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Wize High School Algebra I Textbook (Common Core) > Foundations of Algebra

The Language of Algebra

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The Language of Algebra

In math we like to use symbols to help represent real-world problems.

Example
A small cup of coffee is $1.75. Let's say you want to buy coffee for you and all of your friends, and you want to calculate how much you will have to pay in total.

In English words:

You'll have to pay $1.75×the number of cups of coffee you want to buy\boxed{\$1.75\times{\colorFive{\bm{\text{the number of cups of coffee you want to buy}}}}}.

Using Symbols:

If we let x\bcfi x be the number of cups of coffee you want to buy, then you'll have to pay $1.75x\boxed{\$1.75\bcfi x}


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The building blocks of the English Language→AlphabetsThe building blocks of the Algebra Language→Numbers  and  Variables\boxed{\begin{array}{ccc} \text{The building blocks of the English Language} &\to& \bold{{Alphabets}}\\\\ \text{The building blocks of the Algebra Language} &\to& \colorFour{\bold{{Numbers~~and~~Variables}}} \end{array}}


A number is used to represent a quantity or amount. If we have a number that is alone, we call it a constant.
Examples
  • 33 can represent 3 cats, 3 meters, 3 grams, $3, etc.
  • −5-5 can represent −5°-5\degree, owing $5, 5 steps left, etc.
  • 0.150.15 can represent $0.15, 0.15 ml, 0.15 km, etc.



A variable is any symbol (usually letters) used to represent a quantity (number), this quantity can change and take on many different values!
Examples
  • xx is the number of people on a certain bus
  • hh is the height of a tree
  • ww is the weight of an apple
  • ll is the length of a rectangle and ww is the height of that rectangle

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In English, we can string a bunch of alphabets together→WordsIn Algebra, we can have a product of variables and constants→Terms\boxed{\begin{array}{ccc} \text{In English, we can string a bunch of alphabets together} &\to& \bold{{Words}}\\\\ \text{In Algebra, we can have a product of variables and constants} &\to& \colorFour{\bold{{Terms}}} \end{array}}
Examples
  • 5x5x
  • −0.5xy-0.5xy
  • x3x^3
  • l2w3l^2w^3
  • x6\dfrac{x}{6}

Notes
  • Multiplication is usually shown as 3×y3\times y or (3)(y)(3)(y), but when multiplying by a variable, we can just show it as 3y3y
  • A number that is being multiplied by a variable has a special name called coefficient.

Watch Out!
Is a number or variable by itself a term?
  • Is −5-5 a term?
    Yes!
  • Is xx a term?
    Yes!

What is the coefficient of xx?
We can rewrite xx as 1x1x, and see more clearly that the coefficient is 11 !

Practice: The Language of Algebra

Fill in the blanks with the words variable, constant, coefficient.

−5 x  + 7\Large{\bcfi{-5}~\bcf{x}~~\bct{+~7}}
  • −5\bcfi{-5} is called a
  • x\bcf{x} is called a
  • 7\bct{7} is called a

Practice: Algebra Definitions


Fill in the blanks.
  1. 3x2−4xy+5+a−b3x^2-4xy+5+a-b is called a
  2. The xx in 3x2−4xy+5+a−b3x^2-4xy+5+a-b is called a
  3. The 55 in 3x2−4xy+5+a−b3x^2-4xy+5+a-b is called a
  4. The −4xy-4xy in 3x2−4xy+5+a−b3x^2-4xy+5+a-b is called a
  5. How many terms are there in 3x2−4xy+5+a−b3x^2-4xy+5+a-b?
  6. How many variables are there in 3x2−4xy+5+a−b3x^2-4xy+5+a-b?
  7. The 33 in the term 3x23x^2 is called the
  8. What is the coefficient of the second term in 3x2−4xy+5+a−b3x^2-4xy+5+a-b?
  9. What is the coefficient of the fourth term in 3x2−4xy+5+a−b3x^2-4xy+5+a-b?
  10. What is the coefficient of the fifth term in 3x2−4xy+5+a−b3x^2-4xy+5+a-b?

*If your answer is a negative number, don't forget to type the - sign (for example -7)

*If your answer is a positive number, you don't have to type the + sign (for example 10)