Math · Algebra & word problems
Expressions and Order of Operations
Estimated time includes reading and one quiz. Take the time you need.
Start here: key ideas
- A variable represents an unknown or changeable number.
- Grouping and exponents come before multiplication and addition.
- Like terms have identical variable parts.
- Distribution multiplies every term inside the grouping.
What you’ll be able to do
- Translate phrases into expressions.
- Evaluate in correct order.
- Combine like terms and distribute.
Build the expression from relationships, then simplify without changing its meaning.
Read an expression
An expression combines numbers, operations, and perhaps a variable, a symbol for a number. In 4x+7, 4 is the coefficient multiplying x. A term is a part separated by addition or subtraction, so 4x and 7 are terms.
Words signal relationships: “five more than n” is n+5, while “five less than n” is n−5. Order reverses after “less than”: “five less than twice n” is 2n−5.
Worked translation: “5 less than three times n” becomes 3n − 5: build three times n first, then subtract 5. “Less than” reverses the order of the two named quantities; it does not change the sign of the variable.
Evaluate in a defensible order
Use grouping first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right. Multiplication is not always before division; they share priority. For 18÷3×2, work left to right: 6×2=12.
Scroll the table sideways if needed.
| Stage | Expression |
|---|---|
| Start | 3+2(5−1)² |
| Group | 3+2(4)² |
| Exponent | 3+2(16) |
| Multiply/add | 35 |
Like terms and distribution
Like terms have the same variable raised to the same power: 3x and −5x are like, but 3x and 3x² are not. Combine coefficients: 3x−5x=−2x. The distributive property multiplies an outside factor by every term: 4(2x−3)=8x−12. A negative outside changes both signs: −2(x−5)=−2x+10.
Worked examples
Worked example 1: Translate “three times the sum of y and 4”: 3(y+4). The grouping matters; 3y+4 would add only four after tripling y.
Worked example 2: Simplify 2(3x−1)+4x. Distribute to get 6x−2+4x, then combine like terms: 10x−2.
A reliable way to reason
Order of operations is a grammar for mathematical meaning. Grouping says which collection acts as one quantity. Compare 3(x+4) with 3x+4: the first triples the entire sum, while the second triples only x. Substituting x=2 makes the difference visible: 18 versus 10. Exponents describe repeated factors before surrounding multiplication is performed.
When simplifying, preserve equivalence at every step. A coefficient can combine only with another coefficient attached to the identical variable part. In 2x²+5x−x², the x² terms combine to x², but 5x remains separate. Distribution can be checked by substituting an easy value into the original and simplified forms. For −3(2y−4), distribution gives −6y+12. At y=1, both forms equal 6, confirming both signs. Translation deserves the same check: identify the main operation last. In “four times the difference of n and 2,” the difference is grouped first and multiplication is outside, giving 4(n−2). Naming the relationship prevents keyword-driven reversals.
Guided reasoning and error checks
A useful evaluation routine is copy, substitute, group, calculate. For 2a²−3b when a=−3 and b=4, first copy the entire expression. Substitute with parentheses: 2(−3)²−3(4). Evaluate the exponent: 2(9)−12. Multiply: 18−12. Subtract: 6. Parentheses ensure that the negative value, rather than only the 3, is squared.
To check simplification, compare values. The original 3(x+2)+2x and the simplified 5x+6 should match for every x. At x=4, both equal 26. One matching test does not prove an identity, but a mismatch immediately exposes an error. This makes substitution an efficient editing tool for distribution, signs, and like-term work.
Final self-check
Do not invent an equality sign while simplifying. An expression such as 4x+2 has no single solution; it takes different values for different x. Simplifying rewrites it equivalently, while solving requires an equation or inequality. This distinction tells you whether the final response should still contain a variable or should report a value under stated conditions.
Terms to remember
- expression
- Numbers, variables, and operations without an equality sign.
- variable
- A symbol representing a number.
- coefficient
- A numerical factor multiplying a variable.
- term
- A part separated by addition or subtraction.
- like terms
- Terms with identical variable parts.
- distributive property
- Multiplying a factor by every term in a sum.
Your quick summary
- Translate the relationship before inserting values.
- Operations at the same level proceed left to right.
- Combine only matching variable parts and distribute to every term.
- Grouping symbols determine which quantity an outside factor or operation affects.
Lesson quiz
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