Practice Maths

Modelling Relationships 1

Key Terms

Relationship
A connection between two variables where one quantity depends on the other.
Linear relationship
A relationship whose graph is a straight line; y changes by a constant amount for every unit increase in x.
Rule (equation)
An algebraic expression that describes how two variables are connected, e.g. y = 2x + 3.
Table of values
A table showing chosen input (x) values and the corresponding output (y) values calculated from a rule.
Rate of change
How much y increases (or decreases) for each unit increase in x; the coefficient of x in the rule.

Relationships Between Two Variables

A relationship describes how one variable changes as another variable changes. We often use a rule (or equation) to describe this.

A linear relationship has a constant rate of change: equal increases in x always produce equal increases in y. When plotted, linear relationships produce a straight line.

Building a Table of Values

  1. Choose values of x (usually 0, 1, 2, 3, 4).
  2. Substitute each x-value into the rule to find the corresponding y-value.
  3. Record in a table.
Hot Tip
For a linear relationship, equal increases in x always give equal increases in y. To check if a table shows a linear relationship, find the difference between consecutive y-values. If it’s constant, the relationship is linear!

Worked Example — Table of Values

Build a table of values for y = 3x − 1 for x = 0, 1, 2, 3, 4.

x 0 1 2 3 4
y −1 2 5 8 11

Rate of change = 3 (y increases by 3 for each x increase of 1). Linear relationship — plotting these points gives a straight line.

x y 0 1 2 3 4 y=2x+1 (linear) y=x² (non-linear)

Blue = linear (straight line). Red = non-linear (curve — unequal y-increases).

What Is a Mathematical Relationship?

A mathematical relationship describes how two quantities are connected — how changing one affects the other. For example:

  • The more hours you work, the more money you earn (distance and time, cost and items bought)
  • The further you travel at a constant speed, the longer it takes
  • The cost of buying n items at $3 each is 3 × n dollars

When we write this connection as an equation, we call it a mathematical model. Models help us make predictions without having to measure everything directly.

Writing a Rule as an Equation

To write a rule, identify the two quantities (variables) and the relationship between them:

  • A taxi charges $5 per kilometre plus a $3 flagfall. Cost (C) for d km: C = 5d + 3
  • A plant is 2 cm tall and grows 1.5 cm per week. Height (h) after w weeks: h = 1.5w + 2
  • Buying bags of lollies at $2.50 each. Total cost (T) for n bags: T = 2.5n

Once you have the rule, you can substitute values to make predictions.

Plotting a Relationship on a Graph

To graph a relationship, create a table of values first: choose several values of one variable, calculate the other using the rule, then plot the pairs as coordinates.

  • For T = 2.5n: when n=0, T=0; n=1, T=2.5; n=2, T=5; n=3, T=7.5
  • Plot points (0,0), (1,2.5), (2,5), (3,7.5) and draw a line through them

A straight-line graph indicates a linear relationship — the two quantities change at a constant rate relative to each other.

Linear vs Non-linear Patterns

A relationship is linear if the graph is a straight line (e.g. T = 2.5n). It is non-linear if the graph is a curve (e.g. A = n² for the area of a square with side n). Linear relationships have a constant rate of change; non-linear relationships have a changing rate of change. At Grade 7, most relationships you model are linear.

Key tip: When writing a rule from a table of values, check if the output increases by the same amount each time the input increases by 1. If yes, it's linear — that constant increase is the coefficient of n in your equation. Also check the value when n = 0: that's the constant (the starting value or y-intercept).

Mastery Practice

  1. Complete Tables of Values Fluency

    Complete a table of values for each rule using x = 0, 1, 2, 3, 4.

    1. y = 2x
    2. y = x + 5
    3. y = 3x − 2
    4. y = 4x + 1
    5. y = ½x + 3

  2. Is It Linear? Fluency

    For each table, decide whether the relationship is linear or non-linear. Justify your answer by checking the differences between consecutive y-values.

    1. x: 0, 1, 2, 3, 4  —  y: 1, 3, 5, 7, 9
    2. x: 0, 1, 2, 3, 4  —  y: 0, 1, 4, 9, 16
    3. x: 0, 1, 2, 3, 4  —  y: 10, 8, 6, 4, 2
    4. x: 0, 1, 2, 3, 4  —  y: 3, 6, 12, 24, 48
    5. x: 0, 1, 2, 3, 4  —  y: 5, 5, 5, 5, 5
    6. x: 0, 1, 2, 3, 4  —  y: 2, 5, 9, 14, 20

  3. Find the Rule Understanding

    Find the rule for y in terms of x. State the rate of change and describe the relationship in words.

    1. x: 0, 1, 2, 3  —  y: 0, 4, 8, 12
    2. x: 0, 1, 2, 3  —  y: 3, 5, 7, 9
    3. x: 0, 1, 2, 3  —  y: 10, 7, 4, 1
    4. x: 0, 1, 2, 3  —  y: 1, 4, 7, 10
    5. x: 0, 1, 2, 3  —  y: 6, 6, 6, 6
    6. x: 0, 1, 2, 3  —  y: 0, 1.5, 3, 4.5

  4. Real-World Modelling Problem Solving

    Each green line shows the linear relationship. Y-intercept (red dot) is the starting value; the slope is the rate of change.

    1. A plumber charges a $60 call-out fee plus $40 per hour.
      60 1 2 3 4 t (hr) C ($)
      1. Write a rule for the total cost C in dollars in terms of the number of hours t: C = …
      2. Complete a table for t = 0, 1, 2, 3, 4.
      3. Find the cost for 3 hours of work.
      4. Find the number of hours worked if the total cost was $260.
    2. A taxi charges a $3.50 flag-fall plus $2.20 per kilometre.
      3.50 1 2 3 4 d (km) F ($)
      1. Write a rule for the total fare F in dollars in terms of the distance d km: F = …
      2. Complete a table for d = 0, 1, 2, 3, 4.
      3. Find the fare for an 8 km trip.
    3. A phone starts with $50 credit and uses $3 of credit per day.
      50 credit out d (days) C ($)
      1. Write a rule for the remaining credit C in dollars after d days: C = …
      2. On which day does the credit run out (C = 0)?
    4. A swimming pool starts empty and is filled at 200 litres per minute.
      0 V = 3000 t (min) V (L) proportional
      1. Write a rule for the volume V in litres after t minutes: V = …
      2. How long will it take to fill the pool to 3 000 litres?

  5. Substituting into Rules Fluency

    For each rule, find y when x = 0, 3, and 6.

    1. y = 4x
    2. y = 2x + 7
    3. y = 10 − 2x
    4. y = ½x + 1
    5. y = 3x − 3
    6. y = 20 − 3x

  6. Rate of Change Fluency

    For each rule, state the rate of change (how much y increases for each increase of 1 in x) and whether the relationship is increasing or decreasing.

    1. y = 5x + 2
    2. y = −3x + 10
    3. y = x + 9
    4. y = −x + 4
    5. y = 7
    6. y = 0.5x
    7. y = 100 − 10x
    8. y = 4x − 1

  7. Proportional Relationships Understanding

    A relationship is proportional if y = kx for some constant k (the ratio y÷x is always the same, and the y-intercept is 0).

    1. Is y = 3x proportional? What is the constant of proportionality?
    2. Is y = 3x + 2 proportional? Explain why or why not.
    3. A table shows x: 2, 4, 6, 8 and y: 6, 12, 18, 24. Is this proportional? Write the rule.
    4. A table shows x: 1, 2, 3, 4 and y: 4, 7, 10, 13. Is this proportional? Write the rule.
    5. Compare y = 4x and y = 4x + 5. Both increase at the same rate. What is different about them? Which starts higher when x = 0?
    6. A car travels at constant speed. After 1 hour it has travelled 80 km, after 2 hours 160 km, after 3 hours 240 km. Is this proportional? Write the rule and state the constant of proportionality. What does it represent?

  8. Interpreting Tables and Rules Understanding

    1. The rule y = 2x + 10 represents a mobile phone plan where y is the total cost in dollars and x is the number of GB of data used. What does the “10” represent? What does the “2” represent?
    2. Two rules: y = 5x + 3 and y = 3x + 5. When x = 1, which rule gives the larger y? When x = 5, which gives the larger y? At what value of x are they equal?
    3. A table of values is given: x: 0, 1, 2, 3, 4 and y: 8, 11, 14, 17, 20. Find the rule. What is the rate of change? What is the starting value?
    4. Which of these rules represents a decreasing relationship: y = 3x − 1, y = −2x + 6, y = x + 4, y = 8 − 3x? List all that decrease.

  9. Creating Tables from Contexts Understanding

    1. A candle is 20 cm tall and burns at 2 cm per hour. Write a rule for the height H cm after t hours. Complete a table for t = 0, 1, 2, 3, 4, 5. When does the candle burn out?
    2. A baker makes 12 muffins per batch. Write a rule for the total muffins M after b batches: M = ___. Complete a table for b = 0, 1, 2, 3, 4, 5. How many batches to make 60 muffins?
    3. A cyclist starts 5 km from home and cycles towards home at 3 km per hour. Write a rule for their distance D km from home after t hours. Complete a table for t = 0, 1, 2. When are they home?
    4. Two workers paint a house. Worker A paints 4 walls per hour. Worker B paints 2 walls per hour but already painted 6 walls before starting. Write a rule for each worker’s total walls W after t hours. After how many hours have they painted the same number of walls?

  10. Modelling Extended Contexts Problem Solving

    1. A marathon runner is at the 10 km mark and runs at 12 km/h. Write a rule for the runner’s distance D km from the start after t hours of running. How far has the runner gone after 1.5 hours of running from the 10 km mark? At what time t does the runner reach 46 km?
    2. A bank account has $200 and the owner deposits $50 per week. Write a rule for the balance B after w weeks. How many weeks until the balance reaches $700? Write a second rule if instead the owner withdraws $30 per week. When does the account hit $0?
    3. A school is planning a camp. The venue costs $300 (fixed) plus $15 per student. Write a rule for total cost C in terms of n students. The school’s budget is $750. What is the maximum number of students who can attend? If 30 students attend, what is the cost per student?
    4. Two mobile phone plans: Plan X charges $20 per month plus $0.10 per text. Plan Y charges $35 per month with unlimited texts. Write a rule for each plan’s monthly cost. For how many texts per month do the plans cost the same? For someone who sends 200 texts per month, which plan is cheaper?
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