Rearranging Formulas and Letter Coefficients

In this lesson you will: Solve a formula for any variable it contains; Use the same reasoning on letters that you use on numbers; Solve an equation whose coefficients are letters.

Treat every other letter as a number

To solve P=2l+2w for w, pretend P and l are numbers you happen not to know yet. Then it is a two-step equation: subtract 2l, divide by 2.

P-2l=2w, so w=(P-2l)/(2). Nothing new happened. The only difficulty is psychological.

Why you would want to

A formula is usually written to compute one particular quantity. Rearranging it lets you compute a different one without redoing the algebra every time.

If you are checking twenty rectangles that all have perimeter 60 and you want each width, solving for w once is far less work than solving twenty separate equations.

Letters as coefficients

Ohio's standards include equations \"with coefficients represented by letters.\" To solve ax+b=c for x, subtract b and divide by a: x=(c-b)/(a).

The one genuinely new idea is the condition at the end. That division is only legal when a != 0, because the division property of equality requires a nonzero divisor. If a=0 the equation is b=c, which has no x in it at all.

Common mistake — Dividing only part of the numerator

Solving P=2l+2w for w, a student writes w=(P)/(2)-2l.

The slip is a natural one: the 2 in front of w has to go, and dividing P by 2 feels like the move. The 2l then gets carried across unchanged.

But once you subtract 2l, the entire left side is P-2l, and all of it must be divided by 2. Dividing only the first term changes the value.

Test it with numbers. If P=20 and l=4, the true width is (20-8)/(2)=6. The wrong formula gives (20)/(2)-8=2.

How to catch it: Substitute easy numbers into both your rearranged formula and the original. If they disagree, the rearrangement is wrong. This takes fifteen seconds and catches almost every error of this kind.