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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.


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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