Practice Maths

Like Terms and Simplifying

Key Ideas

term
A number, a pronumeral, or a product of both (e.g. 5x, 3y2, 7).
coefficient
The number in front of the pronumeral — in 5x, the coefficient is 5.
pronumeral
A letter used to represent an unknown or changing quantity (e.g. x, y, n).
like terms
Terms with the same pronumeral(s) raised to the same power — 3x and 7x are like terms.
simplify
Collect all like terms to write an expression in its shortest equivalent form.
Hot Tip 5x + 3y ≠ 8xy. You cannot add unlike terms. Think of it like adding apples and oranges — they stay separate.

Worked Example

Simplify: 4x + 3y − 2x + 7y − 5

Step 1 — Group like terms.
(4x − 2x) + (3y + 7y) − 5

Step 2 — Collect.
= 2x + 10y − 5

What Makes Terms "Like"?

Algebra uses letters (called pronumerals or variables) to represent unknown or changing quantities. A term is a product of numbers and variables: 5x, −3y2, 7ab, and 4 are all terms. The coefficient is the numerical part: in −3y2, the coefficient is −3. Like terms must have exactly the same variable part — same letters and same powers. So 4x and 7x are like terms, but 4x and 7x2 are not (different powers), and 4x and 7y are not (different variables).

Think of variables like units of measurement. 3 metres + 5 metres = 8 metres (same unit, can add). But 3 metres + 5 seconds cannot be combined — different units. Similarly, 3x + 5y cannot be simplified because x and y are different "units" of algebra.

Collecting Like Terms

To simplify 5x + 3y − 2x + 8y − 4: group the x terms together and the y terms together. (5x − 2x) + (3y + 8y) − 4 = 3x + 11y − 4. The constant term (−4) is its own like-term group (it has no variable). Always look for all groups of like terms systematically before combining.

With squared terms: 3x2 + 5x − 2x2 + 1. The x2 terms: (3x2 − 2x2) = x2. The x terms: 5x. The constants: 1. Result: x2 + 5x + 1. Note that x2 and 5x are NOT like terms, even though both contain x — the powers differ.

Key Tip: 3x + 2x2 cannot be simplified. The x term and the x2 term are different — like trying to add metres and metres-squared. The power makes them different types, even if the variable letter is the same.

Substitution: Giving Variables Values

Substitution means replacing each variable with a given number and evaluating. If x = 4 and y = −2, evaluate 3x2 − 5y + 1: replace x with 4 and y with −2: 3(4)2 − 5(−2) + 1 = 3(16) + 10 + 1 = 48 + 10 + 1 = 59. Key step: always use brackets when substituting to avoid sign errors, especially with negatives.

Substitution is how formulas work. The area formula A = ½bh becomes practical when we substitute b = 8 and h = 5 to get A = ½ × 8 × 5 = 20 cm². Every formula in science, engineering, and finance uses substitution.

Key Tip: Always use brackets when substituting negative numbers. If x = −3, then x2 = (−3)2 = 9 (positive), NOT −32 = −9. The bracket ensures the negative is squared along with the 3.

Common Misconceptions to Watch Out For

Several errors trip up students with like terms. First: 5x + 3x ≠ 8x2 — you add coefficients, you do NOT multiply the variable. Second: 5x × 3x = 15x2 (multiplication IS different — coefficients multiply and powers add). Third: expressions like 3x + 5y + 7 cannot be "simplified" any further — it already is simplified; three unlike terms written together is the final answer. Recognising when something is fully simplified is itself an important skill.

Mastery Practice

  1. From the list below, identify all the like terms. Group them together. Fluency

    3x,   5y,   7x,   2y2,   −4x,   9y,   3x2,   6y2,   −2y,   8,   −5,   4x

    1. Which terms are like terms with 3x?
    2. Which terms are like terms with 5y?
    3. Which terms are like terms with 2y2?
    4. Which terms are like terms with 8?

  2. Simplify by collecting like terms. (Single variable) Fluency

    1. 4a + 7a
    2. 9b − 3b
    3. 5m + 2m − 3m
    4. 8x − x + 4x
    5. 12n − 5n − 4n
    6. 3p + 4p − 8p + 2p
    7. 6t + 0 − 3t
    8. −2k + 9k − 4k

  3. Simplify by collecting like terms. (Multiple variables) Fluency

    1. 3x + 2y − x + 5y
    2. 6a + 3b − 2a − b
    3. 4m − 3n + 7m + 6n
    4. 5p + 2q − 3p − 2q
    5. 8x − 4y + 3 − 2x + y + 1
    6. 2a2 + 5a − 3a2 + 2a
    7. x2 + 3x + 4 − 2x + x2 − 1
    8. 7c + 4d − 3c + 2d − 5c

  4. Each simplification below contains an error. Find the mistake and write the correct answer. Understanding

    1. 5x + 3y = 8xy
    2. 4a + 2a2 = 6a3
    3. 7m − 4m + 3 = 10m
    4. 2x2 + 5x2 = 7x4
    5. 8p − 8p = 8
    6. 3ab + 2ab − ba = 4ab

  5. Find the perimeter of each shape by simplifying the algebraic expressions. Understanding

    1. A triangle with sides 3x, 5x, and 4x.
    2. A rectangle with length (3a + 2) and width (2a − 1).
    3. A pentagon with sides: 2n, 3n, n + 4, 2n − 1, and n + 3.

  6. Set up and simplify algebraic expressions for each situation. Problem Solving

    1. A school canteen sells x meat pies at $4 each and y sausage rolls at $3 each. Write an expression for the total sales. If x = 25 and y = 40, find the total.
    2. Sophia has 3 more pencils than Marcus. Marcus has m pencils. Write an expression for: (a) the total pencils they have together, and (b) the total if their friend Lena also joins, who has twice as many pencils as Marcus.
    3. A builder uses (2n + 5) bricks for one wall and (3n − 2) bricks for a second wall. Write and simplify an expression for the total bricks used, then find the total when n = 10.

  7. Simplify each expression. Take care with negative signs. Fluency

    1. −3x + 7x − 2x
    2. 5a − 8a + a
    3. −4m + 3m − m + 6m
    4. 2p − 7p + 4p − p
    5. −6n − n + 9n
    6. −5t + 2t − 3t + t
    7. 8k − 12k + 3k
    8. −2b − 3b + 10b − 4b

  8. Simplify each expression involving multiple types of terms. Understanding

    1. 4x + 3 − 2x + 7
    2. 5a2 − 3a + a2 + 4a − 2
    3. 6mn + 2m − 3mn + 5n − m
    4. 3x2 + x − 4 − x2 + 3x + 6
    5. 8p + 2q − 5p + 4 − q − 3
    6. 4ab + 3a − 2ab − a + 5b

  9. Simplify each expression first, then evaluate using the given values. Understanding

    1. 3x + 5x − 2x   when x = 4
    2. 4a − a + 6a   when a = −2
    3. 7m + 2n − 3m + n   when m = 3, n = −1
    4. 2x2 + x − x2 + 4x   when x = 2
    5. 5p − 8 + 2p + 3   when p = 5
    6. 9y − 4z − 3y + z   when y = 2, z = 3

  10. Apply simplification to geometry and real-world problems. Problem Solving

    1. A rectangular sports field has length (5x + 3) m and width (2x + 1) m. Write and simplify an expression for the perimeter. Find the perimeter when x = 4.
    2. Three friends each run different distances. Aiden runs (4n − 1) km, Bree runs (2n + 3) km, and Carlos runs (3n + 2) km. Write and simplify an expression for the total distance. Find the total when n = 5.
    3. A shop sells small boxes weighing 3w grams and large boxes weighing 7w grams. On Monday 4 small boxes and 2 large boxes are shipped. On Tuesday, 1 small box and 5 large boxes are shipped. Write and simplify a single expression for the total weight shipped over both days.
    4. A student writes: “3x2 + 5x = 8x3”. Identify the two errors and write the correct simplified form.
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