Algebraic Expressions
Translate the situation first, preserve the order of operations, and simplify only compatible terms.
Algebraic Expressions and Mathematical Translation
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An algebraic expression records quantities and relationships
An expression contains numbers, variables, and operations but no statement that two sides are equal. Translate phrases in their stated order and use parentheses when an operation affects a group.
five less than three times n → 3n−5Only like terms—terms with identical variable parts—can be combined by adding or subtracting their coefficients.
Order changes meaning
“Five less than three times n” is 3n−5, while “three times five less than n” is 3(n−5).
Parentheses protect groups
When subtracting or multiplying an expression, apply the operation to every term inside the group.
Name each quantity before building the expression
Underline the fixed amount, circle every per-unit rate, and define the variable. Then build fixed amount + (rate)(quantity) or the corresponding geometric relationship.
Why it works
Variables represent quantities, while coefficients and operations describe how those quantities are related. Keeping those roles visible prevents translation errors.
Five forms you should recognize
Problem: Simplify 5x−3 + 2x + 8.
(5x + 2x) + (−3 + 8) = 7x + 5Why: Variable terms combine with variable terms; constants combine with constants.
Problem: Write five less than three times n.
3n−5Why: Build three times n first, then subtract five.
Problem: Simplify −3(2x−5).
−6x + 15Why: The negative multiplier acts on both terms, changing the second sign.
Problem: A rectangle has width w and length w + 3. Find its perimeter expression.
2w + 2(w + 3) = 4w + 6Why: Perimeter includes two widths and two lengths.
Problem: Interpret C = 2,500 + 75n for an event with n guests.
fixed cost = ₱2,500; guest-dependent cost = ₱75nWhy: The constant does not change with attendance; the variable term does.
Check before you commit
- Combining terms whose variables or exponents do not match
- Reversing phrases involving “less than” or “subtracted from”
- Distributing a negative sign to only the first term
- Squaring each term but omitting the middle term in a binomial square
- Confusing a fixed charge with a per-unit rate
- Dividing only one term when an entire remaining amount must be shared
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Algebraic Expressions FAQ
What makes two terms like terms?
They must contain exactly the same variables raised to the same powers; only their coefficients may differ.
Why do translation problems need parentheses?
Parentheses preserve which quantity an operation affects, especially with subtraction, multiplication, and division.
Is an expression the same as an equation?
No. An equation states that two expressions are equal; an expression alone has no equals sign.
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