How can I use a table to check a linear relationship?

Start by keeping each input-output pair together: one row is one observation. Then use the stated rule to calculate an output for each listed input and compare it with the table. OpenStax describes using paired values, looking for an operation connecting the quantities, and comparing how the columns change when investigating a relationship. [S1]

For a small made-up example, suppose the proposed rule is y = 2x + 1. The table gives:

1. x = 0, y = 1
2. x = 1, y = 3
3. x = 2, y = 5

Check each row: 2(0) + 1 = 1; 2(1) + 1 = 3; and 2(2) + 1 = 5. All three listed pairs fit. The point is to test the rule against the actual pairs—not just decide whether the numbers look like they are increasing.

Read down the input column and note each step. Then read down the output column and compare the corresponding changes. In the example, x increases by 1 each time, while y increases by 2 each time. That repeated change agrees with the rule y = 2x + 1: for every increase of 1 in x, the rule adds 2 to y.

Check whether the input steps are equal before comparing output changes. If the input goes up by 2 in one step and by 1 in another, the output changes do not have to match each other, even when a linear rule fits. Instead, compare the output change with the size of its matching input change. Looking for patterns and comparing column changes are useful ways to investigate a proposed relationship. [S1]

Treat this as a cross-check, not a substitute for testing the rule on each row. A transcribed value, skipped row, or mismatched pair can make a pattern misleading.

First recalculate that row carefully: substitute its input into the proposed rule, and compare the result with the output that is actually listed. Also check that you have read the columns correctly and kept the original pairs together. If the mismatch remains, report it plainly: the proposed rule matches these listed rows but not that one.

Do not quietly adjust the rule just to make the table look consistent. If the values come from a situation, revisit what each quantity represents and whether the stated rule includes all relevant parts. OpenStax describes identifying quantities and known values, organizing them in words or a table, and translating the description into an equation; its example uses a fixed fee plus a per-minute amount. [S2] That is a way to build or check a rule for that particular situation, not evidence that any arbitrary table must follow the same kind of rule.

A mismatch tells you the stated rule does not fit every listed pair as written. By itself, it does not tell you which value or assumption is wrong.

You can say that the rule fits all the pairs shown in your table. Keep that conclusion limited to the displayed values. A finite table cannot establish that the same rule holds for every possible input; different rules can agree on a small set of pairs and differ elsewhere. [S1]

For a useful write-up, include the proposed equation, the rows you checked, and your conclusion. For the example, you could write: “I tested y = 2x + 1 at x = 0, 1, and 2. It gives the listed outputs 1, 3, and 5, so the rule fits all three table entries. This table alone does not prove it applies to inputs not shown.”

If you are deriving a rule rather than checking a stated one, identify what the quantities mean and use the known values to describe their relationship before writing an equation. A table can help organize that reasoning, but the evidence here does not establish a rule beyond the values or situation you have examined. [S2]