I used boolean algebra to simplify an expression with $3$ variables. After simplifying, it reduces to $2$ variables. The first truth table has $8$ rows and the second one has $4$. How to prove that they're logically equivalent?
Considering $+$ as or, . as and $\sim$ as not.
The original expression is:
$$F = \sim x (xy + \sim(x + y) + z) + z\sim x + \sim(zx)$$
Reduced to:
$$(\sim x) + (\sim z)$$
Instead of making a separate truth-table for each statement, you create a single table for both statements. This is what is called a joint truth-table, that has as its reference columns all variables that occur in either of the statements you are interested in, and in which you work out the truth-values of all those statements.
\begin{array}{ccc|c|c} x&y&z&x'(xy+(x+y)'+z)+zx'+(zx)'&x'+z'\\ \hline 0&0&0&bla&bla\\ 0&0&1&bla&bla\\ ...&&&...&... \end{array}