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What Slope Means in SAT Word Problems

A whole family of SAT questions gives you a linear model like C of t equals 35t plus 90 and asks what the 35 or the 90 represents. No calculation is involved. The work is attaching units to each number and matching them against answer choices that are all written to sound plausible.

The rule

The coefficient on the variable is a change per one unit of the input, and the constant is the output value when the input is zero.

In any linear model written as output equals rate times input plus starting amount, there are exactly two numbers with meaning. The coefficient on the variable is a rate of change: how much the output moves for each increase of one in the input. The constant is the value of the output when the input is zero, which in context is the starting amount, the fixed fee, or the initial population.

Make units explicit before you look at the choices. If C of t gives the total cost in dollars of renting equipment for t days, then 35 is measured in dollars per day and 90 is measured in dollars. Saying it in one sentence, the rental costs 35 dollars for each additional day plus a one-time fee of 90 dollars. That sentence will match exactly one answer choice.

The wrong answers are built by shuffling those units. One choice will call 35 the total cost, which is wrong because a total has no per-day attached. Another will call 90 the cost per day, swapping the two roles. A third will describe 35 as the cost of the first day, which is off because the first day costs 35 plus the 90 fee. Reading the units filters all three.

Watch the direction of the rate. If a model is P of x equals 1200 minus 45x for the number of pages left after x hours of reading, then 45 is a decrease, so the correct interpretation is that the number of remaining pages drops by 45 each hour. A choice saying the pages increase by 45 an hour is testing whether you noticed the minus sign in front of the coefficient.

The same reasoning covers models with two letters, such as a line written 4x plus 5y equals 200 where x is hours of one job and y is hours of another. Here neither coefficient is a slope until you solve for one variable. Rewriting as y equals negative four fifths x plus 40 makes the trade-off visible: each extra hour of the first job costs you four fifths of an hour of the second.

The trap

Describing the coefficient as a total amount rather than a per-unit rate, which is exactly how the most tempting wrong answer is written.

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Try one
Real question

The cost to rent a bike is $5 plus $3 per hour. After how many hours does the cost equal $26?

A city's recycling center processed 8,400 tons in its 3rd year of operation and 14,700 tons in its 10th year. Tonnage has grown at a constant rate. How many tons does the model predict for the 16th year?

  • A) 14,400
  • B) 19,200
  • C) 20,100 ✓
  • D) 21,000

Why: Tonnage rose 6,300 over 7 years, which is 900 tons per year. From year 10 to year 16 is 6 years, so add 5,400: 14,700 + 5,400 = 20,100. Choice D counts 7 years instead of 6. Answering 14,400 means you multiplied the annual growth by 16 and ignored the model's starting value. Answering 19,200 means you counted 12 years of growth between the 3rd and 16th years instead of 13. Answering 21,000 means you counted 7 years of growth after the 10th year instead of 6.

Practice more of these free

Questions students ask

How do I tell the slope from the starting value in a word problem?

Set the input to zero. Whatever the output becomes is the starting value, and it is the number with no variable attached to it. The other number is multiplied by the variable, so it describes change per unit and always carries a per-something in its units.

What if the model is written with the constant first?

It changes nothing. In f of x equals 90 plus 35x the 35 is still the rate and the 90 is still the starting amount. Identify which number is multiplied by the variable rather than reading left to right.

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