Answer:
Question: What is the orthocenter of a triangle with the vertices (-1,2) (5,2) and (2,1)?
The coordinates of point A are (-1,2), point B are (5,2), and point C are (2,1).
The orthocent is the intersection of the three altitudes. An altitude goes from a vertex and is perpendicular to the line containing the opposite side.
In the coordinate plane the equations of the altitudes can be found and then a system of equations can be solved.
Altitude 1. From point C perpendicular to the line containing side AB.
Slope of line AB is 0 (horizontal line), a vertical line is perpendicular to a horizontal line. Thus, the equation of altitude 1 is  x=2 .
Altitude 2. From point B perpendicular to the line containing side AC.
Slope of line AC is  −13 , the slope of a line perpendicular to line AC is 3. The equation of altitude 2 is  y=3x−13  
Altitude 3. From point A perpendicular to the line containing side BC.
Slope of line BC is  13 , the slope of a line perpendicular to line BC is  −3 . The equation of altitude 3 is  y=−3x−1  
The orthocenter is the point where all three altitudes intersect.
x=2  
y=3x−13  
y=−3x−1  
Use substitution to solve the first two equations  y=3(2)−13=−7  
The orthocenter is the point  (2,−7)  
we did not need the third equation, but we can use it as a check, plug the coordinates into the third equation:
−7=−3(2)−1  
−7=−6−1  
−7=−7  it works.