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.)
Best Answer
"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.