General form (an equation of a line)!?

Question

There are different forms of equations of lines, right?

Point-slope, slope-intercept, etc. I get all of them, but what is the general form?

What is A? What's B? What's C? (Ax+By=C) I know they're a 'constants', but...I just don't get it. They're not a point on the line, obviously, so what are they? I hope you get me. :D

Answer 1

The values of A, B and C have to be given to you to define the line

The value of the constant C gives the amount that the line is displaced vertically when it crosses the Y intercept

The slope of the line is given by -A/B

You can convert Ax+By=C into

Ax + By + C = 0     or    

y = (-A/B)x + (-C/B)

In the example given by Elliot P,

2x+4y=6

4y = -2x+6

so

y = -2/4x +6/4

in this case m= -0.5, c=1.5

Multiplying both sides by 2 does not introduce a constant.

and doesn't change the values of m and c

when y=0  -0.5x+1.5=0

0.5x = 1.5

so

x = 3

This means the line crosses the x intercept at +3,0

When x=0, y = 1.5

This means the line crosses the y intercept at 0,1.5

Multiplying the whole equation by 2 or any other value does not alter the slope or the position of the line, it just complicates things

The value "2" is not a constant, it is just a value.

The constant is already defined as c

Answer 2

If I'm not mistaken, A, B, and C represents a number on the equation. For example..

A-3

B-2

C-4

Therefore,

3x+2y=4

or

4x+5y=6

A-4

B-5

C-6

Hope I helped!!

Answer 3

The "general form" of an equation of the line which I have learned is the form y=mx+c. x and y are variables which we are using, m is the gradient of the line and c is the y-intercept. For any straight line, these numbers do not change for that line. This is why they are called constants.

In the other form, ax+by=c, we see certain relationships form. This form uses the intercepts of that line to form the equation. For example, on a certain graph the x-intercept is when y=0. So, when y=0, we get ax+0=c, and therefore x=c/a. So, the x-intercept = c/a. Similarily, the y-intercept y=c/b. So, if the line is 2x+4y=6, the x-int is c/a, 6/3=2 and the y-int is c/b, 6/2=3, we can also say that the y-int=3 can not only be made up of 6/2 but also 12/4, 15/5, 18/6 which all equal three.

So, in ax+by=c, if the enitre equation is multipled by a constant, say 2, the equation holds true, and if ax+by=c then 2ax+2by=2c

These numbers a, b and c tell us more to the line than there is. They can tell us the lines intercepts, and then, using m=rise/run, the gradient of that line. Without having values for a, b and c, all we can tell is that there is a line, and nothing else.

Answer 4

They're a description of how the x-values and y-values on the axes relate to each other.

if x+y =5 then when x is 2, y is 3 and the point (2,3) is on the line

the x intercept occurs at x =5 because that's when y is zero and when y is zero the point must be somewhere on the x-axis. Similarly for y-intercept.