Sample Problem: Solving Algebraic Equations

Example:

• a + 1 = 3
• a = 3 - 1
• If you transpose a certain value to the other side of the equation, it "becomes the opposite of the original," so plus becomes minus, minus becomes plus, multiply becomes divide and vice versa.
• a = 2
• 2 + 1 = 3
• 3 = 3
• a = 2

Sample Problems:

• Solve for a:
• 5 + a = 7
• a - 13 = 4
• 3a  = 9
• a/2 = 10
• Solve for b:
•  9 + b = 15
• b - 2 = 1
• 2b = 2
• b/3 = 4
• Solve for c:
• 6 + c = 13
• c - 3 = 2
• 2c = 20
• 6/c = 3
• Solve for x: 
• 2 + x = 24
• x - 3 = 20
• 8x = 16
• x/7 = 5
• Solve for y:
• 8 + y = 28
• y - 4 = 12
• 3y = 15
• y/8 = 2

Sample Problem: Substituting Variables

Example Problem:

• What is the value of a + 1 if:
• a = 1
• a = 2
• a = 3
• a = 4
• a = 5

Answer to Example Problem:

• a + 1 =
• 2
• 3
• 4
• 5
• 6

Sample Problems:

• What is j + 7 if...
• j = 5
• j = 9
• j = 6
• What is u + 5 if...
• u = 4
• u = 2
• u = 8
• What is d + 9 if...
• d = 5
• d = 2
• d = 11
• What is b + 6 if...
• b = 1
• b = 4
• b = 3
• What is x + 8 if...
• x = 1
• x = 2
• x = 3

Sample Problems: Variables

Here are some sample problems regarding the previous discussed topic: Variables.
Try to answer the following orally and as fast as you can. Ready... get set... go!

• Try to identify the variable(s) in the following expressions:
• a) 7j + 8
• b) 5x - 2
• c) 9a + 3b
• If x represents the number of meters Nick sprinted for gym class, make an expression for:
• a) The distance Nick sprinted, doubled.
• b) Half of the distance Nick sprinted.
• c) The distance Nick sprinted, plus an additional 10 meters.
• If y represents the number of hours Jane spent sleeping, make an expression for:
• a) 5 hours more than the number of hours Jane slept
• b) Twice the time Jane spent sleeping
• c) Twice Jane's time sleeping, plus 5 more hours
• If z represents the number of gun shots James fired, make an expression for:
• a) The number of shots James would have made if she fired 8 more shots.
• b) The number of shots James would have made if she only fired half of the total shots she did.
• c) The number of shots James would have made if she fired 3 less shots.
• If x represents the amount of money (in PHP) Julia spent at the bookstore, make an expression for:
• a) The amount of money she would have spent if she bought one more book worth 280 pesos.
• b) The amount of money she would have spent if she didn't buy the book worth 150 pesos.
• c) The amount of money she would have spent if she had spent twice as much, and bought an additional book on sale for 75 pesos.

Done with the excercise? Still remember your answers? Wanna know how many you got correct? Wanna check if you even managed to get all the questions right?

Read ahead to see the answers!

Variables

Variables are symbols that represent a number in algebraic equations, problems, etc. Usually, they are letters but sometimes they could also be letters of the Greek alphabet, or if you'd like, they can be drawings, if you're really that creative! As I've said, a variable can represent any number, so...


Example:

• In the algebraic expression 5 + a , a is a variable and represents any number.
• In the algebraic equation 7 (a) = 28 , a is a variable that, based on the equation, represents the number 4.
• To check if this is correct, we replace the a with 4, thus making the equation 7 · 4 = 28

Usually, we use variables for when we are talking about unknown quantities.

Example:

• I know that John is 5 years older than his sister Jane. However, I do not know how old Jane is. So, if I were told to write John's age based solely on what I know, I would have to write it as 5 + x , with x being Jane's age, which is the unknown quantity keeping me from finding out how old John is.
• If a friend tells me my book weighs 3 times more than hers, but I do not know how heavy her book actually is, then I guess I'd have to write it down as:
• x representing her book's weight
• 3x for my book's weight, which is three times heavier than hers.
• If Sam grows 5 cm more, he will finally be as tall as Pam who is 160 cm tall. This statement says that Sam's current height plus 5 cm is equal to 160 cm . Thus, we shall write
• x + 5 = 160 which represents Sam's current height
• and, upon solving the problem, we come to find out that x = 155
• Therefore, we now know that Sam is 155 cm tall.

Note:

• Remember that algebraic expressions and equations can contain more than one variable.
• Example:
• 2 × (a · b)
• This algebraic expression contains two variables: a and b

QOTW from Richard van der Merwe

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. :)

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.