Practice Maths

Finding the Rule for a Linear Relationship

Key Ideas

Key Terms

y = mx + c
the standard form for a linear equation; every straight-line rule can be written this way.
gradient (m)
the constant amount y changes each time x increases by 1; found from a table as the constant difference in y ÷ constant difference in x, or from two points as (y2 − y1) ÷ (x2 − x1).
y-intercept (c)
the value of y when x = 0; read directly from a table where x = 0, or calculated by substituting a known point after finding m.
constant difference
the fixed change in y for each equal step in x; if the difference is constant the relationship is linear.

Finding the rule from a table:

x: 0, 1, 2, 3  |  y: 5, 8, 11, 14
Constant difference in y = 3, so m = 3.
When x = 0, y = 5, so c = 5.
Rule: y = 3x + 5

Hot Tip The gradient m = constant difference in y ÷ constant difference in x from the table. Always check your rule by substituting one of the given points back in.

The Rule y = mx + b

Every linear relationship can be written in the form y = mx + b, where:

• m is the gradient (how steeply y changes when x increases by 1)
• b is the y-intercept (the value of y when x = 0)

Your job is to find these two values from a table of values or a graph. Once you have them, you can write the full rule.

Finding m from a Table of Values

Look at how y changes each time x increases by 1. This constant change is the gradient m.

Example table:
x: 0, 1, 2, 3, 4
y: 3, 7, 11, 15, 19

Each time x goes up by 1, y goes up by 4. So m = 4.

If x doesn't increase by 1 each time, find the change in y divided by the change in x: m = (change in y) ÷ (change in x).

Finding b from a Table or Graph

b is the value of y when x = 0. Look at the table — find the row where x = 0 and read off y.

From the example above: when x = 0, y = 3. So b = 3.

The rule is: y = 4x + 3.

Check: when x = 2, y = 4(2) + 3 = 11. ✓

If x = 0 is not in the table, use the gradient to work backwards. From x = 1, y = 7: going back one step, y = 7 − 4 = 3 when x = 0. So b = 3.

Finding m from a Graph (Rise over Run)

On a graph, pick two clear points on the line. Count how far the line goes up (rise) and how far it goes across (run).

m = rise ÷ run

Example: The line goes through (1, 5) and (3, 11). Rise = 11 − 5 = 6. Run = 3 − 1 = 2. m = 6 ÷ 2 = 3. So for every 1 unit across, the line goes up 3 units.

Putting It Together: Writing the Rule

Step 1: Find m (from the table or graph).
Step 2: Find b (y-value when x = 0).
Step 3: Write y = mx + b.
Step 4: Check by substituting another (x, y) pair from the table — both sides should be equal.

Example: A taxi charges $3.50 per km plus a $5 base fee. The table shows:
km: 0, 1, 2, 3  |  Cost ($): 5, 8.50, 12, 15.50
m = 3.50 (cost goes up by $3.50 each km), b = 5 (base fee). Rule: Cost = 3.50 × km + 5.

Key tip: Always verify your rule by substituting at least two (x, y) pairs from the table. If your rule doesn't work for all the values in the table, you've made an error in finding m or b. Finding the rule is just detective work — look for the pattern in the y-values first.

Mastery Practice

  1. Find the rule (y = mx + c) for each table of values. Fluency

    1. x: 0, 1, 2, 3  |  y: 2, 5, 8, 11
    2. x: 0, 1, 2, 3  |  y: 4, 2, 0, −2
    3. x: 0, 1, 2, 3  |  y: −1, 3, 7, 11
    4. x: 0, 2, 4, 6  |  y: 1, 5, 9, 13
    5. x: −1, 0, 1, 2  |  y: −4, −1, 2, 5
    6. x: 0, 1, 2, 3  |  y: 10, 7, 4, 1
    7. x: 0, 1, 2, 3  |  y: 0, 6, 12, 18
    8. x: 1, 2, 3, 4  |  y: 4, 7, 10, 13

  2. Write the rule y = mx + c given the gradient and y-intercept. Fluency

    1. m = 4, c = −3
    2. m = −2, c = 5
    3. m = 1, c = 0
    4. m = −5, c = −1
    5. m = 3, c = 7
    6. m = ½, c = 4
    7. m = −1, c = 8
    8. m = 0, c = −6

  3. Find the rule given two points on the line. Fluency

    1. (0, 3) and (2, 7)
    2. (0, −1) and (3, 5)
    3. (1, 4) and (3, 10)
    4. (2, 5) and (5, 11)
    5. (0, 8) and (4, 0)
    6. (−1, 6) and (3, −2)
    7. (2, 3) and (6, 7)
    8. (1, −5) and (4, 4)

  4. Which rule matches each verbal description? Choose from the options given or write your own. Understanding

    1. “Starts at 6 and decreases by 2 for every 1 unit increase in x.”
    2. “Passes through the origin and has a gradient of 5.”
    3. “Has a y-intercept of −3 and a gradient of 4.”
    4. “A horizontal line that crosses the y-axis at −2.”
    5. “The gradient is −1 and the line passes through (3, 0).”
    6. “Crosses the x-axis at 4 and has gradient 2.”

  5. For each pair, determine if both tables represent the same linear rule. Explain your reasoning. Understanding

    1. Table A: x: 0,1,2 / y: 3,5,7  |  Table B: x: 3,4,5 / y: 9,11,13
    2. Table A: x: 1,2,3 / y: 4,7,10  |  Table B: x: 0,2,4 / y: 1,7,13
    3. Table A: x: 0,1,2 / y: −2,2,6  |  Table B: x: 3,5,7 / y: 10,18,26
    4. Table A: x: 0,3,6 / y: 5,2,−1  |  Table B: x: 1,4,7 / y: 4,1,−2

  6. Find linear rules in real-world situations. Problem Solving

    1. A mobile phone plan charges a connection fee plus a rate per minute. After 5 minutes the cost is $2.25, and after 10 minutes it is $3.50. Find the rule for cost C in terms of minutes m.
    2. A taxi charges a flag fall plus a rate per kilometre. A 4 km trip costs $10.20 and an 8 km trip costs $15.80. Find the per-kilometre rate and the flag fall. Write the rule.
    3. Zara is saving money. She already has some savings and adds a fixed amount each week. After 3 weeks she has $85, and after 7 weeks she has $125. How much did she start with? Write the rule for total savings S after w weeks.
    4. A table of values for a linear relationship is partly given. Find the missing values and state the rule: x: 0, ?, 4, 6 / y: −3, 1, ?, 9.

  7. Investigate a pattern and write a linear rule. Problem Solving

    Tile Pattern. A pattern of tiles is built using rows. Row 1 has 5 tiles, Row 2 has 8 tiles, Row 3 has 11 tiles, and Row 4 has 14 tiles.
    1. Write the rule for the number of tiles T in row n.
    2. How many tiles are in Row 20?
    3. Which row has exactly 50 tiles?
    4. Explain what the gradient and y-intercept represent in the context of this tile pattern.

  8. Two linear rules — find where they give the same value. Problem Solving

    Competing Plans. Plan A for streaming costs $5 per month plus $2 per movie watched. Plan B costs $15 per month with no extra charge per movie.
    1. Write a rule for the total cost C of each plan after watching m movies.
    2. Complete a table of values for m = 0, 2, 4, 6, 8, 10 for both plans.
    3. For how many movies watched do both plans cost the same?
    4. Which plan is cheaper if you watch 15 movies per month? Show working.

  9. Find the rule from incomplete information and extend the pattern. Problem Solving

    Mystery Table. A table of values for a linear relationship has some values missing: x: −2, 0, 2, ?, 6 / y: 11, 5, ?, −7, −13.
    1. Find the gradient of the relationship.
    2. Find the rule in y = mx + c form.
    3. Find the two missing values in the table.
    4. What value of x gives y = −25?

  10. Create and compare linear rules from a real-world scenario. Problem Solving

    Water Tank. Tank A starts with 200 litres and loses 15 litres per hour. Tank B starts with 50 litres and gains 10 litres per hour.
    1. Write the rule for the volume V of water in each tank after h hours.
    2. After how many hours do both tanks contain the same volume? Show working.
    3. What is the volume in each tank at that point?
    4. After 8 hours, which tank has more water? How much more?
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