Line equations

[Gamer30]

Hi everyone,
I wish to seek maths enthusiastic helpers out there in providing me solution to the following:
1) As we know, line equation is in the form of y=m*x+b, so, if we were to draw a line equation y=3, straight away we just need to draw a horizontal straight line cutting across y=3, right? Is it bcos we know that in horizontal line the gradient would be 0.
So, when we substitute gradient 0 into y=m*x+b, we would get y=b right? But if we were to ask to draw line equation x=3, gradient of the line would be undefined right? Therefore, y=undefined right? So my question is that how come in this case, the line is drawn at x=3 vertically cutting across x-axis at 3? Can someone kindly explain why x=3 is a straight vertical line (since gradient of a vertical line is undefined), so y=undefined*x+b should be undefined also, right?

 

[Harry_the_cat]

y=mx+b is the equation of a line with a defined gradient.

If the gradient is undefined, the equation has the form x=k.
The line x=3, for example, is a vertical line which goes through all the points where x=3, eg (3,0), (3, 1), (3,2), (3,2.45), (3, 78) ... Plotting all possible points results in a vertical line cutting the x-axis at (3, 0). This is the only type of line that does NOT take the form y=mx+b. This is because a vertical line is NOT a function in the strict sense of the word.

You seem to be trying to fit a vertical line into an equation of a line with a defined gradient. Sort of like trying to fit a square peg into a round hole!!

Back to your point about, for example, y=3 which can be written in the form y=mx+3 where m=0. This is consistent with a horizontal line having a gradient of 0. Also, the line y=3 passes through all the points where y=3, eg (0, 3), (1,3), (2,3), (7.6, 3), (-52.75, 3) etc. Plot these points, you get a horizontal line.

 

[LCKurtz]

Also, if you want to avoid what seems like an unsymmetrical situation with the equation of a straight line because you have y as a function of x, you can avoid the "infinite" or undefined slope situation by using a parametric equation for the line. You can write the equation of a straight line through [imath](a,b)[/imath] going in any direction by specifying a direction vector [imath]\langle c,d \rangle[/imath]. Then you get[math]x = a+ct,~y=b+dt[/math] and it doesn't matter what direction the line is in, including vertical.