Finding the gradient of a straight line when given coordinates (2m, n) and (3, -4)

[MathsFormula]

Question is :

(i) Find the gradient of the line joining points (2m, n) and (3, -4)

This bit I did correct:

Gradient = (difference in y-coordinates) / (difference in x-coordinates)

(2m, n) = the difference in coordinates (subtraction sum)
- (3, -4)

(2m-3), (n+4) = the difference in coordinates


Gradient = (difference in y-coordinates) / (difference in x-coordinates)

(n+4) = gradient CORRECT ANSWER
(2m-3)


(ii) Find the value of n if the line is parallel to the x-axis

This bit I did correct:

The line is therefore horizontal so gradient = 0

(n+4) = 0
(2m-3)

n = -4 CORRECT ANSWER


(iii) Find the value of m if the line is parallel to the y-axis

Here is where I'm confused. I got the correct answer but don't understand why. Please help

The line is therefore vertical so gradient = 1 ????


(n+4) = gradient
(2m-3)


(n+4) = 1
(2m-3)

We know n = -4 from part (ii) then

(-4+4) = 1
(2m-3)


0 ___ = 1 THE LEFT SIDE IS ZERO
(2m-3)

0 = 1 The 'm' vanishes


What's going on? Please help. Answer = 1.5

If I ignore that the left side is zero and just say 0 =(2m - 3) then I can get the correct answer


Thank you

 

[Mrspi]

Question is : (i) Find the gradient of the line joining points (2m, n) and (3, -4) This bit I did correct: Gradient = (difference in y-coordinates) / (difference in x-coordinates) (2m, n) = the difference in coordinates (subtraction sum) - (3, -4) (2m-3), (n+4) = the difference in coordinates Gradient = (difference in y-coordinates) / (difference in x-coordinates) (n+4) = gradient CORRECT ANSWER (2m-3) (ii) Find the value of n if the line is parallel to the x-axis This bit I did correct: The line is therefore horizontal so gradient = 0 (n+4) = 0 (2m-3) n = -4 CORRECT ANSWER (iii) Find the value of m if the line is parallel to the y-axis Here is where I'm confused. I got the correct answer but don't understand why. Please help The line is therefore vertical so gradient = 1 ???? (n+4) = gradient (2m-3) (n+4) = 1 (2m-3) We know n = -4 from part (ii) then (-4+4) = 1 (2m-3) 0 ___ = 1 THE LEFT SIDE IS ZERO (2m-3) 0 = 1 The 'm' vanishes What's going on? Please help. Answer = 1.5 If I ignore that the left side is zero and just say 0 =(2m - 3) then I can get the correct answer Thank you

A vertical line (one that is parallel to the y-axis) has an undefined gradient. You've found an expression for the gradient: (n + 4) / (2m - 3) Remember that division by zero is undefined, so (n + 4) / (2m - 3) will be undefined if the denominator of the gradient fraction is 0. So.....when is 2m - 3 = 0? That should give you the answer you're looking for.

 

[MathsFormula]

A vertical line (one that is parallel to the y-axis) has an undefined gradient. GOT THAT BIT

You've found an expression for the gradient: (n + 4) / (2m - 3) GOT THAT

Remember that division by zero is undefined, so (n + 4) / (2m - 3) will be undefined if the denominator of the gradient fraction is 0. DON'T UNDERSTAND THIS

The denominator is the (2m - 3). Now what?

So.....when is 2m - 3 = 0? How did you get to here?

Thanks


I realised I was wrong when I said "The line is therefore vertical so gradient = 1"

I have :

(n+4) = gradient
(2m-3)

So all I can see is:

(n+4) = undefined
(2m-3)

What numerical value do we assign to something that is undefined?

 

[Deleted member 4993]

Thanks


I realised I was wrong when I said "The line is therefore vertical so gradient = 1"

I have :

(n+4) = gradient
(2m-3)

So all I can see is:

(n+4) = undefined → 2m - 3 = 0 → and continue....
(2m-3)

What numerical value do we assign to something that is undefined?

.

 

[MathsFormula]

(n+4) = undefined for a vertical line
(2m-3)

Are we just going to assume that gradient of a vertical line is 1?

And we know n = -4 ..... so

-4 +4__ = 1
(2m -3)


0 = 2m -3

m = 3/2


Gradient of a horizontal line is 0. The opposite of horizontal is vertical so vertical line gradient must be 1? Is that what we're saying?

 

[ksdhart2]

No, the gradient/slope of a vertical line is most definitely not 1. Think about what you know about lines, and what slope really means. Slope is defined as "rise over run." If the slope of a line were to be 1, what would that imply about the rise and the run of that line? What would such a line look like? Because that line is not vertical, you can see that the slope of a vertical line cannot possibly be 1. But, then, the question becomes what is the slope? Well, let's again return to the definition of "rise over run." In a horizontal line, whose slope you know is 0, it only has run but no rise, making the slope 0/run. That is, of course, equal to 0.

Now, what happens if we consider a vertical line instead? It has only rise and no run, so the slope would be rise/0. Dividing by 0 is undefined, so a vertical line's slope is undefined. This also fits with your intuition that the slope of a vertical line ought to be the "opposite" of the slope of a horizontal line. The only logic error you made there was assuming that the opposite of 0 is 1. It's not. The slope of a horizontal line is 0, which can be written as 0/1. The "opposite" (aka reciprocal) of that is then 1/0.

 

[JeffM]

(n+4) = undefined for a vertical line
(2m-3)

Are we just going to assume that gradient of a vertical line is 1?

And we know n = -4 ..... so

-4 +4__ = 1
(2m -3)


0 = 2m -3

m = 3/2


Gradient of a horizontal line is 0. The opposite of horizontal is vertical so vertical line gradient must be 1? Is that what we're saying?

What I am about to say is a little advanced.

\(\displaystyle a \ne 0\ and\ b = \dfrac{a}{0} \implies b\ is\ not\ a\ real\ number.\)

This is normally expressed as "a / 0 and a \(\displaystyle \ne\) 0 means a / 0 is not defined." However, it is possible to define a / 0 outside the real number system, and, being very informal, you might call that quotient infinity. Whether you call it undefined or infinity or thingamabob is not, in my opinion, terribly important. What is important is that whatever you call it, it is in no way, shape, or form a real number like 1. And, again, whatever you call it, it is the slope of a vertical line. Now for a while, it is probably safest for you to call it undefined, meaning not a real number. Teachers like everybody to use the same vocabulary.

To get back to the problem at hand. You want the slope of a vertical line, which means it must be undefined, which means the denominator must be zero.

\(\displaystyle gradient = \dfrac{n + 4}{2m - 3}\).

We want the denominator to be 0 for the gradient to be undefefined / infinity / thingamabob / whatever you call it. What is the denominator? Under what circumstance will it be zero?

 

[MathsFormula]

You want the slope of a vertical line, which means it must be undefined, which means the denominator must be zero.

\(\displaystyle gradient = \dfrac{n + 4}{2m - 3}\).


We want the denominator to be 0 for the gradient to be undefined / infinity / thingamabob / whatever you call it. What is the denominator? Under what circumstance will it be zero?

\(\displaystyle gradient = \dfrac{rise}{run}\)


For a vertical line \(\displaystyle gradient = \dfrac{1}{0}\) = undefinied


So I need to find a way of making \(\displaystyle \dfrac{n + 4}{2m - 3}\) equal 'undefined'

\(\displaystyle \dfrac{rise}{run}\) can only be 'undefined' if the denominator equals ZERO

So I need to find what 'm' must be to make the denominator of the fraction \(\displaystyle gradient = \dfrac{n + 4}{2m - 3}\) equal to ZERO so that the fraction becomes 'undefined'

2m - 3 = 0

Therefore m = 2/3 THIS IS THE ANSWER

\(\displaystyle gradient = \dfrac{n+4}{0}\) = undefined = INFINITY


THAT WAS TOUGH. BASICALLY I needed to 'know' that the gradient of a line is undefined and then just make the equation equal undefined too.


Thank you everyone for your help. I think I've understood it correctly.

Is INFINITY the same thing as saying UNDEFINED?

 

[ksdhart2]

Is INFINITY the same thing as saying UNDEFINED?

In some sense, yes. But not really. Dividing by 0 will always produce undefined as an answer. However, if we're looking at the behavior of a fraction as the denominator gets closer and closer to 0, then we can say that the limit is infinity. That is to say, no matter how small you make the denominator, there will always be a smaller denominator that will make the fraction bigger. But if you were to ever actually reach zero, then the expression would be undefined.

 

[y2kevin]

\(\displaystyle gradient = \dfrac{rise}{run}\)


For a vertical line \(\displaystyle gradient = \dfrac{1}{0}\) = undefinied


So I need to find a way of making \(\displaystyle \dfrac{n + 4}{2m - 3}\) equal 'undefined'

\(\displaystyle \dfrac{rise}{run}\) can only be 'undefined' if the denominator equals ZERO

So I need to find what 'm' must be to make the denominator of the fraction \(\displaystyle gradient = \dfrac{n + 4}{2m - 3}\) equal to ZERO so that the fraction becomes 'undefined'

2m - 3 = 0

Therefore m = 2/3 THIS IS THE ANSWER

\(\displaystyle gradient = \dfrac{n+4}{0}\) = undefined = INFINITY


THAT WAS TOUGH. BASICALLY I needed to 'know' that the gradient of a line is undefined and then just make the equation equal undefined too.


Thank you everyone for your help. I think I've understood it correctly.

Is INFINITY the same thing as saying UNDEFINED?

if 2m - 3=0

Why wouldn't m = 3/2 ?