6.4 NORMAL PATH calculations with letters

Theory 

You can calculate with letters just as you can calculate with numbers. In order to calculate with letters, you need to know what like terms are. 

Like terms

Sophie is going to the shop to buy avocados and boxes with blueberries. Her mom texts her what she bought so, she can transfer money back to the bank account of Sophie.  Sophie always uses abbreviations when texting, this time as well. She gives the boxes with blueberries the letter b and the avocados the letter a.

Blueberries 

Avocado

Sophie texts her mom that she bought 3 boxes with blueberries and 2 avocados. 

She uses the abbreviations and texts her mom: 

3b + 2a

She did not tell her mother that she bought 5ab because than her mom wouldn't know how many avocados she bought and how many boxes of blueberries. It could be 4 avocados and 1 box of blueberries etc. 

Blueberries and avocados are totally different. This means that they are not alike. 

In the language of math: They are non like terms. 

You may not add or subtract non like terms.

Imagine Sophie buying 5 boxes of blueberries. She gets home but she forgot to buy another box of blueberries for the old lady in her street. So, she goes back to the shop and buys another box of blueberries. 

She bought: 5b + b 

You are now allowed to simplify the sum because the boxes of blueberries are the same. They are like terms. 

5b + b = 6b 

Sophie bought 6 boxes of blueberries. 

The sum of like terms 

You may only add or subtract like terms. 

Imagine Sophie buying 3 apples

She could say that Sophie bought an  apple + apple + apple (sum) or with her abbreviations a + a + a.

She can write this as a product so, 3 apples = 3 . a = 3a  (product)

Simplifying products 

Unfortunately, using pictures or making stories with simplifying products makes no sense. Therefore, you are going to learn how to simplify products in a more abstract way. 

7a .3z = 21az 

There is an invisible multiplication sign between the numbers and the letters. 

Therefore, you can also write the calculation as 7 . a . 3 . z 

You are allowed to swap the numbers and the letters because it is multiplication: 2 . 3 gives the same answer as 3 . 2. 

You will get 7 . 3 . a . z  or 3 . 7 . a . z or a different order.  Now, it is easier to calculate because the numbers and the letters are next to each other. 

a . a = a 2 ,  x . x = x 2 ,  l . l = l 2 ,  f . f = f 2  ...

If you are going to  multiply a letter by itself, the answer is going to be squared. 

This could be explained by looking at the area of a square 

Area square = length . width

Area square = 4 . 4 = 4 2  

The same works with letters.

Area square = length . width

Area square = h. h = h2  

Examples

  •    x + x = 2x

x - x = 0 

x . x = x2

  • 7ab + ab = 8ab 
  • 5t + 8q - t + 2q = 4t + 10q 
  • 4 . p . k . 5 = 20kp 
  • 7be . 2b = 14b2e 

Exercises 

Calculate

a) 5a . 2b 

b) 7b . 9b 

c) 18y . 0 . 3b 

d) -s . u . -p 

e) 3a + 7a + 101b - a

f) 10pq + pq . 43 - 2 

g) 5gh . g . 4 . a 

h) 6a + 7 . 2a - 5 

i) 5ab + 6a + 7b + 3ab + 2a . -3b + 4a