Given two equations with the same variable, how can they be combined? If the equations are consistent, they may be solved as *simultaneous equations*. But what if the equations are inconsistent? There are two ways to combine them in that case, one is OR, the other is AND.

Consider the equations *x = a* and *x = b*, where *a ≠ b*. If we multiply these equations together, we get

*x*² = *ab*,

in which the solution is *x* = √*ab*, so that *x* is the geometric mean of *a* and *b*.

If we make the equations homogeneous first, then multiply them together, we get: 0 = *x − a* and 0 = *x − b*, so that

0 = (*x − a*) (*x − b*) = *x*² − (a + b) *x* + *ab*.

The solution of the combined equation is *either x = a* or *x = b*. To combine equations with AND, multiply homogeneous equations together.

Another way to combine equations is to add them together. In this case, we get

*x + x* = 2*x* = a + b, or *x* = (*a + b*)/2,

so that *x* is the arithmetic mean of *a* and *b*. Homogeneous equations added produce the same result: 0 = *x − a + x − b* = 2*x* − (*a + b*), so that *x* = (*a + b*)/2.