1.01 Algebraic expressions

An expression is a mathematical statement that contains one or more numbers and variables joined together by operators and grouping symbols. An expression does not contain an equal sign or inequality symbol.

An algebraic expression is an expression that includes at least one variable.

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.

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:

Examples

Example 1

Create a model using algebra tiles for the following algebraic expression:

-7 + 5(1-3x)

Worked Solution

Create a strategy

Algebra tiles use tiles with +1, -1, +x, and -x to represent the individual terms. The number outside of the parentheses for the distributive property represents how many groups of identical expressions will be made.

Apply the idea

We will need the -x, +1, and -1 tiles to make our model. The 5 outside of the parentheses means we will need 5 groups of 1-3x.

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Example 2

Vincenzo runs a removalist company that charges \$ 37.50 per hour plus a one-off truck hire fee of \$ 150.00.

Write an expression that models how much he charges for a job that lasts a hours.

Worked Solution

Create a strategy

We need to look at the two values that affect the price of the job; the cost per hour of \$ 37.50 and the truck hire fee of \$ 150.00.

For each hour worked, Vincenzo charges an additional \$37.50. Let's consider a few cases:

Cost of working 1 hour: \$150+\$37.50

Cost of working 2 hours: \$150+\$37.50+\$37.50

Cost of working 3 hours: \$150+\$37.50+\$37.50+\$37.50

Notice that for each additional hour worked, we add an additional \$37.50. Multiplication is repeated addition, so we can multiply \$37.50 by the number of hours instead of adding repeatedly.

Apply the idea

Since the \$150 is a one-time fee, this value will remain constant. Next, we multiply \$37.50 by the number of hours which is a.

\text{Cost: } 37.5a+150

Reflect and check

For this problem, we would replace a with the number of hours Vincenzo works on a particular job, and the result would be the amount of money he makes on that job.

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Example 3

If the area of a square is given by the expression (2x - 1)^2, explain what 2x-1 represents in the context of the problem.

Worked Solution

Create a strategy

First, we need to remember the formula for the area of a square.

We know that the area of a square is given by s^2 where s represents the side length.

Apply the idea

\displaystyle \text{Area}	\displaystyle =	\displaystyle s^2
	\displaystyle =	\displaystyle (2x-1)^2

Therefore, we have:

\displaystyle s

\displaystyle =

\displaystyle 2x-1

This shows that s which represents the side length is given by 2x-1. Thus, the side length of the square is 2x-1.

Reflect and check

From here, we could easily use algebraic expressions to determine the perimeter or other useful measurements of the square.

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In life, the order in which we do things is important. For example, we put on socks then shoes, rather than shoes and then socks.

The order of operations tells us the steps to evaluate expressions with multiple operations, so that the same numerical result is achieved. The order goes:

1. Complete all operations within grouping symbols such as brackets [\ldots], parentheses (\ldots) or absolute values |\ldots|. If there are grouping symbols within other grouping symbols, do the innermost operation first.
2. Evaluate all exponents, such as squares and cubes.
3. Multiply and/or divide in order from left to right.
4. Add or subtract in order from left to right.

We often want to substitute values for the variables in an algebraic expression. That way we can evaluate the expression to yield a numerical result.

As an example, a concession stand sells bags of popcorn for \$2.50 each and hot dogs for \$1.50 each. The total cost of buying p bags of popcorn and h hot dogs can be represented by the expression 2.50p + 1.50h.

Riley goes the concession stand every Saturday and buys 2 bags of popcorn and 4 hot dogs for her friends, and wants to determine the total cost of her order 3 Saturdays in a row.

Since the same order will be repeated 3 times, the distributive property will be used to rewrite the expression. By substituting the values p=2 and h=4 into the new expression, we see that

The total cost of Riley's purchase is \$33.

Examples