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1. Equations & Inequalities in One Variable
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Algebra 1

1.01 Interpreting algebraic expressions

Lesson
Practice
Lesson

Concept summary

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 which includes at least one variable.

Algebraic operation

A mathematical process, such as addition, subtraction, multiplication, or division

Example:

+, -, \times, and \div

Variable

A symbol, usually a letter, used to represent an unknown value.

Example:

Expression: 3x-y+2

Variables: x and y

Term

One part of an expression. Terms are separated by addition or subtraction.

Example:

Expression: 3x-y+2

Terms: 3x, -y, and 2

Coefficient

The number or constant that multiplies a variable in an algebraic term. If no number is specified, the coefficient is 1.

Example:

Term: 3x

Coefficient: 3

Constant term

A term that has a fixed value and does not contain a variable. Sometimes just called a constant.

Example:

Expression: 3x-y+2

Constant: 2

Like terms are terms that have the same variables with the same exponents. An example is shown below:

\displaystyle 3x+5x^2 -2x -8
\bm{\text{Terms:}}
3x, 5x^2, -2x, and -8
\bm{\text{Like terms:}}
3x and -2x

Worked examples

Example 1

Consider the algebraic expression: 15y^2+7y^3-3y^2-5

Identify:

  • The constant term

  • The like terms

  • The coefficient of y^3

Approach

The constant term is the numeric term without any variable factor, the like terms are the algebraic terms that have the same algebraic factors, and the coefficient of y^3 is the numeric factor of the term containing a factor of y^3.

Solution

The constant term is -5, as it is the term without any variable part.

The like terms are 15y^2 and -3y^2, as they have the same variables with the same exponents.

The coefficient of y^3 is 7, as this is the numeric factor of the term 7y^3.

Reflection

Notice that we include the sign with all terms. For example, one of the like terms is -3y^2, which has a coefficient of -3.

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.

Approach

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. Once we determine how to write the cost per job in terms of a hours, we need to write an expression that is the sum of these two values.

Solution

\text{Cost: } 37.5a+150

Example 3

Bernie and Paula are painting the walls of their house. Bernie can paint x square meters an hour while Paula can paint y square meters an hour. Bernie spends 4 hours painting while Paula spends 6 hours over the weekend. Together they paint a total of 4x+6y square meters over the weekend.

Determine what 6y represents in the expression.

Approach

We can look at the units that are included in the term 6y. Paula paints for 6 hours and y is the number of square meters she can paint per hour. Consider what will result from multiplying the units:

\text{h} \times \dfrac{\text{m}^2}{\text{h}}=\text{m}^2

We can use this to help us determine what 6y represents.

Solution

The number of square meters Paula painted over the weekend.

Outcomes

A1.N.Q.A.1

Use units as a way to understand real-world problems.*

A1.N.Q.A.1.A

Choose and interpret the scale and the origin in graphs and data displays.*

A1.A.SSE.A.1

Interpret expressions that represent a quantity in terms of its context.*

A1.A.SSE.A.1.A

Interpret parts of an expression, such as terms, factors, and coefficients.

A1.MP2

Reason abstractly and quantitatively.

A1.MP3

Construct viable arguments and critique the reasoning of others.

A1.MP6

Attend to precision.

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