STUDYROOM AUSTRALIA Calm, structured practice
Personalised tutoring & learning support
Year 7–8 Algebra Diagnostic (60-minute): Expressions, Equations & Linear Relationships
Mathematics
Year 7–8 bridge
AC9M7A02, AC9M7A03, AC9M7A04, AC9M8A02 (expressions, equations, linear relations)
Use variables, simplify & expand expressions, solve 1–2-step equations, and describe linear patterns from tables & stories
Bar models & trial tables | Graphing, generalising & explaining structure

Print via your browser’s print dialog (Ctrl/Cmd + P). This worksheet is designed as a calm diagnostic for Years 7–8.

Learning intention & success criteria Start here

Learning intention

Today I am learning to represent unknowns with variables, simplify and expand algebraic expressions, and solve linear equations from worded situations .

Success criteria

I can:

Tutor note: Invite the student to rewrite the learning intention in their own words.
Warm-up / Prior knowledge Quick check

Complete the questions below to get your brain ready.

  1. Evaluate 2n + 5 when n = 7.
  2. Simplify: 5x + 3x − 4.
  3. Expand: 3(4y − 2).
  4. Write an expression for: “ten minus four times a number p”.
Vocabulary / Key ideas Language
Teaching input – What you need to know Read & notice

We can use algebra to model real-life situations. We choose a variable for the unknown, write an expression or equation, then use inverse operations to solve and check our work.

Steps for solving a simple linear equation:

  1. Represent: Choose a variable for the unknown and write an equation from the story or pattern.
  2. Solve: Use inverse operations to undo the equation step-by-step on both sides.
  3. Check & interpret: Substitute back in, check the equation is true, then explain what the answer means in the story.
Write your own one-sentence story and matching equation below (include a joining fee or starting amount).
Worked example Model

Example question:

A music app charges a $6 sign-up fee plus $4 per month. Noah paid $38 in total. For how many months did he have the app?

Step-by-step solution:

  1. Let m be the number of months. Total cost: 4m + 6 = 38.
  2. Subtract 6 from both sides: 4m = 32. Divide both sides by 4: m = 8.
  3. Check: 4 × 8 + 6 = 32 + 6 = 38. This matches the total. So Noah had the app for 8 months.

Why this works:

We represent a constant rate ($4 per month) with a coefficient and the sign-up fee as a constant. Using inverse operations keeps both sides of the equation balanced so the solution is still true.

In your own words, explain why we subtract 6 before dividing by 4.
Guided practice (We do) Together

Use the worked example to help you answer these questions.

  1. A taxi charges a $5 flagfall plus $3 per kilometre. The fare was $26. How many kilometres did the taxi travel? Let k be kilometres.
  2. A sports club charges a $12 joining fee plus $7 per visit. Mia paid $54 in total. How many visits did she make? Let v be visits.
  3. You start with $10 on a travel card and lose $2 for each bus trip. After some trips you have $ 2 left. How many trips did you take? Let t be trips.

Adjustments:

Support: For each question, make a small trial table (input → output) to explore values before writing the equation.

Challenge: For Q1 and Q2, write the rule as an equation and sketch a quick graph showing the intercept and rate of change.

Independent practice (You do) Your turn

Now try these on your own. Use the example and your notes to help.

  1. Simplify: 6a + 2a − 9.
  2. Expand: 4(3b − 1).
  3. Solve: 2x + 7 = 19.
  4. Solve: 5y − 3 = 2.
  5. Solve: t/4 + 6 = 11.
  6. Cost story: A game costs $15 plus $4 per sticker pack. You pay $47 in total. How many sticker packs did you buy? Let s be the number of sticker packs.
  7. Table → rule:
    Items (x): 0, 1, 2, 3
    Total (y): −2, 1, 4, 7

    (a) What is the rule for y in terms of x?
    (b) Use your rule to find y when x = 10.
  8. Perimeter story: A rectangle has width 5 cm and length (3m + 2) cm. Its perimeter is 26 cm. Find the value of m.
Extension / Enrichment (Optional) Stretch

Create your own linear story involving a starting amount and a “per” amount (for example, streaming, rides, savings, or fundraising).

  1. Write a short story for the situation.
  2. Write an equation using a variable to represent the unknown.
  3. Choose a total and solve your equation.
  4. Explain what the solution means in your story and sketch a quick graph of your rule.
Reflection Check-in

Prep–Year 2

Circle how you feel about today’s work:

😊
😐
🙁

Today I learned…

Years 3–12

What did I do well?

What was challenging?

What strategy helped me succeed?

Materials (if needed) Set up

Answer key — For parents & tutors

Warm-up answers

  1. 2n + 5 when n = 7: 2 × 7 + 5 = 14 + 5 = 19.
  2. 5x + 3x − 4 = 8x − 4.
  3. 3(4y − 2) = 12y − 6.
  4. “ten minus four times a number p”: 10 − 4p.

Guided practice answers

  1. Taxi: 5 + 3k = 26 ⇒ 3k = 21 ⇒ k = 7 km.
  2. Sports club: 12 + 7v = 54 ⇒ 7v = 42 ⇒ v = 6 visits.
  3. Travel card: 10 − 2t = 2 ⇒ −2t = −8 ⇒ t = 4 trips.

Independent practice answers

  1. 6a + 2a − 9 = 8a − 9.
  2. 4(3b − 1) = 12b − 4.
  3. 2x + 7 = 19 ⇒ 2x = 12 ⇒ x = 6.
  4. 5y − 3 = 2 ⇒ 5y = 5 ⇒ y = 1.
  5. t/4 + 6 = 11 ⇒ t/4 = 5 ⇒ t = 20.
  6. 15 + 4s = 47 ⇒ 4s = 32 ⇒ s = 8 sticker packs.
  7. Table: difference in y is +3 each time, so y = 3x − 2.
    (a) Rule: y = 3x − 2.
    (b) When x = 10, y = 3 × 10 − 2 = 30 − 2 = 28.
  8. Perimeter P = 2L + 2W.
    L = (3m + 2), W = 5, P = 26.
    2(3m + 2) + 2(5) = 26
    6m + 4 + 10 = 26 ⇒ 6m + 14 = 26 ⇒ 6m = 12 ⇒ m = 2.
    (Length is then 3 × 2 + 2 = 8 cm.)

Extension model response (sample)

Example: “A rideshare app charges a $5 booking fee plus $3 per kilometre.” Rule: C = 3k + 5.
If the total cost is $26, then 3k + 5 = 26 ⇒ 3k = 21 ⇒ k = 7 km.
On a graph, the y-intercept (5) shows the booking fee and the slope (3) shows the cost per kilometre.