6
2
x7
2x  8
6 6  6 2  6  4
1 x  7  3 2x  8
Example 3 Solve the Equation
x  7 2x  8

 4
6
2
x7
2x  8
6 6  6 2  6  4
1 x  7  3 2x  8 6 4
Example 3 Solve the Equation
1 x  7  3 2x  8 6  4
Example 3 Solve the Equation
1 x  7  3 2x  8 6  4
x 7
Example 3 Solve the Equation
1 x  7  3 2x  8 6  4
x  7  6x
Example 3 Solve the Equation
1 x  7  3 2x  8 6  4
x  7  6x  24
Example 3 Solve the Equation
1 x  7  3 2x  8 6  4
x  7  6x  24   24
Example 3 Solve the Equation
1 x  7  3 2x  8 6  4
x  7  6x  24   24
 7x
Example 3 Solve the Equation
1 x  7  3 2x  8 6  4
x  7  6x  24   24
 7x  17
Example 3 Solve the Equation
1 x  7  3 2x  8 6  4
x  7  6x  24   24
 7x  17   24
Example 3 Solve the Equation
1 x  7  3 2x  8 6  4
x  7  6x  24   24
 7x  17   24
 17  17
Example 3 Solve the Equation
1 x  7  3 2x  8 6  4
x  7  6x  24   24
 7x  17   24
 17  17
7x
Example 3 Solve the Equation
1 x  7  3 2x  8 6  4
x  7  6x  24   24
 7x  17   24
 17  17
7x
 7
Example 3 Solve the Equation
1 x  7  3 2x  8 6  4
x  7  6x  24   24
 7x  17   24
 17  17
7x
 7
7x   7
7
7
Example 3 Solve the Equation
1 x  7  3 2x  8 6  4
x  7  6x  24   24
 7x  17   24
 17  17
7x
 7
7x   7
7
7
x
Example 3 Solve the Equation
1 x  7  3 2x  8 6  4
x  7  6x  24   24
 7x  17   24
 17  17
7x
 7
7x   7
7
7
x  1
Example 4 Solve
.06x + .09(15-x)=.07(15)
Example 4 Solve
.06x
.06x + .09(15-x)=.07(15)
.0915 - x  
.0715
Example 4 Solve
.06x + .09(15-x)=.07(15)
100.06x 100 .0915 - x  100 .0715
Example 4 Solve
.06x + .09(15-x)=.07(15)
100.06x 100 .0915 - x  100 .0715
6x
Example 4 Solve
.06x + .09(15-x)=.07(15)
100.06x 100 .0915 - x  100 .0715
6x
 9(15 - x)
Example 4 Solve
.06x + .09(15-x)=.07(15)
100.06x 100 .0915 - x  100 .0715
6x
 9(15 - x)
 715
Example 4 Solve
.06x + .09(15-x)=.07(15)
100.06x 100 .0915 - x  100 .0715
6x  9(15 - x)
6x
 715
Example 4 Solve
.06x + .09(15-x)=.07(15)
100.06x 100 .0915 - x  100 .0715
6x  9(15 - x)
6x  135
 715
Example 4 Solve
.06x + .09(15-x)=.07(15)
100.06x 100 .0915 - x  100 .0715
6x  9(15 - x)
6x  135  9x
 715
Example 4 Solve
.06x + .09(15-x)=.07(15)
100.06x 100 .0915 - x  100 .0715
6x  9(15 - x)
6x  135  9x
 715
 105
Example 4 Solve
.06x + .09(15-x)=.07(15)
100.06x 100 .0915 - x  100 .0715
6x  9(15 - x)
6x  135  9x
 3x
 715
 105
Example 4 Solve
.06x + .09(15-x)=.07(15)
100.06x 100 .0915 - x  100 .0715
6x  9(15 - x)  715
6x  135  9x  105
 3x  135
Example 4 Solve
.06x + .09(15-x)=.07(15)
100.06x 100 .0915 - x  100 .0715
6x  9(15 - x)  715
6x  135  9x  105
 3x  135  105
Example 4 Solve
.06x + .09(15-x)=.07(15)
100.06x 100 .0915 - x  100 .0715
6x  9(15 - x)  715
6x  135  9x  105
 3x  135  105
 3x
 3x
Example 4 Solve
.06x + .09(15-x)=.07(15)
100.06x 100 .0915 - x  100 .0715
6x  9(15 - x)  715
6x  135  9x  105
 3x  135  105
 3x
 3x
 135
Example 4 Solve
.06x + .09(15-x)=.07(15)
100.06x 100 .0915 - x  100 .0715
6x  9(15 - x)
6x  135  9x
 3x  135
 3x
 135
 715
 105
 105
 3x
 105
Example 4 Solve
.06x + .09(15-x)=.07(15)
100.06x 100 .0915 - x  100 .0715
6x  9(15 - x)
6x  135  9x
 3x  135
 3x
 135
 715
 105
 105
 3x
 105 3x
Example 4 Solve
.06x + .09(15-x)=.07(15)
100.06x 100 .0915 - x  100 .0715
6x  9(15 - x)
6x  135  9x
 3x  135
 3x
 135
 105
 715
 105
 105
 3x
 105 3x
 105
Example 4 Solve
.06x + .09(15-x)=.07(15)
100.06x 100 .0915 - x  100 .0715
6x  9(15 - x)
6x  135  9x
 3x  135
 3x
 135
 105
30
 715
 105
 105
 3x
 105 3x
 105
Example 4 Solve
.06x + .09(15-x)=.07(15)
100.06x 100 .0915 - x  100 .0715
6x  9(15 - x)
6x  135  9x
 3x  135
 3x
 135
 105
30
 715
 105
 105
 3x
 105 3x
 105
  3x
Example 4 Solve
.06x + .09(15-x)=.07(15)
100.06x 100 .0915 - x  100 .0715
6x  9(15 - x)
6x  135  9x
 3x  135
 3x
30  3x
 135
 105
30
 715
 105
 105
 3x
 105 3x
 105
  3x
Example 4 Solve
.06x + .09(15-x)=.07(15)
100.06x 100 .0915 - x  100 .0715
6x  9(15 - x)
6x  135  9x
 3x  135
 3x
30  3x
 135
3x
30

1
05

3
3
30
 715
 105
 105
 3x
 105 3x
 105
  3x
Example 4 Solve
.06x + .09(15-x)=.07(15)
100.06x 100 .0915 - x  100 .0715
6x  9(15 - x)
6x  135  9x
 3x  135
 3x
30  3x
 135
3x
30

1
05

3
3
30
10  x
 715
 105
 105
 3x
 105 3x
 105
  3x
Example 5 Solve
5x-9 = 4(x-3)
Example 5 Solve
5x-9 = 4(x-3)
5x  9  4x  12
Example 5 Solve
5x-9 = 4(x-3)
5x  9  4x  12
 4x
 4x
Example 5 Solve
5x-9 = 4(x-3)
5x  9  4x  12
 4x
 4x
x
Example 5 Solve
5x-9 = 4(x-3)
5x  9  4x  12
 4x
 4x
x 9 
Example 5 Solve
5x-9 = 4(x-3)
5x  9  4x  12
 4x
 4x
x 9 
 12
Example 5 Solve
5x-9 = 4(x-3)
5x  9  4x  12
 4x
 4x
x 9 
 12
9
9
Example 5 Solve
5x-9 = 4(x-3)
5x  9  4x  12
 4x
 4x
x 9 
 12
9
9

x
3
Example 5
Solve
5x - 15 = 5(x-3)
Example 5
Solve
5x - 15 = 5(x-3)
5x -15  5x - 15
Example 5
Solve
5x - 15 = 5(x-3)
5x -15  5x - 15
 5x
 5x
Example 5
Solve
5x - 15 = 5(x-3)
5x -15  5x - 15
 5x
 5x
 15 
Example 5
Solve
5x - 15 = 5(x-3)
5x -15  5x - 15
 5x
 5x
 15 
 15
Example 5
Solve
5x - 15 = 5(x-3)
5x -15  5x - 15
 5x
 5x
 15 
 15
 15
 15
Example 5
Solve
5x - 15 = 5(x-3)
5x -15  5x - 15
 5x
 5x
 15 
 15
 15
 15
0
0
Example 5
Solve
5x - 15 = 5(x-3)
5x -15  5x - 15
 5x
 5x
 15 
 15
 15
 15
0
0
The solution set is
all real numbers.
Example 5
Solve
5x - 15 = 5(x-3)
5x -15  5x - 15
 5x
 5x
 15 
 15
 15
 15
0
0
The solution set is
all real numbers.
Identity
Example 5
Solve
5x - 15 = 5(x-4)
Example 5
Solve
5x - 15 = 5(x-4)
5x -15  5x- 20
Example 5
Solve
5x - 15 = 5(x-4)
5x -15  5x- 20
 5x
 5x
Example 5
Solve
5x - 15 = 5(x-4)
5x -15  5x- 20
 5x
 5x
 15   20
Example 5
Solve
5x - 15 = 5(x-4)
5x -15  5x- 20
 5x
 5x
 15   20
 20
 20
Example 5
Solve
5x - 15 = 5(x-4)
5x -15  5x- 20
 5x
 5x
 15   20
 20
 20
5
0
Example 5
Solve
5x - 15 = 5(x-4)
5x -15  5x- 20
 5x
 5x
 15   20
 20
 20
5
0
Contradiction
Example 5
Solve
5x - 15 = 5(x-4)
5x -15  5x- 20
 5x
 5x
 15   20
 20
 20
5
0
The solution set is

Contradiction
Identity, Conditional Equation, Contradiction
An identity is an equation that is
satisfied by every number for which
both sides are defined.
Identity, Conditional Equation, Contradiction
A conditional equation is an equation
that is satisfied by at least one
number but is not an identity.
Identity, Conditional Equation, Contradiction
A Contradiction is an equation whose
solution is the empty set.
L 1.1 # 1 - 82
Every Other Odd Problem
1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41,
45, 49, 53, 57, etc.
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