Discrete Linear Relations with Integers

🧱 See It

Level 2's savings account went up by a fixed amount every week. A rule can just as easily go down by a fixed amount instead. Suppose a candle starts at 20 cm tall and burns down 3 cm every hour:

Hour	1	2	3	4
Height (cm)	17	14	11	8

Each hour loses exactly 3 cm — the same jump every time, just a negative one. It's still linear.

✏️ Draw It

Plot "hour" along the bottom and "height" up the side. Watch the dots trend downward, left to right, instead of upward:

The dots still line up in a perfectly straight path — just sloping downward this time. The dashed line is only a guide for your eye; the dots are the real data.

🔢 Write It

Using x for the hour and y for the height, the rule has a negative coefficient on x, since the value shrinks each step:

y = −3x + 20

Check it against hour 2: y = −3(2) + 20 = −6 + 20 = 14 — matches the table.

A decreasing pattern isn't a different kind of rule — it's the same y = ___x + ___ shape, just with a negative number multiplying x.

💡 Worked Examples

Example 1 — Use the rule: y = −3x + 20, find y when x = 6

Substitute: y = −3(6) + 20
Multiply: y = −18 + 20
Answer: y = 2

Example 2 — Find the rule for this table:

x	1	2	3
y	30	25	20

Find the jump: 25 − 30 = −5, and 20 − 25 = −5 — a steady jump of −5, so the rule is y = −5x + b.
Find b using x=1: 30 = −5(1) + b → 30 = −5 + b → b = 35.
Rule: y = −5x + 35
Check x=2: −5(2) + 35 = 25 — matches. Check x=3: −5(3) + 35 = 20 — matches.

🧠 Quick Reference

Find the jump first, exactly like before — but now it can be NEGATIVE if the pattern is decreasing. A negative jump means a negative coefficient on x in the rule. Same two-pass method: find the jump, then find what adjusts step 1 to match.