Find a Linear Rule from a Table
Calculate rate over a multi-unit interval and recover the fixed part.
Supplied materials
- Paper or a note app.
- Fictional print table (posters, dollars): (2,11), (5,23), (8,35). A linear rule is assumed.
Learn the idea
A table with gaps requires dividing the change in output by the change in input. From two to five posters, the cost rises twelve dollars over three additional posters. The rate is twelve divided by three, or four dollars per poster.
The fixed part is the cost left after removing the per-poster amount. At two posters, eight of the eleven dollars come from the rate, leaving three dollars fixed. A matching rate across rows supports this supplied linear table; it does not prove every real price follows a line.
- Compute (23 − 11) ÷ (5 − 2) and label dollars per poster.
- Use 11 = 4 × 2 + b to find b = 3.
- Check y = 4x + 3 against (8,35), then examine the new table in the question.
See the method in action
The rate is 12 ÷ 3 = 4. The fixed amount is 11 − 4 × 2 = 3. Thus y = 4x + 3, and at x = 8 it gives 32 + 3 = 35.
Check your reasoning
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See the answer and explanation
y = 3x + 4
The rate is (16 − 7) ÷ (4 − 1) = 9 ÷ 3 = 3. The fixed part is 7 − 3 × 1 = 4, so y = 3x + 4. It also gives 25 at x = 7.
Take it further
Keep rate 3 but raise the fixed part to 6. The rule is y = 3x + 6 and the outputs at x = 1, 4, 7 become 9, 18, 27, each two higher.
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