Hence show that... help!

[12345678]

‘The point P(x,y) lies on the straight line joining A(3,0) and B(5,6). Find the expressions for the gradients of AP and PB. Hence show that y=3x-9’
I understand the expressions for the gradients part; m(AP)= y – 0 / x – 3 = Y / x – 3.
M (PB)= y – 6 / x – 5
(Don’t know if this part helps): Length of AB= √(x₂-x₁)²+(y₂-y₁)² = √(5-3)²+(6-0)²= √4+36= √40
The part I don’t understand is ‘hence show that y=3x-9’. I know that m=3 and y intercept = -9, but how do you calculate/show this from the expressions of the gradient? Any help/ step-by-step will be appreciated.

 

[lookagain]

lookagain, please proofread your posts; thank you.

You are correct. The computer has been showing me a blank screen (instead of my text
that I wanted to edit) when I tried to edit my posts. I deleted my other posts.

 

[HallsofIvy]

The crucial point is that, because this is a straight line, the slope, as calculated using any two points on it, is always the same. Yes, the slope between (x, y) and (3, 0) is \(\displaystyle \dfrac{y- 0}{x- 3}= \dfrac{y}{x- 3}\), the slope between (x, y) and (5, 6) is \(\displaystyle \dfrac{y- 6}{x- 5}\), and the slope between (3, 0) and (5, 6) is \(\displaystyle \dfrac{6- 0}{5- 3}= \dfrac{6}{2}= 3\).

So we must have \(\displaystyle \dfrac{y}{x- 3}= 3\) or y= 3(x- 3)= 3x- 9.

And we must also have \(\displaystyle \dfrac{y- 6}{x- 5}= 3\) or y- 6= 3(x- 5)= 3x- 16 so, adding 6 to both sides, y= 3x- 9 again.

 

[Ted]

I'm not seeing the problem on my end... hmm.

This sentence is an edit... I still see the previous text above.

 

[stapel]

I know that m=3 and y intercept = -9, but how do you calculate/show this from the expressions of the gradient?

Use what you learned back in algebra. The slope-intercept form of a line with slope m and y-intercept at y = b is given by y = mx + b. So plug-n-chug.

Note to other posters in this thread: I've been happily "replying with quote" all morning, until I hit this thread. For this thread, however, the "quote" was blank. (I did a copy-n-paste to do the "quote" above.) Whatever the problem is, I think it's related to particular posts, rather than to the forums as a whole. Perhaps posts during a certain time-window were munged somehow?