Learning Objectives
Identify terms, coefficients, variables, and constants.
Evaluate an expression by substitution.
Use distribution and combine like terms to write an equivalent expression.
Prerequisites: signed arithmetic and order of operations. Algebra uses letters to describe numbers that may be unknown or may vary. An expression represents a value; an equation states that two expressions are equal.
1. Read the Parts of an Expression
In 4x + 7, x is a variable, 4 is its coefficient, and 7 is a constant. The two terms are 4x and 7. Multiplication is often written without a multiplication sign, so 4x means 4 × x.
Expression | Meaning |
|---|---|
x + 5 | Five more than x |
3x | Three times x |
x/4 | x divided by four |
2(x + 1) | Twice the whole quantity x + 1 |
An expression such as 3x + 2 does not have a single numerical value until x is specified. In contrast, the equation 3x + 2 = 11 asks which input makes the equality true.
2. Substitute with Parentheses
Worked Example: A Negative Input
Evaluate 2x² − 3x + 1 when x = −2. Replace each x with (−2).
The expression becomes 2(−2)² − 3(−2) + 1 = 2(4) + 6 + 1 = 15.
The square applies to the whole negative input. Keeping parentheses prevents confusion between (−2)² and −2².
3. Distribute to Every Term
The distributive property says a(b + c) = ab + ac. It also works with subtraction: a(b − c) = ab − ac. Think of a groups, each containing both parts inside the parentheses.
Worked Example: Expand and Check
Expand 3(x + 4). Multiply both terms by 3 to get 3x + 12.
At x = 2, the original expression is 3(6) = 18 and the expanded expression is 6 + 12 = 18.
The property justifies equality for every input. Checking one input is useful for detecting errors, but one successful check alone would not prove a general identity.
A negative sign outside parentheses means multiplication by −1. Therefore −(x − 5) = −x + 5. Both signs change because both terms are multiplied by −1.
4. Combine Only Like Terms
Like terms have the same variable part, including the same powers. For example, 3x and 5x combine to 8x. But 3x and 5x² cannot be combined into one term by adding their coefficients. Constants can be combined with other constants.
Worked Example: Simplify a Whole Expression
Simplify 2(3x − 4) − x + 7. First distribute: 6x − 8 − x + 7.
Combine the x terms: 6x − x = 5x. Combine the constants: −8 + 7 = −1.
The simplified expression is 5x − 1. The term −x has coefficient −1, even though the 1 is not written.
Common Mistakes
Multiply every term inside parentheses when distributing.
Keep a term's sign attached to it when rearranging or combining.
Do not combine terms with different variable powers.
5. Practice
1. Evaluate 5a − 2 when a = −3.
2. Expand −4(x − 2).
3. Simplify 7x + 3 − 2x + 5.
4. Simplify 3(2y + 1) − 2(y − 4).
Worked Solutions
1. 5(−3) − 2 = −15 − 2 = −17.
2. Multiply both terms by −4: −4x + 8.
3. The variable terms give 5x and the constants give 8. The result is 5x + 8.
4. Expand to get 6y + 3 − 2y + 8. Combining like terms gives 4y + 11.
6. Summary
- Expressions describe values; equations make equality claims.
- Use parentheses when substituting a negative value.
- Distribution multiplies every term in a group.
- Combine terms only when their variable parts match.