A reliable method
Compute a rate from the change in output divided by the change in input. Recover the output at input zero only after accounting for where the given observation occurs.
The common trap
The first row of a table is not necessarily the intercept. A table may begin after several hours or several miles have already passed.
Check and transfer
Check the model against more than one given point. Then decide whether the requested prediction stays within a sensible domain for the situation.
Worked comparison: why the alternative fails
A tank has 13 liters at minute 2 and 25 at minute 6. Its slope is (25−13)/(6−2) = 3 liters/minute. Dividing 25 by 6 wrongly assumes the tank began empty; working back gives 7 liters at minute zero.
Transfer and explain
Use V(t)=7+3t to predict V(8)=31. Then ask the reverse question V(t)=22: t=5. Distinguish an input from an output before using the rule.
A service follows a linear cost rule. It charges $24 for 2 visits and $36 for 5 visits. What is the fixed charge at zero visits?
- The charge increases by $12 over three visits, or $4 per visit.
- Move back two visits from the first observation to reach zero visits.
- The requested value is 16. Check its units and meaning against the original question.
Compare the other answer choices
Report the rate per visit rather than the fixed charge.
Report the variable cost of the first two visits.
Treat the first observed total as the intercept.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
GED Mathematical Reasoning overviewSiyavula Grade 10 mathematics foundation reference