# line

#### red and white kop!

##### Junior Member
considering an equation ax + by + c =0, im supposed to give examples of a, b, c for which the equation does not represent a straight line
cant find any

#### mmm4444bot

##### Super Moderator
Staff member

What happens, if all of the parameters are zero?

#### red and white kop!

##### Junior Member
is this the only case? a, b and c being all equal to zero?

#### mmm4444bot

##### Super Moderator
Staff member
red and white kop! said:
is this the only case? Nope, but the cases requested all involve zero(s) somewhere.

Consider all the possibilities that involve zero; there aren't that many cases to consider.

#### red and white kop!

##### Junior Member
if all parameters are 0 then the result is 0=0
if a and b are 0 then the result is c=0
if a is 0 then the result is by=-c, which is a horizontal line
if b is 0 then the result is a vertical line
and then of course there is if a and c are 0 and if b and c are 0 but the truth is i just dont know what to conclude from these results

#### red and white kop!

##### Junior Member
hey i'd really need some helpfum insight on this one

#### chrisr

##### Full Member
Must a, b and c be constants??

If they are not constants, it won't be a line.

A linear equation requires a, b, and c to be constants.
(Well, c could be a multiple of x in theory).

If a, b, or c are functions of x, the function will bend in some way.

$$\displaystyle y=mx+c\$$ is a line, even if m is zero, or if c is zero.

It's the x-axis if both m and c are zero.

If we rearrange that, we get $$\displaystyle mx-y+c=0$$

which is written as ax+by+c=0.

If b and c are zero, that's the y-axis.

a, b and c equal to zero means nothing at all.

So, linear equations have the powers of x and y equal to 1 in that form.
"Linear" means "straight line", a curved line is a curve.

#### mmm4444bot

##### Super Moderator
Staff member

You've been given the General Form of a line.

(With beginning algebra, I think it's clear enough for tutors that the parameters are all constants.)

Ax + By + C = 0, where A, B, and C are all Real numbers AND where A and B are not both zero.

·
red and white kop! said:
if all parameters are 0 then the result is 0=0 This equation contains no variables, so it cannot define a line.

if a and b are 0 then the result is c=0 Since c is non-zero, this equation is never true, so there is no line.

if a is 0 then the result is by=-c, which is a horizontal line Correct.

if b is 0 then the result is a vertical line Correct.

and then of course there is if a and c are 0 and if b and c are 0 but the truth is i just dont know what to conclude from these results

Then try some Real numbers for these last two cases, and see what happens. It takes only a few moments, to experiment.

a = 0
b = 1
c = 0

a = 1
b = 0
c = 0

#### red and white kop!

##### Junior Member
so the one situation in which ax + by + c=0 does not represent a straight line is when a and b are both equal to zero; is this totally correct? i just want to get this thing straightened out for good.

#### mmm4444bot

##### Super Moderator
Staff member
Yes, that's totally correct.

So, you can supply as many examples as needed; make A and B zero, and make C any number.