Linear relationships in tables

A table often hides a pattern. The way to find it is to look at the difference between the rows.

Hours (x)	Pay (y)	Difference
0	0
1	130	+130
2	260	+130
3	390	+130
4	520	+130

The same difference every step → a linear relationship. Here proportional too, since it starts at 0.

Another table:

Months (x)	Saved (y)	Difference
0	500
1	800	+300
2	1 100	+300
3	1 400	+300

Also linear (+300 every time), but it starts at 500. So y = 300x + 500.

Two numbers describe the whole relationship:

• The starting value — what y is when x is 0.
• The change per step — how much y increases when x increases by 1.

The form is always the same:

$y = kx + m$

Letter	Means	Where in the table
k	the change per step	the difference between the rows
m	the starting value	the y value where x = 0

If the table does not start at x = 0:

x	y
2	17
4	25
6	33

The difference is +8 per two steps in x, so k = 8/2 = 4. And m: go back from x = 2 to x = 0, two steps of 4 → 17 − 2·4 = 9. So y = 4x + 9.

A check: 4·6 + 9 = 33. ✓

Not linear — what that looks like:

x	y	Difference
1	2
2	4	+2
3	8	+4
4	16	+8

The difference is not constant — it doubles. That is exponential (y = 2ˣ), not linear.

The rule of thumb: a constant difference → linear. A constant quotient → exponential.

In a coordinate system linear relationships become points on a straight line. That is the fastest way to see which kind you have: plot four points and look.

Find k and m in four tables. Write the answer down before you read the key.

A

x	0	1	2	3
y	7	12	17	22

B

x	1	2	3	4
y	100	90	80	70

C

x	0	2	4	6
y	3	11	19	27

D

x	1	2	3	4
y	3	9	27	81

The key:

• A: k = 5, m = 7 → y = 5x + 7
• B: k = −10, m = 110 → y = −10x + 110 (a negative slope, y decreases)
• C: the difference is +8 per 2 steps → k = 4, m = 3 → y = 4x + 3
• D: not linear — the quotient is 3 every time, so y = 3ˣ

Plot all four in a coordinate system. A, B and C become straight lines. D bends upwards ever more steeply — and that is clearer in the graph than in the table.

The next step: in real measurements the points never lie exactly on a line. Then it is about finding the line that fits best — and that is precisely what the next node and all of linear regression are about.