Tables of Values

A linear rule like y = 2x + 1 works like a machine: you feed in a value of x, the rule processes it, and out comes y. Put in x = 3 and you get 2 times 3 plus 1, which is 7. The x is called the independent variable because you are free to choose it; y is the dependent variable because its value depends on the x you picked. The one thing that trips people is the order — multiply before adding. So y = 2 × 3 + 1 = 7, not 2 × (3 + 1) = 8. There are no brackets in the rule, so the ordinary order of operations applies.

To build a table, substitute each x value into the rule in turn. For y = 2x + 1 with x from −2 to 2, you get y values of −3, −1, 1, 3 and 5. Negative x values need the most care: 2 times negative 2 is negative 4, and negative 4 plus 1 is negative 3 — not negative 5. Writing the substitution out in full before evaluating prevents that. There is also a built-in check worth using every time: as x increases by 1, y should change by the same amount each step. Here it goes up by 2 every time, which matches the coefficient of x. If one of your steps breaks the pattern, that entry is wrong.

When the coefficient of x is negative, the pattern reverses: y decreases as x increases. For y = −2x + 5, the values run 9, 7, 5, 3, 1 — a step of negative 2 each time. The entry to be careful with is x = −2, because you get negative 2 times negative 2, which is positive 4, then plus 5 gives 9. Two negatives multiplied give a positive, so the answer is 9 rather than 1. That combination — a negative coefficient with a negative x value — is where most marks are lost in this topic, so write it out fully rather than doing it in your head.

Each column of the table gives one point: the x value and the y value together make a coordinate pair. Plot all of them and, because the rule is linear, they must fall in a perfectly straight line. Join them with a ruler and extend the line past the plotted points, since the relationship continues beyond the values you happened to choose. The straight-line requirement is also your error check: if one point sits off the line while the others line up, that table entry has an arithmetic mistake in it. Go back and recompute that one rather than drawing a bent line through them all.

Given a table, you can work backwards to the rule in two steps. First, look at how much y changes each time x goes up by 1 — that constant step is the coefficient of x. Here y rises by 3 each time, so the rule begins y = 3x. Second, find the value of y when x = 0, because that is the constant on the end. Here y = 4 at x = 0, so the rule is y = 3x + 4. Always test it on a value you did not use to build it: x = 3 gives 3 times 3 plus 4, which is 13, matching the table ✓. If the table does not include x = 0, step backwards from the first entry by subtracting the step once.