# What does it mean to solve a math problem analytically?

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.)

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It may help to explain what the actual problem was that the text was solving, that was related to the equation $3x+y=7$. –  gt6989b Nov 14 '13 at 17:11
That's all it has. I will put the full paragraph. It's the first page. –  johnny Nov 14 '13 at 17:14
@gt6989b How do I put the "nice" equation font in there? –  johnny Nov 14 '13 at 17:18