Name the input, output, and rate units
A linear model describes equal changes in output for equal changes in input. Write y = mx + b, where m gives the output change for one input unit, and b gives the output when x = 0. In a fare model, x can be distance in miles, y can be price in dollars, and m then has units of dollars per mile. The starting charge b has units of dollars. Naming the units makes it easier to see whether a number describes the initial condition or a rate.
Worked example with two observed fares
Suppose a shuttle ride of four miles costs $26 and a ride of ten miles costs $47. If fares follow a linear rule, six more miles cost $21 more, so the rate is $3.50 per mile. Four miles account for $14 of the $26 fare. The starting charge is $12, and the model is y = 3.50x + 12. At ten miles the variable part is $35. Adding the starting charge gives $47, which checks the model against the second observation.
Keep the direction of both coordinate differences consistent
On a coordinate plane, moving from (1, 8) to (5, 2) increases the input by four and decreases the output by six. The slope is −6/4 or −1.5. Reversing both subtractions gives 6/(−4), which represents the same slope. Reversing only one changes the sign. Before calculating, picture the line from left to right. A decreasing output should have a negative slope, even if the plotted output values happen to be positive.
Choose the equation that matches the question
To find the shuttle distance for a $33 fare, use 3.50x + 12 = 33. First subtract $12 from both sides to isolate the distance charge, then divide by $3.50 per mile. The result is six miles. Dividing the whole fare by the per-mile rate would count the starting charge as extra distance. A different task might compare two fare rules. In that case, set the outputs equal at the same input instead of equating either fare to a preset amount.
Transfer check with two subscriptions
Plan A costs $20 up front plus $6 each week. Plan B costs $8 up front plus $9 each week. Decide when the totals match. Try comparing the fixed gap and the weekly difference before reading on. The starting gap is $12, and the more expensive weekly rate closes it by $3 per week. They match after four weeks at $44. Check both totals at your input, and explain why comparing the weekly rates alone would not locate the match.
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 OpenExam explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
College Board ACCUPLACER Math test contentOpenStax Elementary Algebra 2e