Lesson 6 · from §73 of the 1910 text
Formulas: Primes and Subscripts
Formulas
A formula is a rule written as an equation. A = \frac{ab}{2} says the area of a triangle is half its base times its altitude. Any letter in a formula can be solved for in terms of the others; the work is exactly that of a literal equation.
Primes and subscripts
Often we need the same letter for several related quantities. Two ways are used:
- Primes: a', a'', a''', read "a prime," "a second," "a third."
- Subscripts: a_1, a_2, a_0, read "a sub one," "a sub two," "a sub zero."
W_1 and W_2 are different numbers, as different as a and b. They are not W times 1 and W times 2. Here, type W_1 as W1 or W_1.
Example 1. Solve V = lwh for w. Divide both sides by lh: w = \dfrac{V}{lh}.
Example 2. The area of a trapezoid is A = \frac12 h(b_1 + b_2). Solve for h. Multiply by 2: 2A = h(b_1 + b_2). Divide by b_1 + b_2: h = \frac{2A}{b_1 + b_2}.
Example 3. Solve C = \dfrac{E}{R + r} for r. Multiply by R + r: CR + Cr = E, so Cr = E - CR and r = \dfrac{E - CR}{C}.
When a formula's letters all have values except one, substitute first and then solve the simpler equation (Exercises 21 and 22).
Try one:
Quick check
Solve for V_2: P_1V_1 = P_2V_2. (Type P_1 as P1 or P_1.)
From the history of algebra
Many of these formulas come from physics. C = \frac{E}{R + r} is Ohm's law for a battery of electromotive force E and internal resistance r driving a current C through a resistance R; Georg Ohm published it in 1827. C = \frac59(F - 32) links the scale of Daniel Fahrenheit (1724) to the centigrade scale of Anders Celsius (1742). \frac{V_1}{V_2} = \frac{P_2}{P_1} is Boyle's law for a gas (1662). Writing subscripts to tell related quantities apart became standard in the 1800s; before that, writers ran out of letters fast.
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Solve each formula for the letter named. Type subscripts as plain digits or with an underscore: W_1 is W1 or W_1; type \pi as pi, as in K/(2pi H).
Solve for R: K = 2\pi RH.
Solve for a: A = \dfrac{ab}{2}.
Solve for R: C = 2\pi R.
Solve for r and t: d = rt.
Solve for a and A: \dfrac{a}{A} = \dfrac{D}{360}.
Solve for C: \dfrac{D}{360} = \dfrac{l}{C}.
Solve for r: C = \dfrac{E}{R + r}.
Solve for r and n: C = \dfrac{E}{R + nr}.
Solve for r and n: C = \dfrac{ne}{R + nr}.
Solve for F: C = \frac59(F - 32).
Solve for W_2: \dfrac{W_1}{W_2} = \dfrac{L_1}{L_2}.
Solve for r and t: A = P(1 + rt).
Solve for P_2: \dfrac{V_1}{V_2} = \dfrac{P_2}{P_1}.
Solve for n and l: s = \dfrac{n(a + l)}{2}.
Solve for a, l, and r: s = \dfrac{rl - a}{r - 1}.
Solve for \theta: \dfrac{D}{180} = \dfrac{\theta}{\pi}.
Solve for t_1: V_1 = V_0(1 + .00365\,t_1).
Solve for b_2: A = \dfrac{(b_1 + b_2)a}{2}.
Solve for x: \dfrac{a}{b} = \dfrac{x}{c - x}.
Solve for F, D_1, and D_0: \dfrac1F = \dfrac1{D_1} + \dfrac1{D_0}.
Substitute the numbers first, then solve. Round to two decimal places.
C(t_m - t_1)H_c + W_c(t_m - t_1) = W_h(t_2 - t_m). Find the value of t_m in this equation when C = 80, t_1 = 20, t_2 = 99, H_c = .09, W_c = 1000, and W_h = 800. Round to two decimal places.
C(t_m - t_w)H_c + W(t_m - t_w) = I(t - t_m)H_I. Solve this equation for t when C = 80, t_m = 54, t_w = 18, H_c = .09, W = 100, I = 440, and H_I = .11. Round to two decimal places.