What does it mean to solve a math problem analytically?

Question

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.)

Answer 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.

Answer 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 $x$.

Answer 3

The equation that you state is not a problem. It is an equation that relates the variables x and y. A "problem" (task) might be to solve for y, or solve for x, or put the equation in some other special form, or find x when y is 13, or some such. Apparently the author of the book had something in mind that he didn't state. Unfortunately that doesn't seem to be uncommon. You questioned it which was the exactly right thing to do.

Answer 4

I mean whether you use a numerical approach(by comparing values to get close and close to your desired answer), a graphical approach (looking at the graph to approximately reach your answer, or an analytical approach (using algebra). You will always be given some dependent value (commonly acknowledge as x, but not always), to find the independent value in another set (commonly acknowledge as y, but same, it is not always). So in fact, both x and y are arbitrary variables, you can choose c and e instead of x and y and the equation is going to work. What the more precise definition is that: In the analytical approach, you manipulate the equation algebraically to find the value of the independent variable that you desired to look for.

In lower-level algebra and lower-intermediate calculus analytical approach generally gives you the most accurate answer. Especially when you don't have a powerful computer that is able to get a high level of accurate approximation.

However, there are always equations hard to solve. For example, if you doing integration, some equation is just impossible to integrate, or not without hours of work. So you often approximate by using the numeric integration technique instead of doing it the analytical way. Two examples you could search for are Simpson's rule and Trapezoidal rule, which give a pretty nice approximation when you have sufficient 'cut' made and is not so hard to compare for a large amount of 'cuts' on a modern computer. You can always see this is what the general numerical approach does. (^-^)