Question

Write equation of the line containing (6,3) and (4,1)

Answer 1

There are two ways to approach this.

The first is to just check with equation works with both points by plugging in x=6 and seeing if you get y=3, from the point (6,3), and then repeating that process with (4,1).

Checking (6,3) on y = -1x+5:

    y = -1(6)+5 = -1     -> no good.

Checking (6,3) on y = x-3:

    y = 6-3 = 3           -> check

Checking (6,3) on y = x+5:

    y = 6+5 = 11          -> no good.

Checking (6,3) on y = -1x+3:

    y = -1(6)+3 = -3     -> no good.

So that shows only y = 1x - 3 works, but you should still double check it works for (4,1).

The second option is to construct the equation, which is good to know, in case you don't have a list of equations to check.

For this, you need to calculate the slope, using $m=\dfrac{y_2-y_1}{x_2-x_1}$, and then use that slope to find the y-intercept, (0,b).

Step 1: slope

    $m=\dfrac{1-3}{4-6} = \dfrac{-2}{-2} = 1$

Step 2: find y-intercept

Start by setting up the slope-intercept form with the slope we just found and then plug in one of the points:

    $y=1x+b$, which becomes $3 = 1(6)+b$ using (6,3).

   Solving this for b, you find b=-3.

Step 3: put it all together:

   m = 1 and b = -1 gives you

   y = 1x -3

Both ways work and are valid.  Context will indicate which one is best.