Practice Maths

★ Topic Review — Linear Equations and Their Graphs

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This review covers all four lessons: Solving Linear Equations and Inequalities, Graphing Linear Functions, Linear Models and Applications, and Simultaneous Equations. Questions increase in difficulty.

  1. Fluency — Solving linear equations and inequalities

    1. (a) Solve: 5x − 8 = 2x + 7
    2. (b) Solve: (3x + 1)/4 = (x − 2)/2
    3. (c) Solve the inequality and represent on a number line: 3 − 2x > 11

  2. Fluency — Features of a linear function

    For the line y = −4x + 8:

    1. (a) State the gradient m and y-intercept c.
    2. (b) Find the x-intercept.
    3. (c) Find the y-value when x = −3.
    4. (d) Find the x-value when y = −4.

  3. Fluency — Simultaneous equations

    1. (a) Solve by substitution: 2x + 3y = 15   and   x − y = 0
    2. (b) Solve by elimination: 3x + 2y = 16   and   3x − y = 7

  4. Understanding — Equation of a line

    Find the equation of the line through (−2, 5) with gradient −3.

    1. (a) Write in gradient–intercept form y = mx + c.
    2. (b) Write in general form ax + by + c = 0 (with integer coefficients).

  5. Understanding — Parallel, perpendicular, or same line?

    Are the lines y = 3x − 1 and 2y = 6x + 4 parallel, perpendicular, or the same line? Explain fully.

  6. Understanding — Perpendicular line

    Find the equation of the line perpendicular to y = 4x − 3 that passes through (8, 1).

  7. Understanding — Linear model: mobile data

    A mobile plan includes 10 GB of data. Each additional GB costs $5.

    1. (a) Write a cost model C for extra data, in terms of e (GB used over the 10 GB limit).
    2. (b) Find the extra cost if the total data used in a month is 13.6 GB.

  8. Understanding — Multi-step equation

    Solve: 4(x − 2) − 3(2x + 1) = −5

  9. Understanding — Simultaneous equations: ticket sales

    Adult tickets cost $22 each and child tickets cost $14 each. A group of 12 people paid a total of $204. How many adults and how many children were in the group?

  10. Understanding — Spring model

    A spring is 10 cm at rest and stretches 2 cm for every 1 N of force applied.

    1. (a) Write the equation for length L (cm) in terms of force F (N).
    2. (b) Find the length at F = 6 N.
    3. (c) What force would cause a length of 22 cm?

  11. Understanding — Parallel line from general form

    Find the equation of the line through (2, −3) that is parallel to 4x − 2y = 10.

  12. Problem Solving — Inequality with algebraic fractions

    Solve the inequality (x + 3)/2 ≥ (2x − 1)/3 and represent the solution on a number line.

  13. Problem Solving — Comparing car hire companies

    Luxury Cars: $120/day + $0.35/km.   Budget Wheels: $75/day + $0.52/km.

    1. (a) Write cost equations for each company for one day in terms of km travelled (k).
    2. (b) Find the km per day that makes both companies equal cost.
    3. (c) Which company is cheaper for a 3-day trip with 150 km/day? How much is saved?

  14. Problem Solving — Finding k

    A straight line passes through (1, k) and (k, 9) and has gradient 2.

    1. (a) Use the gradient formula to write an equation in k and solve for k.
    2. (b) Write the coordinates of both points and find the equation of the line.

  15. Problem Solving — Resource allocation

    A factory produces widgets (x) and gadgets (y).

    Each widget uses 4 kg metal + 2 hr labour.
    Each gadget uses 3 kg metal + 5 hr labour.
    Available per day: 48 kg metal, 45 hr labour.

    Solve the simultaneous equations to find how many widgets and gadgets should be produced per day to use all available metal and labour.

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