Basic Algebra and Units

All about how to manipulate equations with variables or symbols. Units such as measures of distance (e.g., meter) or mass (e.g., gram) and so can also be manipulated algebraically. Algebra reduces to multiplying by 1 and/or adding 0 in clever ways! Some examples:

Newton's second law states that force equals mass times acceleration, so

$$F = m\, a.$$

Solving for the acceleration,

$$a = \frac{F}{m}.$$

One that is just math: Given the equation $y^2 + xy = c$, solve for $x$:

$$x = \frac{c-y^2}{y}.$$

(Note that arriving at this solution requires multiple steps.)

One with units: A bike is traveling at 100 cm/sec. How fast is it going in kph?

$$y = 100 \, \frac{\rm cm}{\rm sec}\,\left(\,\frac{\rm hr}{\rm km}\,\frac{\rm km}{\rm hr}\,\right)$$

$$= 100 \, \frac{\rm cm}{\rm sec}\,\left(\,\frac{\rm 3600\, sec}{100,000\, cm}\,\frac{\rm km}{\rm hr}\,\right)$$

$$= 3.6 \, {\rm km/hr}.$$

Just coasting!