Literal Equations

April 30, 2008 at 1:10 pm (basic algebra)
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Follow the solutions to the following three equations. 

EXAMPLE 1

3x=12

\frac{3x}{3}=\frac{12}{3}

x=4

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That was easy, right?

Now let’s look at what we would call a more general case of the same problem:

EXAMPLE 2

ax=12

\frac{ax}{a}=\frac{12}{a}

x=\frac{12}{a}

What are the similarities and differnces between the equations’ in examples 1 and 2?

EX 1: 3x=12           EX 2: ax=12

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Not so bad?  Now let’s look at the most general case of the first equation:

EXAMPLE 3

ax=b

\frac{ax}{a}=\frac{b}{a}

x=\frac{b}{a}

Do you see the similarities between all three examples?

 EX 1: 3x=12           EX 2: ax=12        EX 3:ax=b

The third example is what we call a LITERAL EQUATION because the numbers in example 1 have been replace with letters.

However, the method used to solve them is the same!

Example 3 is an example of solving the literal equation, ax=b, for x in terms of a and b

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OK, let’s try a slightly more complicated case: (solve for r)

Specific case:                      More General case:

2r+3=7                       ar+3=7

Even more General case:   The most General case (literal equation)

ar+b=7                       ar+b=c

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What Makes Mathematics Hard to Learn?

April 15, 2008 at 11:32 am (Uncategorized)

Check out this thought provoking essay from Dr. Marvin Minsky.

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Distributive Property

April 15, 2008 at 9:56 am (basic algebra)
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IF

3 \times 4 implies repeated addition of the number 4, 3 times.

For example: 3 \times 4=4+4+4

THEN

a(b+c) would imply repeated addition of the number (b+c), a times.

Note: a(b+c) is exactly the same as a \times (b+c)

That would be:

(b+c)

+

(b+c)

+

(b+c)

+

\vdots

+

(b+c)

_____________

Because you have a of these (b+c)'s,

Adding vertically, you would have a sum of ab+ac

\therefore a(b+c)=ab+ac

If that doesn’t make any sense, check out this numerical example of the same idea:

 3(x+4) implies repeatd addition of (x+4), 3 times.

(x+4)+(x+4)+(x+4) after combining like terms would be, 3x+12 because there are three x's and three 4's

SO

Rather than performing the repeated addition each time we are faced with a multiple of a group, we can just apply the distributive property as shown below.

3(x+4)=3x+3 \cdot 4=3x+12

To look at it using a purely numerical example, consider:

4(2+3) as a 4-time repeated sum of the quantity (2+3)

(2+3)

+

(2+3)

+

(2+3)

+

(2+3)

_____________

4(2)+4(3)

=8+12=20

We can see that if we had computed the above using order of operations we would have obtained the same result of 20.

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Order of Operations

April 15, 2008 at 9:25 am (basic algebra)
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Order of Operations: A way of evaluating expressions with more than one operation. These rules govern precedence in mathematical operations.

For Example: When faced with 4+2 \times 3 , how do you proceed?

There are two apparent options:

4+2 \times 3

=6 \times 3

=18

OR

4+2 \times 3

= 4 + 6

=10

Which is correct?

We must follow the correct order of operations so that this expression has a necessarily unique value.

For the above example, the correct answer is 10.

Now, let’s find out WHY?

The Actual Order

Evaluate numerical expressions in the following order:

  • Grouping Symbols (parentheses, brackets, braces, absolute value)
  • Exponents (includes square roots)
  • Multiplication and/or division from left to right
  • Addition and/or subtraction from left to right

An Easy Way of Remembering

Use this memory tool to help remember the order! Please Excuse My Dear Aunt Sally (PEMDAS)

An alternative form of this is:

Brackets, Indices, Division, Multiplication, Addition, Subtraction (BIDMAS).

Order of Operations – Examples
Expression Evaluation Operation
4 × 2 + 1 = 4 × 2 + 1 Multiplication
= 8 + 1 Addition
= 9
12 – 9 ÷ 3 = 12 – 9 ÷ 3 Division
= 12 – 3 Subtraction
= 9
3 + 12 ÷ (5 – 2) = 3 + 12 ÷ (5 – 2) Parentheses
= 3 + 12 ÷ 3 Division
= 3 + 4 Addition
= 7
7 × 10 – (2 × 4)2 = 7 × 10 – (2 × 4)2 Parentheses
= 7 × 10 – 82 Exponents
= 7 × 10 – 64 Multiplication
= 70 – 64 Subtraction
= 6

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