Dimensional Consistency Math Example 4

Follow the full solution, then compare it with the other examples linked below.

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In the equation

x^2 + 3x = 10

(where

x

is in meters), are all terms dimensionally consistent?

1
$[x^2] = \text{m}^2$ , $[3x] = \text{m}$ , $[10]$ is unitless.
2
No — m², m, and unitless cannot be equal. The 3 and 10 must carry units: $3$ has units m, $10$ has units m².

Answer

Only if 3 has units of m and 10 has units of m².

In pure math, we often ignore units. But in applied contexts, coefficients carry units to maintain dimensional consistency. The '3' would need to be

3\text{ m}

and '10' would be

10\text{ m}^2

About Dimensional Consistency

The principle that every term added or equated in a valid equation must share the same physical dimensions or units.

Is the equation [formula] (velocity = distance + time) dimensionally consistent?

Check: [formula] where [formula] is energy (kg·m²/s²), [formula] is mass (kg), [formula] is speed (m

Is [formula] a valid formula for area?