Simple Simplification

Simple Simplification is just Simply Simplifying.

Students are often told to "simplify their answers" or to provide their answers in their "simplest form" ... What does this mean, exactly? With Algebra, that usually means you have to combine like terms. Like terms are terms with the same variables and possibly exponents, but sometimes with different coefficients. Since they're like that, we can mix 'em together.

Example:

• 7x and 5x are like terms, which means we can combine them to form 12x. Don't worry, this won't change the its value. Here, let me show you:
• 7x + 5x = 12x
• Let x = 9
• 7(9) + 5(9) = 12(9)
• 63 + 45 = 108
• 108 = 108
• But, if in case it were 7x and 5x², we can't combine them because their variables are not like terms, because their variables aren't the same (since x ≠ x²). Get the idea? Let's have more examples.
• 7x² + 5x + 9y² + 6y = 428
• Can you combine any terms in this equation? No. There are no like terms in that equation.
• 7x² + 5x + 9x² + 6x = 428
• Can you combine terms? Yes. We can simplify this equation to come up with: 16x² + 11x = 428
• 7x + 5xy + 9y = 6
• Are there any terms we can combine here to simplify the equation? No. We can't combine 7x with 5xy even though they both have x because 7x doesn't have y and we can't combine 9y with 5xy even though they both have y because 9y doesn't have x.

Try to simplify the following:

• 3x + 4x
• 2y + 3y
• 4a + 5a
• b + 5b
• 4xy + 7x² + 8y² + 2xy + 5x - 6y - 9y

Answers:

• 7x
• 5y
• 9a
• 6b
• 7x² + 5x + 6xy + 8y² - 3y

Note: When faced with more difficult problems, please refer to two of our previous posts regarding Properties of Equality and Properties of Operations & Identities. Who knows, they might help you in ways you never expected. :)

About our Next 'Magical' QOTW

Snapshot taken from Marquee's comment at Simple Words to Magical Algebra.

There you have it, folks! One of our blog contributors had already put up an early "Easter Egg" for y'all about out next Quote of the Week (QOTW)! Isn't this exciting? So, be sure to tune in to LAMEFHM and keep yourself updated every Wednesday for more of our awesome Algebra quotes! And, feel free to just keep those comments comin'. We're more than happy to hear from our readers!

Simple Words to Magical Algebra

In case most of you are wondering what to do when your Algebra teacher gives you a quiz and decides to dictate the questions in sentences... well, don't fret, LAMEFHM is here to help! With this post, we'll show you how to turn simple words to algebraic expressions and/or equations and solve problems!

Don't worry, your teacher is not crazy by dictating stuff instead of writing the number problems on the board during your quiz on Algebra. That's normal. And there's a way to solve it! Mathematical word problems are most commonly solved by translating them into expressions and equations or, as I like to call them, "algebraic sentences."

Here's an example for you:

• Your teacher says... "I bought a coffee maker for $50 after they deducted $20 from its original price because it was on sale and had a discount. How much was the original price of the coffee maker I bought?" (Credits to TutoringMaths for the cool image and to Basic-Mathematics for their awesome word problems!)

To solve this, you have to write it as an algebraic sentence (meaning write it as an expression or equation). To do that, you have to:

1. Distinguish the quantity missing or unknown. Know what you're trying to find. (This is usually what your teacher asks you to find.)
2. Get variables to represent the quantity you're looking for. (Usually, the most common variables used are a, b, c, x, y, and z.)
3. Find out what kind of operations you have to use in solving the given problem.

Following the steps mentioned lets you know that:

1. The original price of the coffee maker my teacher bought is unknown. I must try to find out how much that coffee maker originally cost!
2. I'll get the letter x (since the letter x is so cool) and use it as the variable to my equation (you'll be using an equation instead of an expression with this problem... I'll explain why later) and name the unknown quantity x, making x = original coffee maker price.
3. I'll be using subtraction with this problem ... and let me show you why subtraction is the operation I'll be using... later.

Ready to know the answer? Let's solve!

• x = original price
• x - 20 = 50
• We subtracted 20 from x since x is our original price, and the teacher said he bought his coffee maker with a $20-discount for $50. Subtract $20 from the original price and we get the amount the teacher paid for his coffee maker which was $50.
• x = 50 + 20
• Since $50 is the price of the coffee maker once the $20-discount has already been deducted from its original price, we therefore transpose 20 (since we only need x to be the only one remaining on one particular side of the equation in order to come up with its value) to the other side of the equation (and since we transposed it to the other side, its sign changes, hence the plus sign instead of the minus) and add 20 to 50 to come up with the original price of the product before the $20-discount has already been deducted.
• x = 70
• Here we now have the answer, which is $70 as the coffee maker's original price before its $20-discount has already been deducted.

We here at LAMEFHM hope our posts help--especially this one, since this one was originally created to help--you... so, if you've got questions or more ideas/suggestions/opinions/comments/messages regarding how you think we can be more helpful to you and your mathematical needs... just leave a comment and we'll answer as soon as we possibly can! Thanks! :)

Properties of Operations and Identities

• Commutative Property of Addition
• a + b = b + a
• Example:
• a + b = b + a
• a = 7
• b = 5
• (7) + (5) = (5) + (7)
• 12 = 12
• Commutative Property of Multiplication
• ab = ba
• Example:
• ab = ba
• a = 9
• b = 6
• 9 · 6 = 6 · 9
• 54 = 54
• Associative Property of Addition
• (a + b) + c = a + (b + c)
• Example:
• (a + b) + c = a + (b + c)
• a = 4
• b = 1
• c = 2
• (4 + 1) + 2 = 4 + (1 + 2)
• (5) + 2 = 4 + (3)
• 7 = 7
• Associative Property of Multiplication
• (ab)c = a(bc)
• Example:
• a = 9
• b = 8
• c = 5
• (9 · 8) · 5 = 9 · (8 · 5)
• (72) · 5 = 9 · (40)
• 360 = 360
• Distributive Property of Multiplication Over Addition
• a (b + c) = ab + ac
• Example:
• a (b + c) = ab + ac
• a = 12
• b = 7
• c = 5
• 12 · (7 + 5) = (12 · 7) + (12 · 5)
• 12 · (12) = (84) · (60)
• 144 = 144
• Additive Identity Property
• a + 0 = 0 + a = a
• Example:
• a + 0 = 0 + a = a
• a = 5
• (5) + 0 = 0 + (5) = 5
• 5 = 5 = 5
• Multiplicative Identity Property
• 1(a) = a(1) = a
• Example:
• 1(a) = a(1) = a
• a = 3
• 1(3) = 3(1) = 3
• 3 = 3 = 3
• Quotient Property
• a (1/b) = a/b
• Example:
• a (1/b) = a/b
• a = 1
• b = 2
• 1 (1/2) = 1/2
• ½ = ½
• Multiplicative Inverse Property
• a (1/a) = 1
• Example:
• a (1/a) = 1
• a = 3
• 3 (1/3) = 1
• 1 = 1
• Multiplication Property of Zero
• 0(a) = a(0) = 0
• Example:
• 0(a) = a(0) = 0
• a = 1
• 0(1) = 1(0) = 0
• 0 = 0 = 0

Properties of Equality

• Addition Property of Equality
• If a = b and c = d, then a + c = b + d
• If the value of a is equal to b, and c is similar to the value of d, then adding a and c is equal to the sum of b and d.
• Example:
• a = 1 and b = 1 therefore we can say that a = b ;  c = 2 and d = 2 so we can say that c = d
• a + c = b + d
• 1 + 2 = 1 + 2
• 3 = 3

• Subtraction Property of Equality
• If a = b and c = d, then a - c = d - b
• It's just like how the Addition Property of Equality works except, only this time with this property, you're going to subtract instead of add.
• Example:
• a = 1 and b = 1, a = b ;  c = 2 and d = 2, c = d
• a - c = b - d
• 1 - 2 = 1 - 2
• -1 = -1

• Multiplication Property of Equality
• If a = b and c = d, then ac = bd
• Example:
• a = 1 and b = 1, a = b ;  c = 2 and d = 2, c = d
• ac = bd
• 1 · 2 = 1 · 2
• 2 = 2
• Division Property of Equality
• If a = b and c = d, then a/c = b/d and a/b = c/d
• This property is kinda the same with the other properties previously mentioned since it says that if the value of a is equal to b, and c is similar to the value of d, then a divided by c is equal to b over d. The extra thing that this property has is the a/b = c/d equation. What does that part say? Well, since a = b and c = d, a/b would be equal to 1, and so would c/d. Therefore, since both are equal to 1, a/b = c/d.
• Example:
• a = 1 and b = 1, a = b ;  c = 2 and d = 2, c = d
• a/c = b/d
• ½
   = ½
• 0.5 = 0.5
• a/b = c/d
• 1/1 = 2/2
• 1 = 2
• Reflexive Property of Equality
• This property states that a = a
• Well, this property speaks for itself. If it's a, then it's a. It's kinda like how you are you. a = a. 1 = 1.
• Example:
• a = a
• 1 = 1
• Symmetric Property of Equality
• If a = b, then b = a
• Since a is similar to b, then b is also similar to a.
• Example:
• a = 1 and b = 1
• a = b and b = a
• Transitive Property of Equality
• If a = b and b = c, then a = c
• Example:
• a = 1 and b = 1, a = b ;  c = 1, b = c
• a = 1
• b = 1
• c = 1
• a = b
• a = c
• Substitution Property
• If a = b, neither can be substituted for the other to find the value of the other.
• This one's kinda tricky to explain with just words, so look at this example to understand the property more.
• Example:
• x = y + 5
• If: y = 7
• x = (7) + 5
• x = 12
• Zero Product Property
• If ab = 0, then either a = 0, b = 0 or both a and b = 0
• This property says that if you multiply two variables together and their product is zero, then that means that either one of those variables are equal to zero (since they yielded a product of zero), or both of their values are zero.
• Example:
• ab = 0
• If: a = 9
• b = 0
• If: b = 6
• a = 0

Algebra: T&D

Important T&D (Terms and Definitions) for Algebra:

• Algebraic Equation - an equation that can contain one, two or even more variables.
• Equation - has two expressions that are equal to each other.
• Algebraic Expression - an expression that, like the algebraic equation, can contain one, two or more variables.
• Expression - a phrase representing a number.
• Variable - a symbol (usually a letter) that is used to represent the (missing) value or number.
• Coefficient - the number that is multiplied by the variable.
• Example: In the term 5x, the variable x has a coefficient of 5.
• Constant - a term with no variables
• Like Terms - terms with the same variables and possibly exponents, but sometimes with different coefficients.
• Null Set or Empty Set  - denoted by the symbol Ø, it is a set that has no members, hence its name.
• Replacement Set - set with values from which to find a solution.
• Simplified Form - the result of the expression once all like terms in the expression had already been combined and is still equal to the original expression.
• Reciprocal - the reciprocal of a number n is one over that number or 1/n, and once a number is multiplied by its reciprocal, the resulting product should be equal to 1.
• Solution - any value for the variable that makes the equation true.
• Solution Set - a set that contains the variables that will make the stated equation true.

Quote of the Week

Hello! Welcome to the site (as previously said by Julia).

Here, Linear Algebra becomes normal English instead of mathematical Jargon. That's all I have to say I guess.

Quotes are posted every Wednesday. :)

Back to Basics

Salutations!

So, as you can see, our website is still quite new and since this is just our first post, we thought we'd give crash-course recaps to some of the maths topics you've all probably already encountered (and so have we) before getting into your Linear Algebra classes.

Why did we decide to do so? To make sure that you're already familiarized and/or your mind is refreshed with information regarding most of what we'll talk about here. And, since we're still just beginning our Linear Algebra lessons as well, we still need a lil' bit more time to understand, know the explanations, type, explain and educate, and publish posts all about Linear Algebra!

So, while everything else is still going on, enjoy and have fun with our pre-Linear-Algebra posts, topics, lessons and recaps! :)

Love, The Linear Algebra Made Easy For Hard Math (LAMEFHM) Team!

Credits to Google Images!