Hi there,
Let me give you a small summary on lines/slopes (just for lines in the plane xOy, for sure), because a folk asked me about this in private message...
Every line may be put on the form ax+by = c where a, b, c are real constants, the first two not simmultaneously zero.
This is the general form of a line (in a plane), and we have some important details on that:
> The line passes through the origin, that is, point (0,0) if and only if c is null.
> If a = 0 (then b is not), then y = c/b = constant, that means that the line is horizontal.
Important particular case: when c is zero, we have the "x axis", that may be written as (x,y) in the plane such that y = 0.
> If b = 0 (then a is not), then x = c/a = constant, that means that the line is vertical.
Important part. case: when c is zero, we have the "y axis", that may be written as (x,y) in the plane such that x = 0.
> If b is NOT null, that means the line may be put on the form (called reduced) y = mx +n, where m is the SLOPE of the line and n is the y-intersect ;
I repeat: slopes are defined if and only if the lines are non-vertical.
Slopes are the ratio of the variation on y over the variation on x ("raise over run", as some people say)...
When the slope is (say) 4, that means that going to the RIGHT (say) 1 unit of length, the line goes UP 4 units of length...
When the slope is (say) -0.3, that means that going to the RIGHT (say) 1 unit of length, the line goes DOWN 0.3 units of length...
As far as slopes are concerned, two important properties you should know:
1)
when the slope is positive, the line is "going upwards" (from left to right) ;
when the slope is null (zero), the line is horizontal
when the slope is negative, the line is "going downwards" (from left to right);
P.S.: put your LEFT hand up and down, to "see" slopes positives and negatives....
2)
when the MODULUS of the slope is between 0 and 1, the line is horizontalized, I mean, it´s more "near" horizontal than "near" vertical, considering that slopes 1 and -1 are the "equal" horizontal and vertical (lines with slopes 1 and -1 are parallel to the lines y = x and y = -x...)
when the MODULUS of the slope is greater than 1, the line is verticalized, I mean, it´s more "near" vertical than "near" horizontal, considering that slopes 1 and -1 are... (the same above)
I hope this helps!
As far as the problem itself is concerned, I will solve it below if someone hasn´t already done this while I was typing....
Regards,
Fabio.
