3.03 Represent algebraic expressions

We use algebraic expressions when we want to write a number sentence but we don't know one of the numbers involved.

For example: What is the total weight of a cat and a 10lb weight?

In this case, let's use c for the weight of the cat.

c + 10 is called an algebraic expression which is an expression that contains at least one variable.

c is called a variable. This is a symbol used to represent an unknown quantity.

Coefficients are the numerical factor in a term and are used to show how many variables we have. The variable u with a coefficient of 3 is written as 3u which means 3 \cdot u.

Terms are a number, variable, product, and/or quotient in an expression. They are the building blocks of an expression. Terms are separated by + or - signs.

Consider the expression: -\dfrac{2}{3}y+ 5

• This is an expression with 2 terms.

• The term -\dfrac{2}{3} y has a coefficient of -\dfrac{2}{3}. The negative belongs with the coefficient.

• The term 5 has no variable. It is called a constant term.

Expressions and parts of expressions, like factors and coefficients, all have unique meanings in a given context. Viewing expressions in parts and as a whole while paying attention to the quantities represented by the variables can explain the relationships described by the expressions.

Examples

We can use algebra tiles to help us visualize algebraic expressions.

The tile x represents an unknown number. The tile +1 represents adding one unit and -1 represents subtracting one unit.

This table demonstrates how expressions can be built using the tiles:

Algebra tiles can also help us identify the terms of the equivalent algebraic expression. Let's break down the algebra tiles below.

Notice that there are two different types of algebra tiles. These represent the two terms in the expression.

The first term in blue, are the three tiles with the +x. This represents the term 3x where the coefficient is the 3 and the variable is the x.

The second term in orange, are the six tiles with the -1. This represents the term 6.

When we add them together, we get the algebraic expression 3x-6.

Examples

Example 5

Write an equivalent algebraic expression and identify each term for the following:

a

Worked Solution

Create a strategy

There are many ways to write expressions that are algebraically equivalent by rearranging the terms and combining like terms, but for simplicity, we'll directly reflect the layout shown by the tiles.

Apply the idea

From the image, we have two negative variable tiles, five negative unit tiles plus one positive variable tile, two positive unit tiles. To express this algebraically we can write:

(-x-x-1-1-1-1-1)+(x+1+1)

Another way to write the expression is to count up the same tiles in each terms:(-2x-5)+(x+2)

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b

Worked Solution

Apply the idea

From the image, we have two positive variable tiles, six positive unit tiles, and two negative unit tiles, plus two negative variable tiles, one positive variable tile, four negative unit tiles, and two positive unit tiles. To express this algebraically we can write:

(2x+6-2)+(-2x+x-4+2)

Another way to write the expression is to count up the same tiles in each terms:(2x+4)+(-x-2)

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Example 6

Represent the following expressions using algebra tiles.

a

-7x+2

Worked Solution

Create a strategy

We can use negative variable tiles and positive unit tiles to represent the expression.

Apply the idea

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b

4x-5

Worked Solution

Create a strategy

We can use positive variable tiles and negative unit tiles to represent the expression.

Apply the idea

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