How would you figure this out?

Question

A vertical line, horizontal line and another line that passes through the origin all intersect at (–1,–5). What are the equations of the three lines?

Answer 1

In the Cartesian Coordinate System (x and y axis), the x-axis is horizontal and the y-axis is vertical.  Each point is represented with coordinates in the form (x,y).

 

A vertical line happens when the value of x is a constant. So, all of the points on that line have the form (c,y).  If the point (-1,-5) is on that line, then c=-1.  This line has the equation x=-1 [any value of variable y].

 

A horizontal line happens when the value of y is a constant.  So, all of the points on that line have the form (x,d).  If the point (-1,-5) is on that line, then d=-5.  This line has the equation y=-5 [any value of variable x].

 

Two points are sufficient to determine a line (just connect the dots and keep on going in each direction).  So, if we have two points [the Origin, that is the point (0,0), and the point (-1,-5)], the point-slope form of the equation is quite useful:

     (y-y₁) = ((y₂-y₁)/(x₂-x₁))(x-x₁)

     (y-0)  = ((-5-0)/(-1-0))(x-0)

      y  =  5x

 

Check: 

  0 = 5(0), so the Origin is on the line.

  (-5)=5(-1), so the point (-1,-5) is on the line.