Assistance sought on how to go about solving graphing problem.

The text asks: Find the possible slopes of a line that passes through (4,3) so that the portion of the line in the first quadrant forms a triangle of area 27 with the positive coordinate axes.

I know that the equation for finding the area of a triangle is . How do i go about solving this algebraically? The answers in the book are . I do not know how they arrived at these answers. I don't know how you could graph it without first determining more than one set of points. Any help would be greatly appreciated. Thanks.

Re: Assistance sought on how to go about solving graphing problem.

Let (X, 0) and (0, Y) be the points at which such a line intercepts the x and y axes, respectively. Then the area of the triangle formed is (1/2)XY= 27 we we must have XY= 54.

Any line through (4,3) can be written y= m(x- 4)+ 3. That means we must have y= m(0-4)+ 3= -4m+ 3= Y and 0= m(X- 4)+ 3 so that mX- 1= 0, mX= 1, and X= 1/m. XY= (-4m+ 3)(1/m)= 54. Solve that for m.

Re: Assistance sought on how to go about solving graphing problem.

Re: Assistance sought on how to go about solving graphing problem.

Thanks a ton hallsofivy, and soroban, you guys are the best. Thank you for showing me how to solve this algebraically (the image of the graph helped also), although in hallsofivy's answer I don't see how solving for the x-intercept 0=m(x-4)+3 can be reduced to mx-1=0. I really appreciate it thank you guys.