Compare two options at the right scale

Connect the experiment to the idea

Compare totals for the same input

Suppose plan A costs 2x + 3 credits and plan B costs x + 6 credits. Both must use the same x: the number of uses. At zero uses A is cheaper, costing three instead of six. At four uses A costs eleven and B costs ten.

The lower starting price does not identify the cheaper plan at every input. Separate the fixed fee from the cost of each extra use, and compare totals at the input relevant to the decision.

Find the crossing

At equal cost, 2x + 3 = x + 6. Subtract x from both sides and then subtract three: x = 3. Substituting back into either rule gives nine credits. Below three uses A is cheaper; above three B is cheaper under these exact rules.

For general rules ax + b and cx + d, subtracting gives (a − c)x = d − b. When a and c differ, the crossing is x = (d − b) / (a − c). If they are equal, the lines are parallel when fees differ, and identical when fees also match.

Decide within the model’s limits

A crossing outside the allowed range may have no practical relevance. If usage must be an integer, compare the adjacent allowed integers rather than pretending a fractional order is available. If both rules omit taxes, reliability, waiting time or a capacity limit, the numerical comparison does not settle the whole decision.

The lab draws plan B with a dashed line, so colour is not the only distinction. Its exact crossing is reported even when outside the plotted range. Use the table and equation to check the graph rather than estimating solely from pixels.