Download Introduction to Algebra

Equations
How to Solve Them
What Are Equations ?

Equation

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Statement that two expressions are equal

What do we DO with expressions ?
 Simplify them
 Evaluate them

What do we DO with equations ?
 Solve them
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What Are Equations ?

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Three equation categories

Identity: Logically True
 Example: 2(x + 1) = 2x + 2
 True for ALL values of x

Logically False
 Example: x + 3 = x
 NOT true for ANY x ( would imply 3 = 0,
a contradiction ! )
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What Are Equations ?

The third equation category
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Conditional Equations


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Example: 3x + 1 = 2x – 7
True for SOME values of x
(x = – 6 in this case)
False for other values of x
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Linear Equations in 1 Variable

Standard Form:
ax + b = 0
for some constants a, b with a ≠ 0
WHY a ≠ 0 ?
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Solutions
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Solution is a value of x that makes
the equation TRUE
A solution is a number
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Linear Equations in 1 Variable
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Equivalent Equations
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Equations are equivalent if and only if
they have the same solutions

Solving an equation transforms it into
an equivalent equation of form: x = r

The number r is the solution
– there is only one WHY ?
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Techniques Solving Equations
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Cancellation Rule for Addition
a + c = b + c if and only if a = b
for any numbers a, b and c
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Cancellation Rule for Multiplication
ac = bc if and only if a = b
for any numbers a, b and c with c ≠ 0
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Techniques for Solving Examples

Examples
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Example 1.
2x + 3 = 7
=4+3
then 2x = 4
Cancellation Rule
for Addition
Example 2.
If 2x = 4
= 2• 2
then x = 2
Cancellation Rule
for Multiplication
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Same Rules in Reverse

Addition Rule
a = b if and only if a + c = b + c
for any numbers a, b and c

Multiplication Rule
a = b if and only if ac = bc
for any numbers a, b and c with c ≠ 0
Question:
Why can we add 0 but not multiply by 0 ?
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Same Rules in Reverse
 Example 1. If 2x – 3 = 7
then 2x – 3 + 3 = 7 + 3 Addition Rule
so 2x = 10
 Example 2. If 2x = 10
then
1
(2x) = 1 (10)
2
2
so x = 5
Multiplication
Rule
Question:
Why can we add 0 but not multiply by 0 ?
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Solving Symbolically
Solve:
–3(2x – 1) = 2x
– 6x + 3 = 2x
– 6x + 3 + 6x = 2x + 6x
3 = 8x
(1/8)(3) = (1/8)(8x)
3/8 = x
Solution is 3
8
distributive property
addition rule
simplification
multiplication rule
simplification
Solution Set:
{ }
3
8
Note: The solution is NOT x = 3
8
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Equations
WHY ?
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Solving Symbolically
2–x
3x – 1
–2 =
3
5

Solve:

Simplify by clearing fractions
3x – 1
15
5
(
15
(
3x – 1
5
–
)
2
2–x
= 15
3
(
)
15 (
2 – x)
3
) – 15 (2) = ( )
3(3x – 1) – 30 = 5(2 – x)
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Solving Symbolically

Solve:
2–x
3x – 1
–2 =
3
5
3(3x – 1) – 30 = 5(2 – x)
9x – 33 = 10 – 5x
Equivalent
Equation
43
Solution :
14
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14x = 43
43
x=
14
Solution set :
Equations
{ }
43
14
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Solving Equations Graphically
Solve: 3x + 1 = –2x + 11
Consider this as the
equality of two functions
y1 and y2 with
y1 = 3x + 1
and
y2 = –2x + 11
Lines intersect where
(x, y1) = (x, y2)
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Equations
y
y1 = 3x + 1
y1

x
y2
x
y2 = –2x + 11
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Solving Equations Graphically
Solve: 3x + 1 = –2x + 11
Lines intersect where
(x, y1) = (x, y2)
In this case
(x, y1) = (x, y2) = (2, 7)
So
x=2
Solution is 2
y
y1 = 3x + 1
y1

x
y2
x
y2 = –2x + 11
Solution set is { 2 }
Question: How do we find 2 and 7 graphically ?
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Solving Equations Numerically
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Solve: 14x – 36 = 7
This can be written as
y(x) = 14x – 36 = 7
x
y
0
–36
Want x value where y = 7
1
–22
Desired x between 3 and 4
2
–8
Increase resolution
between 3 and 4
3
6
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20
5
34
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Solving Equations Numerically
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Solve: 14x – 36 = 7
Increased resolution
y = 7 for x between
3.0 and 3.1
Expand 3.0 – 3.1 into
new table (3.00 – 3.09)
for next decimal on y
Continue refining x
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x
3.0
3.1
3.2
3.3
3.4
3.5
4.0
y
6.0
7.4
8.8
10.2
11.6
13.0
20.0
*
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Solving Equations Numerically
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Solve: 14x – 36 = 7
Continue refining x
to force y closer to 7
Question:
How accurate is
this method ?
How long does
it take ?
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7
x
y
3.00 6.00
3.01 6.14
3.02 6.28
….
….
3.07 6.98
3.08 7.12
3.09 7.26
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Solving Equations: Review
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Graphical Solution
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Least accurate, visual solution
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Can be automated via computer/calculator
Makes trends more obvious
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Numerical Solution
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Approximate solution but refinable
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Natural for collected data
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Easily automated
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Solving Equations: Review
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Symbolic Solution
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The most accurate
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Purely algebraic
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Good for predictions
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Think about it !
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