Equations

How to Solve Them

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What Are Equations ?

Equation 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 ?

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

Conditional Equations 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:

for some constants a, b with a ≠ 0

Solutions Solution is a value of x that makes

the equation TRUE A solution is a number

WHY a ≠ 0 ?

ax + b = 0

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Linear Equations in 1 Variable Equivalent Equations

Equations are equivalent if and only if they have the same solutions

Solving an equation transforms it into an equivalent equation of form:

The number r is the solution

x = r

WHY ?– there is only one

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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

Cancellation Rule for Multiplication

ac = bc if and only if a = b

for any numbers a, b and c with c ≠ 0

Techniques Solving Equations

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Examples Example 1.

2x + 3 = 7

Example 2.

•= 2 2

Techniques for Solving Examples

= 4 + 3then 2x = 4

then x = 2

If 2x = 4

Cancellation Rule for Addition

Cancellation Rule for Multiplication

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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

Same Rules in Reverse

Question:

Why can we add 0 but not multiply by 0 ?

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Example 1. If 2x – 3 = 7

Example 2. If 2x = 10

Same Rules in Reverse

then 2x – 3 + 3 = 7 + 3

so 2x = 10

so x = 5

Question:

Why can we add 0 but not multiply by 0 ?

then (2x) = (10)21

21

Addition Rule

Multiplication Rule

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Solve:

–3(2x – 1) = 2x

– 6x + 3 = 2x distributive property

– 6x + 3 + 6x = 2x + 6x addition rule

3 = 8x simplification (1/8)(3) = (1/8)(8x) multiplication rule 3/8 = x simplification

Solution is

Solving Symbolically

38

The solution is NOT

WHY ?

Solution Set: { } 38

Note: 38x =

Equations 127/8/2013

Solve:

Simplify by clearing fractions

Solving Symbolically

– 2 3x – 1 5

= 2 – x

3

( )– 2 3x – 1

5 15 = ( )2 – x

3 15

3(3x – 1) – 30 = 5(2 – x)

= ( )3 15 2 – x ( )( )3x – 1

5 15 15 2 – ( )

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Solve:

Solving Symbolically

– 2 3x – 1 5

= 2 – x

3

3(3x – 1) – 30 = 5(2 – x)

9x – 33 = 10 – 5x

14x = 43Equivalent Equation

1443

Solution :

= 1443x

Solution set :1443{ }

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Solve: 3x + 1 = –2x + 11

Consider this as the equality of two functions y1 and y2 with

and

Lines intersect where (x, y1) = (x, y2)

Solving Equations Graphically

y

xy1 = 3x + 1

y1 = 3x + 1

y2 = –2x + 11 y2 = –2x + 11

y1 y2

x

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Solve: 3x + 1 = –2x + 11

Lines intersect where (x, y1) = (x, y2)

In this case (x, y1) = (x, y2) = (2, 7)

Solving Equations Graphically

y

x

y1 = 3x + 1

y2 = –2x + 11

y1 y2

x

So x = 2Solution is 2

Solution set is { 2 }

Question: How do we find 2 and 7 graphically ?

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Solve: 14x – 36 = 7

This can be written as

y(x) = 14x – 36 = 7

Want x value where y = 7

Desired x between 3 and 4

Increase resolution between 3 and 4

Solving Equations Numerically

0 –36

1 –22

2 –8

3 6

4 20

5 34

x y

7

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Solve: 14x – 36 = 7

Increased resolution

Solving Equations Numerically

x y

3.0 6.0 3.1 7.4 3.2 8.8 3.3 10.2 3.4 11.6 3.5 13.0 4.0 20.0

7

y = 7 for x between 3.0 and 3.1

Continue refining x

Expand 3.0 – 3.1 into new table (3.00 – 3.09) for next decimal on y

*

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*

Solve: 14x – 36 = 7

Solving Equations Numerically

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

7

Continue refining x

Question:How accurate is this method ?How long does it take ?

to force y closer to 7

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Graphical Solution Least accurate, visual solution Can be automated via computer/calculator Makes trends more obvious

Numerical Solution Approximate solution but refinable Natural for collected data Easily automated

Solving Equations: Review

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Symbolic Solution The most accurate Purely algebraic Good for predictions

Solving Equations: Review

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Think about it !