6: Straight Line y=mx

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The straight line is made of many points. And each line is different. So, what is the common feature that these points have which other points in other lines do not have. 

This common feature that distinguishes the points on a particular line is given by the relationship between x and y. 

Each point on the x-y axis has two unique numbers (called coordinates). So, point A (a,b) or pt. B (aCos z, aSin z). The points on a line all have a certain relationship. Since there are infinite number of points on the line and we can’t show all of them, we take a general coordinate and call it (x,y) and then show the relationship between them which will then hold for all the points on that line. Let us see the simplest example: y = x which is the equation of a straight line. In the diagram two points are shown: (a,a) & (b,b). Since the equation is y=x so the two coordinates will be equal. In fact any number like (1,1) or 2,2 or 7.1,7.1 will all fall on this line.

We end up with an isosceles right angled triangle and 
So, the angle the line y = x makes with the x axis is 45 degrees. 


what if we change the angle from 45 to something else
Let us increase the angle to 60 degrees and see the diagram and we find the equation of the line thus
Next let us change the angle to 30 degrees and see the diagram and we find the equation of the line 

Basic Equation of the Straight Line

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And that gives us the general equation of the Straight line

y = m x
where m is tan z
where z is the angle the line makes with the x axis

m is also called the slope of the line

​​Imagine the line shows you a mountain slope and now it will be clear why the word slope is used. And mathematically it is precisely measured by tan z,
z being the angle of the slope.