What this chapter is about
In mathematics, we often use letters to stand for numbers. These letters are called variables because their value can vary or change. When we combine variables with numbers using operations like addition, subtraction, multiplication and division, we create algebraic expressions.
This chapter teaches you how to form expressions using letter-numbers, understand what each part of an expression means, and work with these expressions. You learn to identify terms, coefficients and constants. You also learn to simplify expressions by combining like terms.
After studying this chapter, you should be able to write real-life situations as algebraic expressions, identify the parts of any expression, and simplify expressions by adding or subtracting like terms.
Key ideas
- A variable is a letter (like x, y, n or p) that represents an unknown or changing number.
- An algebraic expression is a combination of variables, numbers and operations. For example, 3x + 5 is an expression.
- A term is a single part of an expression separated by + or − signs. In 4a + 7b − 2, the terms are 4a, 7b and −2.
- The coefficient is the number multiplied with a variable. In 6m, the coefficient is 6.
- A constant is a term without any variable. In 5n + 9, the constant is 9.
- Like terms have the same variable raised to the same power. 3x and 7x are like terms. 3x and 3y are not like terms.
- We can only add or subtract like terms. Unlike terms stay separate.
- Expressions help us describe patterns and real-life situations in a short, general way.
Formulas and facts to remember
- Expression = terms joined by + or − signs.
- Term = coefficient × variable, or just a constant.
- Like terms = terms with exactly the same variable part. Example: 2p and 5p are like terms.
- To add like terms: add their coefficients. Example: 4x + 3x = 7x.
- To subtract like terms: subtract their coefficients. Example: 9y − 4y = 5y.
- Unlike terms cannot be combined. Example: 3a + 2b stays as 3a + 2b.
- A variable alone (like x) has coefficient 1. So x means 1 × x.
Worked examples
Example 1: Writing an expression from a situation
Ravi has some pencils. His sister gives him 8 more pencils. Write an expression for the total number of pencils Ravi has now.
Solution: Let p stand for the number of pencils Ravi had at first. After getting 8 more, total pencils = p + 8. The expression is p + 8.
Example 2: Identifying parts of an expression
Find the terms, coefficients and constant in the expression 5m + 3n − 12.
Solution: The expression has three terms: 5m, 3n and −12. In 5m, the coefficient of m is 5. In 3n, the coefficient of n is 3. The constant term is −12 (it has no variable).
Example 3: Simplifying by combining like terms
Simplify: 7a + 4b + 2a − b + 6.
Solution: Step 1: Group like terms together. Terms with a: 7a and 2a. Terms with b: 4b and −b. Constant: 6.
Step 2: Combine like terms. 7a + 2a = 9a. 4b − b = 4b − 1b = 3b. Constant stays as 6.
Step 3: Write the simplified expression. 9a + 3b + 6.
Common mistakes
- Treating unlike terms as like terms → Only combine terms that have the same variable. 3x + 2y cannot become 5xy.
- Forgetting that a variable alone has coefficient 1 → Remember that y means 1y, so y + 5y = 6y.
- Ignoring the sign before a term → The sign belongs to the term. In 8 − 3p, the term is −3p, not 3p.
- Adding coefficients of unlike terms → 4a + 5b is already simplified. Do not write 9ab.
- Confusing expressions with equations → An expression has no equal sign. 2x + 3 is an expression, but 2x + 3 = 11 is an equation.
Quick revision
- A variable is a letter that stands for a number.
- Terms are separated by + or − signs.
- Coefficient is the number in front of a variable.
- Like terms have the same variable; only these can be combined.
- To combine like terms, add or subtract their coefficients.
- Constants are terms with no variable.