1. ## coordinate

In a coordinate system, if three points (5, 3), (x, 4) and (3, 2) lie on a same line, then find a value of x?

Now, the slope appears to be 1/2 from the two given points.

Does this mean that x will be a number than when used to calculate the slope of the line with any of the other two points will still be 1/2? In this case, 7? Or, do we have to develop the equation of the line and substitute (x,4) into the equation of the line in which case I get 5.

2. They should both give you the same answer. Can you show a bit more work?

3. You should get the same answer from both methods, because they are somewhat related. The formula I use for finding the gradient is:

$\displaystyle \frac{y_2 - y_1}{x_2 - x_1} = m$

Then, I would use:

$\displaystyle \frac{y - y_1}{x - x_1} = m$

To get the equation of the line. But, since both equals to m, I can directly say that:

$\displaystyle \frac{y_2 - y_1}{x_2 - x_1} = \frac{y - y_1}{x - x_1}$

So, it's the same as making both gradients equal.

A shorter way that I came to.

Rearrange your coordinates so that the y coordinates are in ascending order.

(3, 2) , (5, 3) , (x , 4)

Looking at the y coordinates, it increases by 1 each time. And since all the points lie on a straight line, you can conclude that (5, 3) is the midpoint of (3, 2) and (x, 4). In which case, you will have an arithmetic sequence for the x coordinate too. It's 3, 5 then x. Since it increases by 2, you add 2 to 5 to give 7, the value of x.

In any case, you should get x = 7.

4. Wow nevermind. Simple arithmetic kills :-p

5. Hello, sfspitfire23!

Did you make a sketch?

I did . . . and baby-talked my way to the answer.

In a coordinate system, points $\displaystyle A(5,3),\;B(x, 4),\;C(3, 2)$ are collinear.

Find a value of $\displaystyle x.$
Code:
      |                                       B
|                                  (x,4)*
|                                   *   :
|                               *       :1
|                       A   *           :
|                  (5,3)o - - - - - - - +
|                   *   :       ?
|               *       :1
|       c   *           :
|  (3,2)o - - - - - - - +
|               2
|
- - + - - - + - - - - - - - + - - - - - - - +
|       3               5

Going from $\displaystyle C$ to $\displaystyle A$, we move 2 units right and 1 unit up.

Going from $\displaystyle A$ to $\displaystyle B$, we move ? units right and 1 unit up.

It is obvious (I hope) that: .$\displaystyle ? \,=\,2$

Therefore: .$\displaystyle B(7,4)\,\text{ and }\,x = 7$