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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
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To use nice fonts, typeset them with LaTeX -- simple $ before and after -- e.g. $3x+7$ yields $3x+7$ –  gt6989b Nov 14 '13 at 19:42
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3 Answers 3

up vote 2 down vote accepted

"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.

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Ok. I guess I need to get Algebra One :). I took Calc 2 several years back, but always took for granted the rules I had memorized. what would I look at to know why I can do y=7-3x? –  johnny Nov 14 '13 at 18:39
    
@johnny turning $3x+y = 7$ into $y = 7-3x$ is just application of the field axioms of the reals and the properties of equivalence relations. Basically, what you are doing is subtracting $3x$ from both sides: $3x+y-3x = 7-3x$. Then, noting that addition is commutative, we have $3x+y-3x = y+3x-3x = y$. –  Arkamis Nov 14 '13 at 19:18
    
I see what you are saying, but my question is even more fundamental. Where do I read about "field axioms of the reals"? I can see it work. I must know more about why it works. I don't know if that makes sense. I want to know about the axioms. Some guy just discovered this one day? Something else? Thanks for your time and patience. –  johnny Nov 14 '13 at 21:21
    
You can read about it in any analysis book, or by reading about the [en.wikipedia.org/wiki/… numbers). In short, the axioms that make it work are rules that have been "made up." However, they are rules that make sense with ordinary life. If you wanted to construct the real numbers, you might start with the natural numbers (1, 2, etc) and then start with some basic operations (addition, multiplication), and then derive properties from there. –  Arkamis Nov 14 '13 at 23:20
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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 $x$.

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Why did they pick y? –  johnny Nov 14 '13 at 18:36
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Traditionally, $y$ is used to represent the variable we want to know about or predict and $x$ the variable whose value we know. In analytic solutions, we want to express the variable whose value we want to know in terms of the variables whose values we already have. –  response Nov 14 '13 at 19:14
    
But we don't already know x. I guess it is just tradition. –  johnny Nov 14 '13 at 21:22
    
Well, in your case it is a 'made-up' problem. But, in some situations, $x$ represents a value we want to change (say, velocity of a missile) and we want to predict where it will land which is then denoted by $y$. So, then the goal is to obtain $y$ as a function of $x$ and then understand how 'where it will land' changes as we change velocity with which we launch it. –  response Nov 14 '13 at 22:38
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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.

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