Tag: collecting like terms

Simplifying Algebraic Expressions: Terms, Coefficients, Like Terms and Algebra Tiles

Aven Brisail2 Comments

FORM 2 • TERM 1 • LESSON 09

In the last lesson you wrote and evaluated expressions. Now you will learn to simplify them – write them in the shortest possible way. You will name the parts of an expression, model terms with algebra tiles, multiply and divide terms, and collect like terms. This builds on the Form 1 lesson on simplifying expressions and like terms.

Learning outcomes

By the end of this lesson you should be able to:

  • 2.6.2.1 – Explain what an algebraic expression is.
  • 2.6.2.2 – Show terms with objects (algebra tiles), pictures and symbols.
  • 2.6.2.3 – Identify like and unlike terms.
  • 2.6.2.4 – Tell the difference between a coefficient and an operational sign.
  • 2.6.2.5 – Add, subtract, multiply and divide terms using tiles, pictures and symbols.
  • 2.6.2.6 – Simplify algebraic expressions.

What is an algebraic expression?

An algebraic expression is a collection of numbers and letters joined by operations, such as 4x − 3y + 7. It has no equals sign, so it cannot be “solved” – but it can be simplified, and it can be evaluated if you know the letters.

  • A term is one part of the expression: 4x, 3y and 7 are the three terms of 4x − 3y + 7.
  • A coefficient is the number multiplying a letter: in 4x the coefficient of x is 4. In x on its own, the coefficient is 1.
  • A constant is a term with no letter, such as 7.
  • An operational sign is the + or − written between terms. It tells you whether to add or subtract the next term.

Coefficient or operational sign?

In 4x − 3y + 7, the coefficient of y is 3, and the operational sign in front of it is −. The sign is not part of the coefficient when you name it, but it always travels with the term after it when you rearrange or collect terms. So the term is really “−3y”, and you can write the expression as 4x + 7 − 3y, but not as 4x + 3y − 7.

Algebra tiles

Algebra tiles are a concrete way to see terms. A long tile stands for x, a small square stands for 1, and a large square stands for x². Shaded (orange) tiles are negative.

Algebra tiles: a large square for x squared, a long tile for x, a small square for 1, and orange negative tiles for minus x and minus 1 x² x 1 −x −1 x² x 1 −x −1
Figure 1 – The algebra tiles: x², x and 1, with their negative tiles −x and −1.

A positive tile and its negative tile make a zero pair, because x + (−x) = 0 and 1 + (−1) = 0. Taking out zero pairs is how tiles show adding and subtracting:

Tiles for 2x plus 3 and for minus x minus 2 combine; after removing zero pairs, one x tile and one unit tile are left 2x + 3 x x 1 1 1 −x − 2 −x −1 −1 Remove the zero pairs (x with −x, 1 with −1): Left over: x + 1 = x + 1
Figure 2 – 2x + 3 add −x − 2. One x and −x make a zero pair, and two 1s and −1s make zero pairs, leaving x + 1.

Like and unlike terms

Like terms have exactly the same letters, with the same powers. Only like terms can be added or subtracted into a single term.

Like termsUnlike terms
3x and −5x3x and 3y
2ab and 7ba (ab = ba)4a and 4a²
x² and −6x²5 and 5x

To collect like terms, add or subtract the coefficients and keep the letters: 5a + 3b − 2a + b = (5 − 2)a + (3 + 1)b = 3a + 4b.

Multiplying and dividing terms

You can multiply or divide any terms, like or unlike. Multiply (or divide) the numbers, then the letters.

  • 3 × 2x = 6x (three groups of 2x tiles is 6x tiles).
  • 2a × 3b = 6ab.
  • x × x = x², and 4x × 5x = 20x².
  • 12x ÷ 4 = 3x.
  • 10ab ÷ 5a = 2b.
  • −6y × 2 = −12y, using the integer sign rules.

Worked examples

Example 1: Naming the parts

Question: For 6m − 4n + 9, write down
(a) the terms
(b) the coefficient of n
(c) the operational sign before the n-term
(d) the constant.

  1. (a) The parts separated by + and − are 6m, 4n and 9 (the n-term is −4n).
  2. (b) The number multiplying n is 4.
  3. (c) The sign in front of it is −.
  4. (d) The term with no letter is 9.

Answer:
(a) 6m, −4n, 9
(b) 4
(c) −
(d) 9.

Example 2: Like terms

Question: From 4p, −2q, p², 7p, 3qp, −pq, 5, pick out the groups of like terms.

  1. 4p and 7p have only p.
  2. 3qp and −pq both have p × q.
  3. −2q, p² and 5 have no partners.

Answer: 4p and 7p; 3qp and −pq.

Example 3: Collecting like terms

Question: Simplify 7p − 3q − 9p + 8q.

  1. Gather the p-terms: 7p − 9p = −2p.
  2. Gather the q-terms: −3q + 8q = 5q.

Answer: −2p + 5q (or 5q − 2p).

Example 4: Terms with squares

Question: Simplify 4x² + 3x − x² − 5x + 2.

  1. x²-terms: 4x² − x² = 3x².
  2. x-terms: 3x − 5x = −2x.
  3. The constant 2 has no partner.

Answer: 3x² − 2x + 2.

Example 5: Multiplying and dividing

Question: Simplify
(a) 4a × 3b
(b) −5x × 2x
(c) 18pq ÷ 6q.

  1. (a) 4 × 3 = 12 and a × b = ab.
  2. (b) −5 × 2 = −10 and x × x = x².
  3. (c) 18 ÷ 6 = 3 and pq ÷ q = p.

Answer:
(a) 12ab
(b) −10x²
(c) 3p.

Example 6: Using tiles

Question: Use tiles to simplify (3x − 2) + (−x + 5).

  1. Lay out 3 x-tiles and 2 negative unit tiles, then 1 negative x-tile and 5 unit tiles.
  2. Remove zero pairs: one x with one −x, and two 1s with two −1s.
  3. Left: 2 x-tiles and 3 unit tiles.

Answer: 2x + 3.

Common mistakes

  • Adding unlike terms. 3x + 2y cannot be simplified; it is not 5xy.
  • Losing the sign. In 8a − 3b + 2b, the −3b keeps its minus: the answer is 8a − b, not 8a + 5b.
  • Mixing up x² and 2x. x × x = x², but x + x = 2x.
  • Forgetting the hidden 1. x means 1x, so 5x − x = 4x, not 5.
  • Treating x and x² as like terms. 4x + 3x² stays as it is.

Practice questions

Try these yourself first, then open each answer to check your working.

1. In 9k − 2m + 5 − k:
(a) how many terms are there?
(b) what is the coefficient of m?
(c) which terms are like terms?

Show answer

(a) 4
(b) 2, with the operational sign − in front
(c) 9k and −k.

2. Which pairs are like terms?
(a) 6y and −y
(b) 2x and 2x²
(c) 3mn and 4nm
(d) 7 and 7a.

Show answer

(a) and (c).

3. Simplify:
(a) 5x + 3x − 2x
(b) 4a − 9a
(c) 6b + 2c − b + 3c.

Show answer

(a) 6x
(b) −5a
(c) 5b + 5c.

4. Simplify:
(a) 3m − 4n − 5m + n
(b) 2x² + 5x − 3x² + x
(c) 8 − 3y + 2y − 10.

Show answer

(a) −2m − 3n
(b) −x² + 6x
(c) −y − 2.

5. Simplify:
(a) 5 × 3t
(b) 2y × 7y
(c) −3a × 4b
(d) 6m × (−2n).

Show answer

(a) 15t
(b) 14y²
(c) −12ab
(d) −12mn.

6. Simplify:
(a) 20x ÷ 5
(b) 14ab ÷ 7b
(c) −24k² ÷ 6k.

Show answer

(a) 4x
(b) 2a
(c) −4k.

7. Use tiles (or a drawing of them) to simplify (2x + 4) + (3x − 6).

Show answer

5x − 2.

8. A rectangle is 3x cm long and 2x cm wide. Write and simplify expressions for
(a) its perimeter
(b) its area.

Show answer

(a) 3x + 2x + 3x + 2x = 10x cm
(b) 3x × 2x = 6x² cm².

Quick check

1. What is the coefficient of y in 7x − 4y?

Show answer

4 (the operational sign in front of it is −).

2. Are 5ab and −2ba like terms?

Show answer

Yes.

3. Simplify 6x − 8x + 3.

Show answer

−2x + 3.

Previous: Algebra Review: Variables, Word Statements and Substituting Integers

Next: Linear Equations: Letters on Both Sides, Brackets and Word Problems

Curriculum link: This post is relevant to the Trinidad and Tobago National Lower Secondary (NCSE) Mathematics Curriculum – Form 2, Term 1, Strand 2.6 Algebra, Topic 2.6.2 Simplification of Algebraic Expressions – Learning Outcomes 2.6.2.1 (algebraic expressions), 2.6.2.2 (representing terms concretely, pictorially and symbolically), 2.6.2.3 (like and unlike terms), 2.6.2.4 (coefficient and operational sign), 2.6.2.5 (operations on terms) and 2.6.2.6 (simplifying algebraic expressions).

Updates

2026/10/23

– First version of this lesson.

2026/10/25

– Put each lettered part of the questions and answers on its own line.

© 2026 Vedesh Kungebeharry. All rights reserved.

Simplifying Algebraic Expressions: Like Terms, the Four Operations and the Distributive Law

Aven BrisailLeave a comment

To simplify an algebraic expression means to write it in a shorter, neater form that has exactly the same value for every number you substitute. 3x + 5x and 8x are equal for every value of x, but 8x is simpler. In this lesson you will learn to spot like terms, collect them, multiply and divide terms, and use the distributive law to expand brackets.

This lesson builds on Introduction to Algebra: Variables, Constants, Expressions and Substitution.

Like and unlike terms

Like terms have exactly the same letters, with the same powers. Only the numbers in front (the coefficients) may differ.

Like termsUnlike terms
3x and 7x3x and 3y (different letters)
2ab and 5ba (ab = ba)x and x2 (different powers)
y2 and 4y24a and 4 (one has a letter, one does not)

Think of it like money: you can add 3 five-dollar notes and 7 five-dollar notes to get 10 five-dollar notes, but 3 five-dollar notes plus 3 ten-dollar notes is not “6 fifteen-dollar notes”.

Adding and subtracting: collect like terms

  • Add or subtract the coefficients of like terms; the letters stay the same: 3x + 5x = 8x and 7y − 2y = 5y.
  • Remember an invisible 1: x means 1x, so 5x − x = 4x.
  • Group like terms together, keeping each term’s sign with it: 4a + 3b + 2a − b = (4a + 2a) + (3b − b) = 6a + 2b.
  • Unlike terms cannot be combined: 6a + 2b is already as simple as it gets.

Multiplying and dividing terms

Unlike adding, you can multiply or divide any terms. Multiply (or divide) the numbers, then deal with the letters.

  • 3a × 4b = 3 × 4 × a × b = 12ab
  • 2x × 5x = 2 × 5 × x × x = 10x2
  • 12ab ÷ 3a = 12ab3a = 4b (12 ÷ 3 = 4, and the a on top cancels with the a below)
  • 10x ÷ 5 = 2x

The distributive law: expanding brackets

The distributive law says that a number outside a bracket multiplies every term inside it: a(b + c) = ab + ac. You already use it in mental arithmetic: 3 × 14 = 3 × (10 + 4) = 30 + 12 = 42.

Area model: a rectangle 3 high and x plus 4 long is split into a 3 by x part and a 3 by 4 partx433x123(x + 4) = 3x + 12
Figure 1 – The area model: a rectangle 3 units tall and (x + 4) units long has area 3 × x plus 3 × 4.

Worked examples

Example 1: Expanding one bracket

Question: Expand
(a) 3(x + 4)
(b) 5(2y − 3)
(c) a(a + 3)

  1. (a) 3 × x + 3 × 4 = 3x + 12
  2. (b) 5 × 2y − 5 × 3 = 10y − 15
  3. (c) a × a + a × 3 = a2 + 3a

Answer: 3x + 12; 10y − 15; a2 + 3a.

Example 2: Expand and simplify

Question: Simplify 2(3x + 1) + 4(x + 5).

  1. Expand each bracket: 2(3x + 1) = 6x + 2 and 4(x + 5) = 4x + 20.
  2. Write it all out: 6x + 2 + 4x + 20.
  3. Collect like terms: (6x + 4x) + (2 + 20) = 10x + 22.
  4. Check by substituting x = 1: the original gives 2 × 4 + 4 × 6 = 8 + 24 = 32, and 10 × 1 + 22 = 32 ✔

Answer: 10x + 22.

Example 3: Two letters

Question: Simplify 3(2a + b) + 2(a + 4b).

  1. Expand: 6a + 3b + 2a + 8b.
  2. Collect the a terms: 6a + 2a = 8a. Collect the b terms: 3b + 8b = 11b.

Answer: 8a + 11b.

Example 4: Subtracting a bracket

Question: Simplify 4(2m + 3) − 3(m + 1).

  1. Expand the first bracket: 8m + 12.
  2. The minus sign in front of the second bracket means we subtract all of 3(m + 1) = 3m + 3. So we subtract 3m and subtract 3: 8m + 12 − 3m − 3.
  3. Collect like terms: 8m − 3m = 5m and 12 − 3 = 9.

Answer: 5m + 9.

Common mistakes

  • Adding unlike terms. 3x + 2y cannot be simplified – it is not 5xy.
  • Adding letters as powers. x + x = 2x, but x × x = x2.
  • Multiplying only the first term. 3(x + 4) = 3x + 12, not 3x + 4.
  • Forgetting the power. 2x × 3x = 6x2, not 6x.
  • Losing the invisible 1. 5x − x = 4x, not 5.

Practice questions

Try these yourself first, then open each answer to check your working.

1. From the list 4p, 3q, 7p, pq, 2qp, p2, pick out the pairs of like terms.

Show answer

4p and 7p; pq and 2qp. (3q and p2 have no partner.)

2. Simplify 6m + 3n − 2m + 5n.

Show answer

4m + 8n.

3. Simplify
(a) 4x × 3y
(b) 5a × 2a
(c) 18pq ÷ 6q

Show answer

(a) 12xy
(b) 10a2
(c) 3p

4. Expand
(a) 4(3k + 2)
(b) 7(2 − x)

Show answer

(a) 12k + 8
(b) 14 − 7x

5. Expand and simplify 3(2x + 5) + 2(x + 1).

Show answer

6x + 15 + 2x + 2 = 8x + 17.

6. Expand and simplify 5(a + 2b) − 2(a + b).

Show answer

5a + 10b − 2a − 2b = 3a + 8b.

7. Use x = 2 to show that 3(x + 4) is not equal to 3x + 4.

Show answer

3(2 + 4) = 3 × 6 = 18, but 3 × 2 + 4 = 10. Different values, so they are not equal. (The correct expansion is 3x + 12, which gives 18.)

8. A rectangle has length 3x + 4 and width x + 2. Write and simplify an expression for its perimeter, then find it when x = 5.

Show answer

2(3x + 4) + 2(x + 2) = 6x + 8 + 2x + 4 = 8x + 12. When x = 5: 52. (Check: length 19, width 7, perimeter 2 × 26 = 52 ✔)

Quick check

1. Are 5xy and 3yx like terms?

Show answer

Yes – xy and yx are the same product.

2. Simplify a + a + a.

Show answer

3a.

3. State the distributive law.

Show answer

a(b + c) = ab + ac: the number outside multiplies every term inside the bracket.

4. How can you check that a simplification is correct?

Show answer

Substitute the same number into the original and the simplified expression – both must give the same value.

Next: Units of Measurement: Metric vs Imperial and Converting mm, cm, m and km

Curriculum link: This post is relevant to the Trinidad and Tobago National Lower Secondary (NCSE) Mathematics Curriculum – Form 1, Term 1, Strand 1.6 Algebra, Topic 1.6.1 Introducing Algebra – Learning Outcomes 1.6.1.7 (identifying like and unlike terms), 1.6.1.8 (simplifying expressions involving the four operations) and 1.6.1.9 (simplifying expressions using the distributive law).

Updates

2026/10/25

– Put each lettered part of the questions and answers on its own line.

© 2026 Vedesh Kungebeharry. All rights reserved.