Overview
A linear model combines a fixed starting amount and a constant amount per unit. Define the unknown with its units before translating a situation. When two quantities are compared, an equation finds equality and an inequality finds a region of possible values.
Consider this example: A delivery service charges $6 plus $2.50 per mile. A trip costs $26. How many miles is the trip? Let x be miles. The charge equation is 6 + 2.50x = 26. Subtract 6 and divide by 2.50 to obtain 8 miles. Dividing 26 directly by 2.50 would treat the fixed fee as mileage.
The method also handles a different case: Three consecutive integers have a sum of 72. What is the largest? Write the integers as x, x + 1, x + 2. Their sum is 3x + 3 = 72, so x = 23 and the largest is 25. The average 24 is the middle integer.
For a further application, consider: Plan A costs $18 plus $4 per visit; Plan B costs $6 per visit. How many nonnegative integer visit counts make Plan A cost at most $90 and save at least $10 compared with Plan B? The budget condition 18 + 4n ≤ 90 gives n ≤ 18. The savings condition 6n − (18 + 4n) ≥ 10 gives 2n − 18 ≥ 10, hence n ≥ 14. The feasible integer counts are 14, 15, 16, 17, 18: 5 counts. Solving only for equal costs n = 9 does not enforce either threshold.
Applying the per-unit charge to the fixed fee changes the model. Finding where plans are equal does not by itself say which plan is cheaper on either side. Write a labeled equation and check that every term has the same unit. For a new comparison, find the boundary and then test a quantity on each side.