Practice Maths

Mixed Algebraic Problems

Key Ideas

like terms
Terms with the same variable(s) raised to the same power — 3x and 7x are like terms; 3x and 7x2 are not.
simplify
Collect all like terms to write an expression in its shortest equivalent form.
expand
Remove brackets using the distributive law — multiply every term inside by the factor outside.
factorise
Write an expression as a product of factors by extracting the HCF — the reverse of expanding.
substitute
Replace a variable with a given number value.
evaluate
Calculate the numerical result after substituting values into an expression.

Choosing the right skill

Read the question carefully. “Simplify” means collect like terms. “Expand” means remove brackets. “Factorise” means put brackets in. “Evaluate” means substitute and calculate.

Hot Tip When simplifying after expanding, first expand all brackets, then collect like terms. Take it step by step!

Worked Example

Question: Given the expression 3(2x + 1) − 4x + 5, expand and simplify, then evaluate when x = 3.

Step 1 — Expand the brackets.
3(2x + 1) − 4x + 5 = 6x + 3 − 4x + 5

Step 2 — Collect like terms.
= (6x − 4x) + (3 + 5) = 2x + 8

Step 3 — Evaluate when x = 3.
= 2(3) + 8 = 6 + 8 = 14

Reading the Question: Choosing the Right Skill

Algebra has four core skills and knowing which to use comes from reading the question carefully. "Simplify" or "collect like terms" means combine like terms without removing or adding brackets. "Expand" means remove brackets by multiplying. "Factorise" means introduce brackets by extracting a common factor. "Evaluate" means substitute a number and calculate. A multi-step question might ask you to expand, then simplify, then evaluate — always do them in that order.

Example of a combined question: "Expand and simplify 4(x + 3) − 2(3x − 1), then evaluate when x = −2." Step 1 (expand): 4x + 12 − 6x + 2. Step 2 (simplify): −2x + 14. Step 3 (evaluate with x = −2): −2(−2) + 14 = 4 + 14 = 18.

Expanding Then Collecting: A Careful Process

When expressions have multiple brackets, expand all brackets first before collecting any like terms. Don't try to do both simultaneously — it leads to errors. Write out each expansion fully, then look for like terms to group. For 2(3x + 1) + 4(x − 5): expand to get 6x + 2 + 4x − 20, then collect: 10x − 18. Notice the sign in front of each bracket tells you what multiplier to use (+4 means multiply by +4).

Be extra careful with subtraction between brackets: 3(2x + 5) − 4(x − 3). The second bracket is subtracted, meaning you multiply by −4: 6x + 15 − 4x + 12 = 2x + 27. Forgetting to change the sign of the second bracket's contents is the most common error here.

Key Tip: Expand ALL brackets before you collect a single like term. Write the fully expanded line first, then on the next line write the simplified answer. Two clean steps are better than one rushed step where errors hide.

Choosing Between Expanding and Factorising

These operations are inverses of each other. If you expand 5x(2x − 3) you get 10x2 − 15x. If you factorise 10x2 − 15x you get 5x(2x − 3). Both are correct expressions for the same quantity — just written differently. The context determines which form is more useful. For solving equations, factorised form is often easier. For substituting values, expanded form might be simpler.

A quick check technique: if you expand a factorised expression and get something different from where you started, you've made an error somewhere. Go back and verify each step.

Key Tip: When substituting negative values, always use brackets: if x = −3, then x2 = (−3)2 = 9. Writing just −32 gives −9, which is wrong. Brackets protect you from sign errors.

Applying Algebra to Word Problems

Algebra becomes truly powerful when used to model real situations. "A rectangle has length (2x + 3) cm and width (x + 1) cm. Write an expression for its perimeter and simplify." Perimeter = 2(2x + 3) + 2(x + 1) = 4x + 6 + 2x + 2 = 6x + 8 cm. Now evaluate when x = 5: 6(5) + 8 = 38 cm. The power of algebra is that one simplified expression works for any value of x.

Mastery Practice

  1. Simplify each expression by collecting like terms. Fluency

    1. 7x + 3x − 2x
    2. 5a − 2b + 3a + 6b
    3. 4m² + 3m − m² + 2m
    4. 9p − 4q + 2p − q
    5. 6x² + 2x − 3x² − 5x + 1
    6. 3xy + 2x − xy + 5x
    7. 8n + 4 − 3n − 9 + n
    8. 2a² − 5a + a² + 7a − 3

  2. Expand each expression. Fluency

    1. 4(3x + 2)
    2. 5(2y − 7)
    3. −3(a + 6)
    4. x(x − 4)
    5. 2m(3m + 5)
    6. −4t(2t − 1)
    7. 6(p + q − 3)
    8. 3k(k − 2) + 5k

  3. Factorise each expression fully. Fluency

    1. 8x + 12
    2. 15y − 10
    3. 6a² + 9a
    4. 14m² − 21m
    5. 4p² + 8p + 12
    6. −5t² + 10t

  4. Expand and simplify each expression. Understanding

    1. 3(x + 4) + 2(x + 1)
    2. 5(2a − 3) − 4(a − 2)
    3. 4(m + 3) − 2(3m − 5)
    4. 2x(x + 3) − x(x − 1)
    5. 3(y² − 2y + 4) + y(y + 5)
    6. 6(k + 1) − 3(2k − 4) + k

  5. Evaluate each expression using the given values. Understanding

    1. 4x + 7 when x = 3
    2. 2a² − 5a + 1 when a = 4
    3. 3(m − 2) + m when m = −1
    4. 2p + 3q when p = 5 and q = −2
    5. Expand 5(2n + 3) and then evaluate when n = −2.
    6. The expression for the area of a shape is 2x(x + 4). Find the area when x = 3 cm.

  6. Mixed algebraic problem solving. Problem Solving

    1. A rectangle has length (3x + 5) cm and width 4 cm. Write and simplify an expression for its perimeter.
    2. A square has side (2x + 1) cm. Write an expression for its area by expanding. Find the area when x = 3.
    3. Two students simplify 2(3x − 4) + 6x differently:
      • Student A gets: 12x − 4
      • Student B gets: 12x − 8
      Who is correct? Show full working to justify your answer.
    4. The formula for the perimeter of a rectangle is P = 2(l + w). Find P when l = (3x + 1) and w = (x − 2). Expand and simplify your answer.

  7. For each expression, apply the operation stated in brackets. Fluency

    1. 5x + 3y − 2x + y   [simplify]
    2. 3(2a − 5)   [expand]
    3. 4b2 + 6b   [factorise]
    4. 2m(m + 4) − 3m   [expand and simplify]
    5. 10n2 − 15n   [factorise]
    6. −4(x + 3) + 2x   [expand and simplify]
    7. 9p − 3q + 2p + 5q   [simplify]
    8. 6(k2 − 2k + 1)   [expand]

  8. For each item, choose whether to simplify, expand, factorise, or evaluate. Perform the correct operation. Understanding

    1. 4(x + 3) — find the value when x = 2
    2. 6x2 + 9x — write as a product of factors
    3. 5a + 3b − 2a + b — combine like terms
    4. 3(2n − 1) − 2(n + 4) — remove brackets and simplify
    5. 12y − 8 — write as a product
    6. 2x(x − 5) when x = −3 — expand first, then evaluate

  9. Each expression on the left has a simplified equivalent on the right. Match them, then verify by substituting x = 3. Understanding

    1. 3(x + 4) + x
    2. 2x + 8 − x + 1
    3. 4(x − 1) − (x − 7)
    4. 2(3x − 2) − 4x

    Options (match to one of the above):   2x − 4  |  x + 9  |  3x + 3  |  4x + 12

  10. Extended real-world algebra problems. Problem Solving

    1. A landscaper charges $35 per hour for labour and $20 per square metre for paving. A job takes (n + 3) hours to pave an area of (n + 2) m².
      1. Write an expression for the labour cost and expand it.
      2. Write an expression for the paving cost and expand it.
      3. Write and simplify an expression for the total cost.
      4. Find the total cost when n = 5.
    2. A school fundraiser sells small ($x each) and large ($2x each) chocolate bars. On Day 1 they sell 5 small and 3 large. On Day 2 they sell 4 small and 7 large. Write and simplify an expression for the total revenue over both days, then evaluate when x = 3.
    3. The volume of a rectangular prism is given by V = l × w × h. If l = 2x, w = (x + 3) and h = 4:
      1. Write an expression for the volume.
      2. Expand and simplify the expression.
      3. Find the volume when x = 2.
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