From a Situation to an Equation

In this lesson you will: Name the unknown and say what unit it is measured in before writing anything else; Build the expression the way the situation builds the total, then set it equal to the value you were given; Read the units of each coefficient — a per-something number has to multiply the something.

The first line is a definition, not an equation

Most wrong setups can be traced to a variable that was never pinned down. “Let x be delivery” is not a definition. Delivery what? Dollars? Packages? Minutes?

Write m= the number of miles driven in one day. Now every other quantity in the problem has something definite to attach itself to, and the sentence almost translates itself.

Every coefficient carries a unit

The pay model above is P=35+0.12m. The 35 is in dollars. The m is in miles. So the 0.12 cannot be plain — it has to be dollars per mile, because that is the only unit that turns miles into dollars.

That is worth checking on every model you build, because it tells you instantly whether a term belongs where you put it. If a driver covers 150 miles, P=35+0.12(150)=35+18=53 dollars.

When the unknown sits in the exponent

Not every situation grows by the same amount each step. A colony of 250 cells that doubles every hour has 250* 2^t cells after t hours, because each hour multiplies the count by 2 again.

Setting 250* 2^t=8000 gives 2^t=32. No amount of adding and dividing will free t from that exponent. You get it by recognizing 32=2^5, or by running a short table of powers of 2 until you hit 32.

Common mistake — Writing the comparison backwards

A school has three times as many students as teachers. Asked for an equation, a student writes 3s=t.

It is a faithful transcription of the order the words arrive in: three, students, teachers. The sentence says “three times as many students,” so the 3 gets parked next to s.

But the 3 belongs with the smaller group. There are more students than teachers, so the teacher count is what needs multiplying up: s=3t.

The same trap sits inside “8 less than 5 times a number,” which is 5n-8 and never 8-5n.

How to catch it: Substitute a number you can picture. With 10 teachers there should be 30 students. Check s=3t: 30=3(10) works. Check 3s=t: 3(30)=10 is false, so that version is the wrong way round.