What are the vertices of the triangle formed by the lines x + y = 0, x - y = 0, and 2x + y = 1?

To find the vertices of the triangle formed by the three lines given, we first need to determine the points where each pair of lines intersects.

1. Intersection of the first two lines:

The first line is x + y = 0, and the second line is x - y = 0. By solving these equations simultaneously, we can find the point of intersection.

• From x - y = 0, we have y = x. Substituting this into the first equation gives:

( x + x = 0 ) or ( 2x = 0 ), leading to ( x = 0 ) and consequently ( y = 0 ).

Thus, the first vertex is (0, 0).

2. Intersection of the first and third lines:

The first line is x + y = 0, and the third line is 2x + y = 1. To find their intersection:

• From x + y = 0, we can express y as ( y = -x ). Substituting into the third line gives:

( 2x - x = 1 ), simplifying to ( x = 1 )