MandysNotes

Lessons

Lessons
Sunday, 21 March 2010 17:55

Linear Equations in One Variable

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As the name suggests, a linear equation in one variable implies that there is only ONE variable, and that the equation involves only real numbers. A linear equation in one variable can be written in this form: Ax + B = C where A does NOT equal zero.

A linear equation is also a first-degree equation, since the greatest power of any variable is 1.

Here are some examples of linear equations in one variable:

x + 2 = -1

x - 3 = 5

3k + 4 = 10

Equations and Expressions are closely related.

The primary difference between the two is an equals sign. An "equation" has a left side, a right side and an equals sign separating the sides. An "expression," by contrast, doesn't have any "sides" and is simply what the name suggests: An algebraic "expression." Though sometimes it is possible to combine like terms, we are generally not expected to "do" or "solve" anything regarding expressions.

For example:

3x - 7 = 2

This is an EQUATION, because it has a left side, a right side, and an = sign separating the two.

3x - 7

This is an EXPRESSION, because there are no "sides" and no = sign.

Saturday, 20 March 2010 20:14

The Associative Property

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The Associative Property states that for any real numbers a, b and c:

a + (b + c) = (a + b) + c

a(bc) = (ab)c

With the Associative Property, parentheses amongst 3 terms/factors change, but the order of the terms stays the same.

Saturday, 20 March 2010 20:12

Commutative Property

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The Commutative Property states:

For any real numbers a and b:

a + b = b + c

ab = ba

The Commutative Property is when the ORDER CHANGES but the result remains the same.

Saturday, 20 March 2010 20:05

Identity Properties

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For any real number a,

a + 0 = 0 + a = a

a * 1 = 1 * a = a

Notes: An easy way to remember this is that the Identity Property leaves the IDENTITY of a real number unchanged. Adding 0 to any number or multiplying any number times a does not change the value of the number in any way.

EXAMPLES:

12m + m

= 12m + 1m

= (12 + 1)m

= 13m

Saturday, 20 March 2010 00:00

Inverse Properties

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For any real number \( a \neq 0 :\)

\( a + (-a) = 0 \ \) and,

\( a \left( \frac{1}{a} \right) = 1.\)

That is, any number times its reciprocal equals 1.

The reciprocal of zero is not defined.

Very simply stated, the Distributive Property is easy to understand when seen like this:

a(b+c) = ab + ac

OR

(b+c)a = ba + ca

In more complex examples, the Distributive Property can be applied to equations like this one, though the same principles still apply:

distributive property, Mandy's Notes

 

EXAMPLES:

3(x + y)

= 3x + 3y


-2(5 + k)

= -2(5) + (-2)(k)

= -10 - 2k

 

4x + 8x

= (4 + 8)x

= 12x

Saturday, 20 March 2010 19:51

Absolute Value

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Absolute value is the distance on the real number line between any given number and zero.

 

Absolute value can never be negative, because technically distance can NEVER be negative.

 

Distance is always positive.

 

Absolute value is denoted between | number | - so to the reader, it will appear as: |-3| = 3.

Saturday, 20 March 2010 19:43

Negative and Positive Numbers

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{module [79]} 

 

A negative number is any number to the left of 0 on the number line.  

By contrast, a positive number is any number to the right of 0 on the number line. 

Saturday, 20 March 2010 19:38

Double Negatives Equal A Positive

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The technical rule to remember here is that for any real number a:

(-1)(-a) = a.

NOTRad