Understanding Equations

Equations are all about working with equality. As with our last blog, we will use an example involving money. If you had $50 and then spent $50 dollars you would have $0. Right? Now let's say Bob has some money, he spent $50 dollars and now he no longer has any money. How much money did Bob have? Yes, Bob had $50 dollars. Now let's translate this into an equation, using "B" for the unknown amount of money that Bob has.
B - 50 = 0  
To find out the value of "B" we need to perform mathematical operations while maintaining equality. This is done by performing opposite mathematical operations on opposite sides of the equal sign. For our equation, 50 is subtracted on the left side of the equal sign, so instead of subtracting 50 we can add (opposite of subtraction) on the right side of the equal sign to obtain an equivalent equation.
B = 0 + 50
Which simplifies to tell us;
B = 50
Which of course we already knew from before.

Addition and Subtraction are opposite operations and multiplication and division are opposite operations.

Let's look at another example with Bob. This time Bob has $160 until Friday which is 3 days away. If he buys                                                                                       
a Kindle for $139, how much should he spend on lunch each day for the next 3 days. Let's look at an equation using "L" for lunch.
3L + 139 = 160
So 3 lunches plus the Kindle has to equal to $160. Once Bob purchases the Kindle, that amount is deducted from his total amount of money. So now 3 lunches have to equal $160 - 139.
3L = 160 - 139
Bob now has $21 to purchase lunch for 3 days.
3L = 21
To spend the same amount for each day, Bob would divide the total amount of money by the total number of days.
L = 21/3
L = 7
So Bob has $7 to spend on lunch each day.
Now let's solve a similar equation without a story line.
Solve for 'x'.
5x + 120 = 180
          5x = 180 - 120
          5x = 60
            x = 60/5
            x = 12

The process works exactly the same when it's just letters.
Solve for 'a'
ab + c = d
      ab = d - c
        a = (d - c)/b
Though the process is the same, we can't combine unlike variables.

Even when the problem expands and appears more difficult, the process is still the same.
Solve for 'y'.
8y + 12 = 5y + 6
        8y = 5y + 6 - 12
        8y = 5y - 6
 8y - 5y = -6
        3y = -6
          y = -6/3
          y = -2

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Adding and Subtracting Integers

The concepts of adding and subtracting integers are used on a daily basis. I know, I know, you say that you don't use it. Well, you use it more often than you think. Let's take a look into the "daily" shall we. Suppose you go online and purchase the book "Look Young Stay Young" by Johnny Capers using your credit card. After making your purchase, you owe the credit card for the total purchase price. Here's where negative numbers come into play. When you owe, that's a negative number. If the total purchase price was $21.76,  then that's negative $21.76 or -$21.76. Now let's look a different scenarios of making payments on the bill.
If you submit a payment for the entire amount you no longer owe anything.
-21.76 + 21.76 = 0  
                                  
If you submit a payment for more than you owe and there are no other charges on the card then they owe you.
-21.76 + 22.00 = 0.24

If you submit $10.76 then you still owe (not including interest).
-21.76 + 10.76 = -11

Now you want to buy a custom toddler pillow by Bobbleroos using that same credit card for a total purchase price of $18.26. Now you owe an additional $18.26.
-11 - 18.26 = -29.26

Now that you've seen integers in some "daily" transactions, let's talk a little more about some things that may help you in the classroom.
1) When the signs are the same you add the numbers and keep the same sign.
     20 + 10 = 30
     -20 - 10 = - 30
2) A negative sign and a subtraction sign are the same. When the signs are different           you subtract the numbers and take the sign of the "largest number" (technically the number furthest from zero).
    7 - 5 = 2
                                       -5 + 7 = 2
                                       -7 + 5 = -2
                                        5 - 7 = -2
Sometimes things can get tricky when multiple signs and parenthesis are used, so to eliminate confusion rewrite the problem with out unnecessary signs and parenthesis.To do this, whenever there are two signs beside each other replace the two with one sign. If the signs are the same use a "+" sign and if the signs are different use a "-" sign.

Example
-5 + (-9) - (-7)
The underlined signs below can be replaced
-5 + (-9) - (-7)
Becomes
 -5 - 9 + 7
Which is
-14 + 7 = -7

Another Example

3 + (+2) - (+5) - (-6) + (-3)

The underlined signs below can be replaced 

3 + (+2) - (+5) - (-6) + (-3)

Becomes

3 + 2 - 5 + 6 - 3

Which is

5 - 5 + 6 - 3 = 0 + 6 - 3 = 6 - 3 = 3

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Combining Like terms

When children first learn about mathematics, they usually begin by counting objects. For example they may begin to learn by counting apples. Once they become proficient in the process they move away from counting objects to the abstract idea of numbers. Combining like terms puts both of these ideas together. If you have 5 apples and   someone gives you 2 apples, then you have 7 apples in total. If you have 5 apples and someone gives you 4 oranges, then you have 5 apples in total. The same concepts applies to combining like terms. Terms are expressions that can contain numbers and letters that are separated by plus (+) or minus (-) signs. Here are some terms; 5x, 2xyz, 54rs, 65cd, x. Just like the apples and oranges the number in front of the letters indicate the amount of those particular items. There are 5 x's, 2 xyz's, 54 rs's, 65 cd's, and a single x. The single x and the 5x are like terms that can be combined to obtain a total of 6 x's.
5x + x = 6x

So combining like terms is really about condensing expressions by adding or subtracting similar terms. Sometimes people make the mistake of writing 6x^2 (x with an exponent of 2), which implies that there are 6 x^2's instead of 6 x's. The number in front of the variable is just shorthand notation for the amount of these variables in our case we write 6x instead of writing x,x,x,x,x,x. To help keep from changing the variable when combining like terms, think of combining like terms as counting people at a gathering by their last names. Here is a list of some people from the Gray family reunion.
Bobby Gray
Tommy Gray
Sandy John
Rodney John
Susan Young
Anthony Gray
Stacey Gray
Tonya Sullivan
Grant Sullivan

The last name break down is 2 from the Gray family, 2 from the John family, 1 from the Young family, 2 more from the Gray family, and 2 from the Sullivan family.
So this is
2 Gray + 2 John + 1 Young + 2 Gray + 2 Sullivan  = 4 Gray + 2 John + Young + 2 Sullivan

If we use the first letter of each last name to represent the groups we would have
2G + 2J + 1Y + 2G + 2S = 4G + 2J + Y + 2S

More examples.
23xy + 12xy = 35xy

3x - 5y - 4z + 4x + y + 6z = x - 4y + 2z

4x^2 + 2x - 7x^2 = -2x^2 + 2x

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