   SEARCH HOME Math Central Quandaries & Queries  Question from Tiffany, a student: The equation for the line through the points (11,2) and (18,12)  can be written in the form Ax+By=C Find A and B Hi Tiffany,

Are A and B integers?  There is more than one solution!  I'll try to show you how to get two solutions, but there are as many as you want.

It is easy to figure out the slope of the line.  And from that you can get an equation for the line using point-slope form.  Maybe it is possible to find that in your notes or your textbook, or to ask your teacher? You can solve the problem once you have that.

Otherwise, if your points are (x1,y1) and (x2,y2), then the slope is m=(y2-y1) / (x2-x1). The point-slope form of the equation is y - y1 = m(x - x1), where m is the slope calculated above.  If you remember the formula for m, this is y - y1 = [(y2-y1)/(x2-x1)](x - x1).  All that is needed is to multiply both sides by a number so that the denominator cancels, and then simplify.

For example, suppose the points are (1, 2) and (5, 7).

Then the slope m = (7-2)/(5-1) = 5/4.  The point slope form of the equation is y - 2 = (5/4)(x - 1).  When both sides are multiplied by 4 you get 4(y-2) = 5(x-1).  After multiplying everything out and rearranging, this is the same as -5x + 4y = 3, or equivalently 5x - 4y = -3.  As a check on the work, plug in (1,2) and (5,7) and see that they satisfy the equation(s) -- the left hand side and the right hand side are equal.

Now try it with your numbers, and don't forget to check your work!

Victoria

Tiffany,

There are many possible values for A, B and C that would be correct. Conventionally though, we usually choose an A that is positive (multiply both sides of your equation by -1 if A is negative and you'll solve that). As well, whenever possible we choose a set of A, B and C where they are all integers and there is no common divisor among them (in other words, the smallest set of integers that work).

Stephen.     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.