Introduction to algebraic expressions

1Summary 重點摘要

Letters stand for numbers: one expression describes every case. Only like terms combine, and the × sign usually disappears.

2Vocabulary cards Vocabulary

3Step by step Steps

1. 1. What is an algebraic expression?An algebraic expression is a mathematical phrase made of numbers, letters (variables) and operation signs, describing how quantities relate. Take 3x + 2: x is the variable, standing for an unknown number that can change; 3 is the coefficient, the number in front of x; 2 is the constant, whose value never changes. The great advantage is that one short expression covers a whole family of problems: 3x + 2 can mean '3 pens plus 2 dollars' or '3 bags of sweets plus 2 pieces', without writing a new sum for every situation. This is the key step from arithmetic into algebra.
2. 2. What is a variable?A variable is a letter standing for a number that can take different values. In 3x + 2y, both x and y are variables, each free to be any number. When x = 1 and y = 2, 3x + 2y = 3(1) + 2(2) = 3 + 4 = 7; when x = 5 and y = 0, the same expression becomes 3(5) + 2(0) = 15. So one expression gives different results as the variables change — exactly why they are called variables. Note: within one expression the same letter always means the same number, and different letters may stand for different numbers; mixing them up is a sure way to go wrong.
3. 3. SubstitutionSubstitution means replacing each variable with a concrete number, then working out the value in the correct order of operations. For 3x + 2 when x = 4: replace x with 4, giving 3(4) + 2; multiply first, 3 × 4 = 12, then add the constant, 12 + 2 = 14. So when x = 4, the value of 3x + 2 is 14. One more practice: when x = −2, 3(−2) + 2 = −6 + 2 = −4. A common mistake is dropping the brackets and turning 3(−2) into 3 − 2 = 1, a completely different answer; writing the brackets first, then calculating, avoids this slip.
4. 4. Like termsLike terms are terms whose letter parts are identical; only like terms can be combined. For 3x + 5x, both terms contain x, so they are like terms and combine: 3x + 5x = (3 + 5)x = 8x. But in 3x + 2y the letter parts differ (one is x, the other y), so they are not like terms and cannot be merged into 5xy or anything else — they simply stay as they are. A daily-life picture helps: 3 apples plus 5 apples makes 8 apples, but 3 apples plus 2 bananas remain 3 apples and 2 bananas, never '5 apple-bananas'. Constants (such as 2 and 7) are like terms with each other and can be combined.
5. 5. Addition and subtractionTo simplify addition and subtraction with brackets: first remove the brackets, then combine like terms. For (3x + 2) + (5x − 1): the sign before the second bracket is plus, so just drop the brackets, giving 3x + 2 + 5x − 1; combine the x terms, 3x + 5x = 8x; combine the constants, 2 + (−1) = 1; the result is 8x + 1. A trickier case: (3x + 2) − (5x − 1). The minus sign flips every term inside the second bracket: 3x + 2 − 5x + 1 = −2x + 3. The most common error is forgetting to flip the signs, writing −(5x − 1) as −5x − 1; subtracting a bracket changes the sign of every term inside it.
6. 6. MultiplicationMultiplication relies on the distributive law: the factor outside the bracket multiplies every term inside. For 3(x + 2): 3 times x gives 3x, and 3 times 2 gives 6, so 3(x + 2) = 3x + 6. If the outside factor is negative or a variable, the method is the same: −2(x − 4) = −2x + 8 (note that −2 × −4 = +8), and x(x + 5) = x² + 5x. Two mistakes appear most often: multiplying only the first term, writing 3(x + 2) as 3x + 2 and missing the 2; and mishandling the negative sign, writing −2(x − 4) as −2x − 8. After expanding, check that the number of terms matches the number inside the bracket — a quick and effective self-check.
7. 7. SimplifyCombining all the techniques above lets you handle longer expressions. For 5x + 3 + 2x − 1: first identify the like terms — the x terms are 5x and 2x, combining to 7x; the constants are 3 and −1, combining to 2; the result is 7x + 2. Slightly harder: 2(x + 3) + 3(x − 1) = 2x + 6 + 3x − 3 = 5x + 3. Build a fixed routine: remove brackets, mark like terms, combine them, then check that nothing is missing. Writing the answer in order (such as 7x + 2) keeps it clear, easy for teachers to mark and easy to revise later.
8. 8. Common mistakes summaryWhen working with algebraic expressions, avoid four frequent errors. First, combining unlike terms, such as turning 3x + 2y into 5xy — terms with different letters never combine. Second, substituting negative numbers without brackets: putting x = −3 into x² and writing −3² = −9; it should be (−3)² = 9. Third, multiplying only the first term when expanding, as in 3(x + 2) where the 2 is missed. Fourth, confusing coefficients with powers, thinking x + x = x²; in fact x + x = 2x, while x × x = x². After each question, test with a simple number (say x = 1) in both the original and the simplified expression: they match only when the simplification is correct. This habit saves many careless marks.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Variable: letter for unknown
• Coefficient: number in front of variable
• Like terms: same variable to combine
• Substitution: replace variable with number
• Distributive law: 3(x+2) = 3x + 6