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### Geometric Objects and Properties using Algebra

```
Date: 06/29/98 at 00:27:27
From: Suzie Liu
Subject: Algebra 1

My teacher asked us why y = mx+b, and what does it mean.
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```
Date: 06/29/98 at 14:43:14
From: Doctor Rob
Subject: Re: Algebra 1

Look at a plane. Pick a point O called the origin. Through O draw a
line called the x-axis (usually horizontal). At O construct a
perpendicular line called the y-axis. Mark a point on the x-axis,
labeled "1" (usually to the right of O). The distance from O to 1 will
be one unit. The direction moving from O to 1 will be the positive
direction along the x-axis, and the opposite direction will be the
negative one. The points on the x-axis now correspond to real numbers
and vice versa.

In the same way, choose a direction (usually up) and a unit distance
(often the same as on the x-axis) on the y-axis. Points on the y-axis
also correspond to real numbers and vice versa.

This setup is called the Cartesian plane.

y-axis ^
|
|
1-
|
|
---------------o------|-------> x-axis
O|      1
|
|
|
|

Now given any point P in the plane, drop a perpendicular to the
x-axis, meeting it at Q, and a perpendicular to the y-axis, meeting
it at R. Then the real number x corresponding to Q is called the
x-coordinate (or abscissa) of P, and the real number y corresponding
to R is called the y-coordinate (or ordinate) of P. Thus the point P
corresponds to a pair of numbers (x,y). Likewise, a pair of numbers
(x,y) corresponds to a point P, since given x and y, you can find the
point Q corresponding to the number x on the x-axis, and the point R
corresponding to the number y on the y-axis.

Now erect a perpendicular to the x-axis at Q and a perpendicular to
the y-axis at R.  They will intersect in a unique point P.

This is called a Cartesian coordinate system.  Points correspond to
pairs of real numbers called the coordinates of the point.

y-axis ^
|
|
1-
|
|       x      Q
--------------------o------|-------o----> x-axis
O |      1       |
|y             |y
|              |
|              |
R o--------------o P
|       x

Now a line will consist of a set of points, that is, a set of pairs
(x,y). Which set is determined by an equation. All pairs of real
numbers (x,y) that satisfy the equation y = m*x + b, where m and b are
fixed, given real numbers, are the coordinates of points lying on a
line not parallel to the y-axis. The number b is the real number
corresponding to the point on the y-axis where the line crosses it,
called the y-intecept. That point has coordinates (0,b).

These values of x and y do satisfy the above equation, as you can
check for yourself. The number m is called the slope of the line, and
represents the amount the y-coordinate changes when we increase the
x-coordinate by exactly 1. For example, the point (1,b+m) lies on the
line, because these values satisfy the equation (again, check this for
yourself), and the y-coordinate has increased by m while the
x-coordinate has increased by 1 as we move from (0,b) to (1,b+m) along
the line. If m is positive the line rises as we move from left to
right. The bigger m is, the steeper the rise of the line is. If m is
negative, the line falls as we move from left to right. The bigger the
absolute value of m, the steeper the decline of the line is.

Lines are parallel to the y-axis are vertical. They have no slope,
and their equation is of the form x = a, for some fixed, given real
number a.

This is a basic introduction to a subject called Analytical Geometry.
It was invented by Rene' Descartes, a French mathematician of the 17th
century. It amounts to a way to talk about geometric objects and
properties using algebra.

- Doctor Rob, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Basic Algebra
High School Equations, Graphs, Translations
Middle School Algebra
Middle School Graphing Equations

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