TITLE: What does it mean to solve a math problem analytically?
QUESTION [14 upvotes]: I'm reading a Calculus book for my own edification and at the beginning the pre-calculus introduction has the problem,
$3x+y=7$
They talk about solving the problem graphically, analytically, and numerically.  The subject is the basic graph, Rene Descartes, etc.
They have numerical which is just a table of values.  I understand that.
Graph I understand.
But for the analytic approach, they have
"To systematically find other solutions, solve the original equation for $y$
$y=7-3x$
I do not understand how they came up with that.  Why not $x$?  Why is this analytic?  What makes this "analytic"?  Why would it even occur to someone that solving for why is the way to go, the thought process.
I can solve the problem.  That's not the issue.  I want to understand why I'm doing it this way.  Thanks.
edit:
"The Graph of an Equation
Consider the equation $3x+y=7$.  The point $(2,1)$ is a solution point of the equation because the equation is satisfied (is true) when $2$ is substituted for $x$ and $1$ is substituted for $y$.  This equation has many other solutions, such as $(1,4)$ and $(0,7)$.  To systematically find other solutions solve the original equation for $y$.
$y = 7 - 3x$  Analytic approach"
I'm sure this is obvious and maybe I don't understand what the word analytic means in this context.
Calculus of a Single Variable, Sixth Edition, 1998, Larson, Hostetler, Edwards
(I got it a thrift store.)

REPLY [2 votes]: 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).