Constants & Coefficients
The numbers attached to variables — and the numbers that stand alone. Knowing these by sight is non-negotiable.
Building on Lesson 1
In Lesson 1 you learned that a variable is a letter standing in for an unknown or changing number. But variables rarely appear alone. They’re almost always accompanied by numbers — numbers that tell us how many of that variable we have, or numbers that sit alongside them independently.
These numbers have specific names, and recognising them instantly is a foundational skill you’ll use in every single lesson that follows.
Anatomy of an ExpressionLet’s dissect a simple algebraic expression and name every part:
A coefficient is the numerical factor multiplied by a variable. It sits directly in front of the variable, with no operation symbol between them — their closeness is the multiplication sign.
Coefficients can be positive, negative, whole numbers, fractions, or decimals. The key is that they are always attached to a variable.
| Expression | Variable | Coefficient | Meaning |
|---|---|---|---|
| 4x | x | 4 | four lots of x |
| −3y | y | −3 | negative three lots of y |
| ½n | n | ½ | half of n |
| x | x | 1 | one lot of x — the 1 is invisible but always there |
| −x | x | −1 | negative one lot of x — the 1 is invisible but always there |
| 0.5t | t | 0.5 | half of t (same as ½t) |
You met constants briefly in Lesson 1. Now let’s be precise. A constant is any number in an expression that stands entirely alone — it is not multiplied by a variable. Its value never changes within the problem.
· The 6 is a coefficient — it multiplies the variable x
· The 9 is a constant — it stands alone, attached to nothing
Both are “just numbers,” but their roles in the expression are different.
Imagine an expression as a fruit stall. You have 3 apples and 5 oranges, plus a fixed daily fee of R10 to rent the stall. Here, 3 and 5 are coefficients (they tell you how many of each fruit), while R10 is the constant — it doesn’t depend on anything, it’s just always there.
Coefficients work exactly the same way even when the variable has a power (an exponent). The coefficient is always the number multiplying the entire variable term.
| Term | Coefficient | Variable Part |
|---|---|---|
| 2x³ | 2 | x³ |
| −5ab | −5 | ab |
| x²y | 1 | x²y |
| ¾m² | ¾ | m² |
For each expression, state the coefficient(s) and constant(s).
Write the algebraic term described.
Coefficient of x: −5
Constant: 2
Coefficient: 350 (the rate per hour, multiplying h)
Constant: 200 (the fixed call-out fee — always there regardless of hours)
If they are equal: k = −4 A small taste of equation-solving — we’ll formalise this from Lesson 16 onwards.
- A coefficient is the number multiplied by a variable — it tells you “how many” of that variable
- A constant is a number standing alone, not attached to any variable
- When a variable has no visible number in front, its coefficient is the invisible 1
- When a variable has a minus sign in front, its coefficient is −1
- Coefficients can be negative, fractional, or decimal
- The sign (+ or −) before a term belongs to that term’s coefficient or constant