Orthocenter solution

A = (1,1)
B = (3,7)
C = (9,4)

Step 1)
Find slope of AB

7-1
---
3-1

Slope of AB = 6/2 = 3


Step 2)
Find slope of CF (aka Perpendicular slope of AP)

= -1/AB
= -1/3

Step 3)
Now find the equation for CF

y-y1=m(x-x1)
y-4=-1/3(x-9)
3y-12=-x+9
3y=-x+21
x+3y=21 --- Remember this equation

Step 4)
Find slope of BC

4-7
---
9-3

-3
---
6

Slope of BC = -3/6 = -1/2

Step 5)
Find slope of AD (aka perpendicular slope of BC)

= -1/BC
= -1/(-1/2)
= 2

Step 6)
Find the equation for AD

y-y1=m(x-x1)
y-1=2(x-1)
y-1=2x-2
-2x+y=-1 -- This is the second equation you need

Step 7)
Subtract the two equation

x+3y=21
- (-2x+y=-1)

x+3y=21
2x-y=1

We need to eliminate a variable. Choose X

-2(x+3y=21)

-2x-6y=-42
2x-y=1
--
-7y=-42
y=42/7

Now that we have y, replace (y) in another equation

2x-(42/7)=1
2x=1 + 42/7
x = (1+42/7)/2
x = 7/2

ANSWER (7/2,42/7)