# coordinate

• Aug 5th 2010, 12:14 PM
sfspitfire23
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.
• Aug 5th 2010, 12:28 PM
Ackbeet
They should both give you the same answer. Can you show a bit more work?
• Aug 5th 2010, 12:34 PM
Unknown008
You should get the same answer from both methods, because they are somewhat related. The formula I use for finding the gradient is:

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

Then, I would use:

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

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

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

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.
• Aug 5th 2010, 12:40 PM
sfspitfire23
Wow nevermind. Simple arithmetic kills :-p
• Aug 5th 2010, 01:15 PM
Soroban
Hello, sfspitfire23!

Did you make a sketch?

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

Quote:

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

Find a value of $x.$

Code:

      |                                      B       |                                  (x,4)*       |                                  *  :       |                              *      :1       |                      A  *          :       |                  (5,3)o - - - - - - - +       |                  *  :      ?       |              *      :1       |      c  *          :       |  (3,2)o - - - - - - - +       |              2       |   - - + - - - + - - - - - - - + - - - - - - - +       |      3              5

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

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

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

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