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ABC has vertices A(0, 6), B(4, 6), and C(1, 3). Sketch a graph of ABC and use it to find the orthocenter of ABC. Then list the steps you took to find the orthocenter, including any necessary points or slopes you had to derive.

pick a vertex.
find the slope of the perpendicular to the opposite side.
find the equation of the line with that slope, going through the vertex.

pick another vertex and repeat

find the intersection of the two lines.
That's the orthocenter.

Method:
Find the equation of two of those altitudes.
Solve the two equations to find their intersection point.

How?
Make a rough sketch
Pick any point and find the slope of the opposite side.
The slope of the altitude to that side is the negative reciprocal of the slope of that side.
Now you have the slope and a point on that line, find the equation for the line.

Repeat the above for a second altitude, solve the two equations.

PS, just noticed how nice your points are. One of the lines is a horizontal line, so the altitude from (1,3) to that line is x = 1