2.01 Represent algebraic expressions

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 case, let's use c for the weight of the cat and d for the weight of the dog.

c + d is called an algebraic expression.

c and d are called variables. They are used in the place of a numeric value.

Coefficients 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 the building blocks of an expression. They are either a single number or variable or numbers and variables multiplied together. Terms are separated by + or - signs.

Consider the expression: {4x-3y} + \frac{2z}{5} + k + 7

• This is an expression with 5 terms.

• The term 4x has a coefficient of 4.

• The term -3y has a coefficient of -3. The negative belongs with the coefficient.

• The term \dfrac{2z}{5} has a coefficient of \dfrac{2}{5} as \dfrac{2z}{5} = \dfrac{2}{5} \cdot z.

• k is equal to 1 \cdot k which is also equal to 1k, so it has a coefficient of 1.

• The term 7 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:

We can use visual models to rearrange expressions into equivalent expressions. We can see how each of the following equivalent expressions is represented with the algebra tiles.

This model is one group of 6x+6

This model is two groups of 3x+3

This model is arranged into 2x+1 and 4x+5

This model is arranged into -x+3 and 7x+3

All of these expressions are equivalent to 6x+6. Different grouping variations of equivalent expressions can help us understand them better in different mathematical situations.

Examples

Example 5

Write an equivalent algebraic expression for the following:

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 observe two x tiles and five unit tiles. To express this algebraically we can write:

x + x + 1 + 1 + 1 + 1 + 1

Another way to write the expression is:2x + 5

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Example 6

Represent the expression -3x-5+8 using algebra tiles.

Worked Solution

Create a strategy

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

Apply the idea

Reflect and check

We could also use pictorial models to represent the expression.

The box represents the variable, and the strawberries represent the units. The 3 boxes and 5 strawberries are grayed-out to show they have been removed or "subtracted".

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