Apply the order of operations and the properties of real numbers (commutative, associative, distributive, identity, and inverse) to simplify and evaluate numerical and algebraic expressions in one variable (A.EO.1).

What this topic is asking

A.EO.1 asks you to simplify and evaluate expressions in one variable by applying the order of operations and the properties of real numbers. On the Virginia Algebra I SOL these are foundational Expressions and Operations items: simplify a multi-step expression, name the property that justifies a step, or substitute a value and evaluate. They appear as multiple choice, fill-in-the-blank, and drag-and-drop (ordering the steps).

The order of operations

A numerical or algebraic expression is simplified in one fixed order, often remembered as PEMDAS:

1. Parentheses and other grouping symbols (brackets, the bar of a fraction, inside a radical).
2. Exponents.
3. Multiplication and division, left to right, as they appear.
4. Addition and subtraction, left to right.

The pairs matter: multiplication and division share a rank (do them left to right, not all multiplication first), and so do addition and subtraction. For example $20 - 6 \div 2 \cdot 3 = 20 - 3 \cdot 3 = 20 - 9 = 11$, working the division and multiplication left to right before the subtraction.

The properties of real numbers

The properties are the rules that let you rewrite an expression without changing its value. Each has a precise name the SOL may ask you to identify.

• Commutative property. Order does not matter for addition or multiplication: $a + b = b + a$ and $ab = ba$. (Subtraction and division are not commutative.)
• Associative property. Grouping does not matter for addition or multiplication: $(a + b) + c = a + (b + c)$ and $(ab)c = a(bc)$.
• Distributive property. Multiplication distributes over addition: $a(b + c) = ab + ac$. This is the bridge between factored and expanded form.
• Identity property. Adding $0$ leaves a number unchanged ($a + 0 = a$); multiplying by $1$ leaves it unchanged ($a \cdot 1 = a$).
• Inverse property. A number plus its additive inverse (opposite) is $0$: $a + (-a) = 0$. A nonzero number times its multiplicative inverse (reciprocal) is $1$: $a \cdot \frac{1}{a} = 1$.

Simplifying an algebraic expression

To simplify, clear parentheses with the distributive property, then combine like terms. The order of operations still governs which step comes first.

Evaluating an expression

To evaluate, substitute the given value for the variable and simplify using the order of operations. Use parentheses around the substituted value so signs and exponents stay correct. For $-x^2 + 4x$ at $x = -3$: write $-(-3)^2 + 4(-3) = -(9) - 12 = -21$. The exponent applies to $-3$ inside the parentheses (giving $9$), and the leading minus then makes it $-9$; dropping the parentheses and writing $-3^2$ as $9$ is the usual error.

Why the distributive property is the workhorse

The distributive property is the one rule that connects the two ways an expression can look. Read left to right, $a(b + c) = ab + ac$ expands a product into a sum, which is exactly what "simplify" usually wants. Read right to left, $ab + ac = a(b + c)$ factors a sum into a product, which is what later topics (factoring, solving quadratics) need. Every time you remove parentheses to combine like terms, and every time you pull out a common factor, you are using the same property in one direction or the other. That is why a sign error in distribution (forgetting the factor reaches every term, or mishandling a leading minus) propagates through so much of Algebra I: the move is everywhere.

How the SOL examines this topic

• Multiple choice. Name the property that justifies a given step, or pick the correctly simplified expression.
• Fill-in-the-blank. Simplify an expression and type it, or substitute and evaluate to a number.
• Drag-and-drop. Order the steps of a simplification, or drag the property name to each line of a worked solution.

A clarifying idea: simplifying never changes an expression's value, only its form. Every legal step is one of the named properties or the order of operations, so if you cannot name the rule behind a move, the move is probably wrong.

Try this

Q1. Simplify $5x - 2(3x - 7)$. [2 points]

• Cue. $5x - 6x + 14 = -x + 14$.

Q2. Evaluate $4a - a^2$ when $a = 5$. [1 point]

• Cue. $4(5) - (5)^2 = 20 - 25 = -5$.