What does it mean to solve a math problem analytically?

Solution 1:

"Analytically" comes from the same root as "analysis," which in mathematics loosely means the study of the properties of objects.

In this case, analytically solving an equation means finding a solution simply by exploiting known rules: addition and subtraction, associativity, commutativity, etc.

This differs from a "numerical" solution, where a sequence of numbers are used and compared to see if equality is met. Numerical solutions are very similar to graphical solutions, but do not require a pictoral representation.

Solution 2:

Analytic would mean manipulating the equations involved to express one variable in terms of other variables without using numerical computations.

For example, in your case, the value of $y$ was expressed in terms of $x$ without using any explicit value for $x$.

Solution 3:

Analytic in the generic math sense essentially means to solve using Algebra (properties, rules, or theorems, or use trig/functions properties), or in other words without the use of a calculator, graph, or by plugging in values (which is similar to a table of values).