Practice Maths

Topic Review — Linear Models and Further Algebra

Mixed Practice — L45 – L47

This review covers all lessons in the Linear Models and Further Algebra topic. Try each question before checking your answers.

Review Questions

  1. Simplifying and solving equations. Fluency

    1. Solve: 3x + 7 = 22
    2. Solve: 5y − 4 = 16
    3. Solve: 2(m + 3) = 14
    4. Solve: 4n − 9 = n + 6
    5. Simplify: 3a + 5b − a + 2b
    6. Simplify: 4x × 3y
    7. Solve: (t + 1) / 3 = 5
    8. Solve: 2(3k − 4) = 16

  2. Describing patterns algebraically. Fluency

    1. A pattern has tiles arranged in rows. Row 1 has 3 tiles, Row 2 has 5 tiles, Row 3 has 7 tiles. Write a rule for the number of tiles in Row n.
    2. Use your rule from (a) to find the number of tiles in Row 10.
    3. A sequence begins: 4, 9, 14, 19, … Write the algebraic rule for the nth term.
    4. Use the rule T = 3n + 2 to find the 8th term.
    5. Which term of the sequence T = 5n − 3 equals 47?
    6. A sequence has rule T = 2n + 6. Is 30 a term in this sequence? Show your reasoning.

  3. Creating and interpreting linear models. Understanding

    1. A phone plan charges $15 per month plus $0.10 per minute of calls. Write an equation for the total monthly cost C in terms of minutes used m.
    2. Using your equation from (a), find the cost if 120 minutes are used.
    3. If the monthly bill was $32, how many minutes were used?
    4. Plot the linear model from (a) for m = 0, 50, 100, 150. Describe the gradient and y-intercept in context.
    5. A taxi charges a $3.50 flag fall plus $2.20 per kilometre. Write an equation for the total fare F in terms of kilometres k.
    6. Two friends compare taxi costs: Alex travels 8 km; Bella travels 5 km and then adds a $4 luggage fee. Who pays more?

  4. Interpreting and comparing linear models. Understanding

    1. Plumber A charges $50 call-out fee plus $80 per hour. Plumber B charges $90 call-out fee plus $60 per hour. Write equations for both.
    2. After how many hours do both plumbers charge the same amount? Solve algebraically.
    3. For a 3-hour job, which plumber is cheaper?
    4. A graph shows two linear models intersecting at (4, 370). Explain what this point means in the context of a tradesperson hire problem.

  5. Problem solving with linear models and algebra. Problem Solving

    1. A gym has two membership options:
      • Option A: $200 joining fee + $25 per month
      • Option B: No joining fee + $45 per month
      Write equations for both options and find after how many months the costs are equal. Which option is better for someone who plans to attend for 12 months?
    2. A pattern of squares is built using matchsticks. The 1st shape uses 4 matchsticks, the 2nd uses 7, the 3rd uses 10.
      1. Write an algebraic rule for the number of matchsticks M in the nth shape.
      2. How many matchsticks are needed for the 20th shape?
      3. Which shape number uses exactly 100 matchsticks? Show full working.
    3. A container holds 240 litres of water. Water drains at a constant rate of 12 litres per minute. Write a linear model for the volume V remaining after t minutes. Find when the container is empty and when it is half full.
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