3.1 Manipulating algebraic expressions and equations

Achievable Praxis Core: Math (5733)

3. Algebra and geometry

Manipulating algebraic expressions and equations

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This section covers simplifying, expanding, factoring, and rewriting algebraic expressions using the commutative, associative, and distributive properties. These skills help you create equivalent forms that are easier to evaluate or use when solving equations. It also covers following a step-by-step procedure. Solving linear equations is treated in a separate chapter.

Definitions

Term: A single algebraic component made up of a coefficient and variables multiplied together, for example $3 x^{2}$ or $- 5 y$ .
Coefficient: The numerical factor of a term, such as $3$ in $3 x^{2}$ .
Like terms: Terms that have the same variables raised to the same powers, which can be combined by adding or subtracting coefficients.
Distributive property: The rule $a (b + c) = ab + a c$ , which allows expansion or factoring of expressions.
Equivalent expressions: Different forms of an expression that simplify to the same value for all variable inputs.
Expression: Combines numbers, variables, and operations to represent a value (e.g., $3 x - 2$ ).
Equation: Sets two expressions equal, often containing an unknown to solve for (e.g., $3 x - 2 = 7$ ).

An expression has no $=$ sign, so it can only be simplified or evaluated - not solved. An equation includes $=$ and asks for the value(s) of the variable that make the two sides equal.

Translating verbal phrases to expressions

Before you can simplify or solve, you often need to build an expression from a word problem. Most phrases translate directly - but a few trip up even careful readers.

The most common pitfall involves “less than” and “subtracted from.” The key is that these phrases reverse the order you might expect:

“4 less than 7” → $7 - 4$ (not $4 - 7$ )
“x subtracted from 10” → $10 - x$ (not $x - 10$ )
“5 less than a number $n$ ” → $n - 5$

Think of it this way: “4 less than 7” asks you to start at 7 and go down by 4. The number mentioned after “less than” comes first in the expression.

Other common translations:

Verbal phrase	Algebraic expression
the sum of $x$ and $3$	$x + 3$
twice a number $n$	$2 n$
the product of $5$ and $y$	$5 y$
4 less than $x$	$x - 4$
$x$ subtracted from $12$	$12 - x$
the quotient of $x$ and $4$	$\frac{x}{4}$

Simplifying by combining like terms

Simplifying an expression means rewriting it in a cleaner form by combining like terms. Like terms have identical variable parts (including exponents). For example, $3 x^{2}$ and $- 5 x^{2}$ are like terms because both include $x^{2}$ . But $3 x^{2}$ and $3 x$ are not like terms because the exponents on $x$ are different.

How to identify and combine like terms:

Scan for matching variable parts: look for terms with exactly the same variables raised to the same powers (the coefficients can differ).
Group them together: use the commutative and associative properties to rearrange terms so like terms are next to each other.
Add or subtract coefficients: combine only the numbers in front.
Keep the variable part: attach the common variable part to the new coefficient.

Common mistake - combining unlike terms: Only terms with identical variable parts can be combined. Here are two errors to avoid:

$3 x^{2} + 3 x \neq = 6 x^{2}$ and $\neq = 6 x$ - the exponents differ, so these are not like terms and cannot be combined.
$5 x^{2} y$ and $4 x y^{2}$ are not like terms - in $5 x^{2} y$ the exponent on $x$ is 2 and on $y$ is 1, but in $4 x y^{2}$ it’s the reverse. Different exponent patterns mean different terms.

When in doubt, compare the full variable part (letters and their exponents) before combining.

Example: combining like terms

Given the expression

$2 x + 5 x - 3 x + 4 y - y$

Identify like terms:

$x$ -terms: $2 x, 5 x, - 3 x$

$y$ -terms: $4 y, - y$

Combine the $x$ -terms:

Coefficients: $2 + 5 - 3 = 4$

Result: $4 x$

Combine the $y$ -terms:

Coefficients: $4 - 1 = 3$

Result: $3 y$

Final simplified form:

$4 x + 3 y$

Answer: $4 x + 3 y$

Expanding using the distributive property

The distributive property lets you remove parentheses by multiplying a factor across each term inside the parentheses. In symbols:

$a (b + c) = ab + a c .$

How to expand any expression:

Identify the factor (or expression) directly outside the parentheses.
Multiply it by each term inside the parentheses.
Write each product, keeping track of signs.
Combine like terms if any appear.

Common pitfall - distributing a negative: When the factor outside the parentheses is negative, every term inside changes sign. For example:

$- 3 (x - 2) = - 3 x + 6, not - 3 x - 6.$

The $- 2$ inside becomes $- 3 \times (- 2) = + 6$ . A common error is to distribute the negative to the first term only and leave the second term’s sign unchanged. The same care applies when substituting negative values: always use parentheses, e.g., $3 (- 1) = - 3$ ; writing $3 \cdot - 1$ without parentheses invites sign errors.

The same idea works for any number of terms inside the parentheses. Given $k (T_{1} + T_{2} + \dots + T_{n})$ , multiply $k$ by every term to get $k T_{1} + k T_{2} + \dots + k T_{n}$ .

Multiplying two binomials

FOIL - First, Outer, Inner, Last - organizes the distributive steps when multiplying two binomials. You multiply each term of the first binomial by each term of the second, then combine like terms.

Example: expand a binomial product $(x + 2) (x - 5)$

First: multiply the first terms in each binomial
$x \cdot x = x^{2}$

Outer: multiply the outer terms
$x \cdot (- 5) = - 5 x$

Inner: multiply the inner terms
$2 \cdot x = 2 x$

Last: multiply the last terms in each binomial
$2 \cdot (- 5) = - 10$

Combine like terms: add the Outer and Inner results
$- 5 x + 2 x = - 3 x$

Result:

$x^{2} - 3 x - 10$

Answer: $x^{2} - 3 x - 10$

Simplifying more complex expressions

Example: simplify $\frac{2 x ^{2}}{6 x}$ .

Factor coefficients and variables: $\frac{2}{6} \cdot \frac{x ^{2}}{x}$

Simplify $\frac{2}{6} = \frac{1}{3}$ and $\frac{x ^{2}}{x} = x$

Result $\frac{1}{3} x$ - keep the fractional coefficient; canceling variables doesn’t make the $\frac{1}{3}$ disappear.

Answer: $\frac{1}{3} x$ (for $x \neq = 0$ ; the original expression is undefined at $x = 0$ )

Factoring expressions

Factoring reverses expansion. Instead of distributing a factor across terms, you rewrite an expression as a product of factors by pulling out common factors. A factored form is often easier to evaluate, simplify further, or use when solving equations.

The key technique covered here is the greatest common factor (GCF): the largest factor shared by all terms. Factoring it out simplifies the expression.

Example: factor out a common factor

Factor

$6 x + 9$

Find the GCF of the coefficients: $6 = 2 \cdot 3$ and $9 = 3 \cdot 3$ , so the GCF of the coefficients is $3$ .

Check the variable parts: $6 x$ has a factor of $x$ , but $9$ does not, so no variable is shared by all terms.

The GCF of all terms is $3$ .

Divide each term by $3$ to find the co-factor:

$6 x \div 3 = 2 x$

$9 \div 3 = 3$

Write the factored form:

$6 x + 9 = 3 (2 x + 3) .$

Answer: $3 (2 x + 3)$

Plugging in values for variables

When an expression contains one or more variables, you can evaluate it by substituting the given values and then simplifying using order of operations - always wrap substituted negatives in parentheses so you don’t lose a sign.

Identify the variables and their assigned values.
Substitute each variable in the expression with its corresponding number.
Simplify step by step: parentheses and exponents first, then multiplication and division, then addition and subtraction.

Example: evaluate $2 x + 3 y$ when $x = 4$ and $y = - 1$ .

Substitute $x = 4$ and $y = - 1$ :

$2 (4) + 3 (- 1)$

Multiply each term:

$8 + (- 3)$

Add:

$5$

Answer: $5$

Following a procedure

Some questions give a procedure - a list of steps, a flowchart, or a rule with conditions - and ask you to carry it out or to say what it does. Follow it exactly as written, one step at a time, and write down each result.

A conditional rule does different things in different cases, like a shipping policy: “if the order is over $$50$ , shipping is free; otherwise, it costs $$6.95$ .” Check the condition with the given values first, then use only the formula it selects. A flowchart works the same way: each diamond asks a yes-or-no question, and the answer decides which arrow you follow.

Example: a rule with two cases

For a whole number $n$ : if $n$ is divisible by $3$ , let $z = \frac{n}{3} + 4$ ; otherwise, let $z = 2 n - 1$ . Find $z$ for $n = 14$ .

Check the condition: $14 \div 3 = 4$ remainder $2$ , so $14$ is not divisible by $3$ .

Use the second formula: $z = 2 \cdot 14 - 1 = 27$

Answer: $27$

Computing both formulas and choosing the answer that looks familiar is a common mistake; the condition alone decides. Some procedures repeat until they reach a stopping point.

Example: a procedure that repeats

Start with $6$ . If the number is even, divide it by $2$ ; if it is odd, multiply it by $3$ and add $1$ . Repeat with each result, and stop when you reach $1$ . How many steps does it take?

(spoiler)

$6 \to 3 \to 10 \to 5 \to 16 \to 8 \to 4 \to 2 \to 1$

Answer: $8$ steps

A recursive sequence, where a rule builds each term from the ones before it, is a procedure repeated; see Properties of whole numbers.

Combine like terms by adding or subtracting coefficients of matching variable parts.
Use the distributive property to expand parentheses; factor to reverse expansion.
Use FOIL to expand two binomials.
Factor by pulling out the greatest common factor (GCF) of every term.
Verify factoring by expanding back to the original expression.
Rewrite expressions into equivalent forms to simplify, evaluate, or solve.
Evaluate by substituting values and simplifying in order.
To follow a procedure, check each condition before choosing a formula, then carry out the steps in order.

Next | 3.2 Solving equations and inequalities