geometry problem.?

Question

the three lines 8x+3y+12, 6y-7x=24, and x+9y+33=o intersect to form a triangle. find the coordinates of the centroid of the triangle.

Answer 1

Assuming you meant 8x+3y=12, (which seems plausible - this equation will give a nice integral solution)
the answer is (-1, -1)

8x+3y=12
-7x + 6y = 24
multiply the first equation by 2 and subtract the second.
16x + 6y = 24
-7x + 6y = 24
23x = 0
→
x = 0, y = 4

8x+3y=12
x + 9y = -33
multiply the first equation by 3 and subtract the second.
24x + 9y = 36
x + 9y = -33
23x = 69
→
x = 3, y= -4

-7x + 6y = 24
x + 9y = -33
multiply the second equation by 7 and add the first.
-7x + 6y = 24
7x + 63y = -231
69y = -207
→
x = -6, y = -3

So the centroid is located at (the average of the three vertices):

x = (0 + 3 - 6)/3 = -3/3 = -1
y = (4 - 4 - 3)/3 = -3/3 = -1

It is often useful in these cases to go back and double check that the answers found in simultaneous equation solving indeed work in the original equations. You should find that the above points do work. The other numerical answer above contains arithmetic errors (for example, in b/c, 6y + 63y = 66y).

Answer 2

a=8x+3y=12
b=6y-7x=24
c=x+9y+33=0

a=8x+3y=12
b=-7x+6y=24
c=x+9y=-33

a/b
8x+3y=12(*2)
-7x+6y=24
16x+6y=24
-7x+6y=24
23x = 0
x = 0
8x+3y=12
3y=12
y=4
(0,4)

a/c
8x+3y=12(*-3)
x+9y=-33
-24x-9y=-36
x+9y=-33
-23x=-69
x = 3
x+9y=-33
3+9y=-33
9y = -36
y = -4
(3,-4)

b/c
-7x+6y=24
x+9y=-33(*7)
-7x+6y=24
7x+63y=-231
66y = -207
y = -207/66 = -29/22
-7x+6y=24
-7x+6(-29/22)=24
-7x-(174/22)=24
-7x-(87/11)=264/11
-7x = 357/11
x = -357/77 = -51/11
[(-51/11),(-29/22)]

Three points are [(-51/11),(-29/22)], (0,4) and (3,-4)
x = [(-51/11)+0+3]/3 = -18/33
y = [(-29/22)+4+-4]/3 = -29/66
Centroid [ (-18/33), (-29/66) ]

Answer 3

Check your terms; at least 1 is NOT a line.