5.04 Represent algebraic expressions
Introduction
When writing a numerical expression we use numbers and basic operations to build up a number sentence that can be later calculated. Algebraic expressions are the same as numeric expressions except that they also use some new algebraic tools. These new algebraic tools are variables and coefficients.
Variables
We use algebraic expressions when we want to write a number sentence but don't know all the numbers involved.
For example: What is the total weight of a cat and a dog?
In this example we know that the total weight will be the weight of the cat added to the weight of the dog, but we don't know the number for either of these.
What we do instead is pretend that we know what these numbers are and replace them with variables. In this case, let's use c for the weight of the cat and d for the weight of the dog.
Now we can write the number sentence as: \text{Total weight} = c+d
This is an algebraic expression as it is a number sentence that uses variables in the place of some numbers.
Examples
Example 1
There are some red fish and some blue fish in a tank. If 5 yellow fish are added to the tank, how many fish are now in the tank? Write an expression for this scenario.
A variable is a symbol, commonly a letter, that is used in the place of a numeric value.
Coefficients
Coefficients are used in algebraic expressions to represent how many groups of a variable we have. They are written in front of a variable without a multiplication symbol like so: 3p = 3 \times p The coefficient is 3 and the variable is p , so there are 3 groups of p.
We don't need the multiplication symbol because there is no danger of mixing up a coefficient next to a variable. But we can't do this with numbers because if we did, it would be two numbers side by side, and appear to be a 2-digit number. For example: 3\times 4 \neq 34
What about variables that don't appear to have coefficients?
Consider the term x.
Since x is equal to 1 \times x which is also equal to 1x, it actually has a coefficient of 1. Whenever a variable has no written coefficient, its coefficient can be assumed to be 1.
We can also have algebraic terms where the coefficient is a fraction.
Consider: v\div 4
| \displaystyle v \div 4 | \displaystyle = | \displaystyle \dfrac{v}{4} | Assume the variable has a coefficient of 1 |
| \displaystyle \dfrac{1v}{4} | \displaystyle = | \displaystyle \dfrac{1}{4}\times v |
Since dividing by a number is the same as multiplying by its reciprocal, dividing by 4 gives us a coefficient of \dfrac{1}{4}.
Examples
Example 2
Consider the expression 6x+y-2z+5
What does the expression 6x mean?
How many terms does the expression have?
What are the coefficients of variables x, \,y and z in the expression?
A coefficient is a number that is written before a variable. It represents how many groups of the variable we have, and is considered an algebraic term.
Basic operations in algebra
Aside from the use of coefficients in multiplication, the basic operations work almost the same for algebraic terms as they do for numbers.
We use key words in the worded phrase to know what operations to use in the expression:
One way to represent algebraic expressions is with algebra tiles. Terms that can be combined are represented in the same size and shape. Terms that cannot be combined are different sizes.
Examples
Example 3
If x represents the number of peaches then write an expression for the number of peaches minus 17.
Example 4
If x represents the number of pencils then write an expression for the number of pencils divided by 13.
Key words help us to translate worded phrases into expressions:
| Addition | Subtraction | Multiplication | Division |
|---|---|---|---|
| plus | minus | times | divided by |
| sum | difference | product | quotient |
| increase | decrease | multiply | equal parts |
| total | fewer than | of | split |
| more than | less than | groups | equally shared |
| add | subtract | twice | half |